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HANDBOOK OF NUMBER THEORY II by J. S´andor Babes¸-Bolyai University of Cluj Department of Mathematics and Computer Science Cluj-Napoca, Romania

and B. Crstici formerly the Technical University of Timis¸oara Timis¸oara Romania

KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 1-4020-2546-7 (HB) ISBN 1-4020-2547-5 (e-book) Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.

Printed on acid-free paper

All Rights Reserved 2004 Kluwer Academic Publishers No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. C

Printed in the Netherlands.

Contents PREFACE

7

BASIC SYMBOLS

9

BASIC NOTATIONS 1

10

PERFECT NUMBERS: OLD AND NEW ISSUES; PERSPECTIVES 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Some historical facts . . . . . . . . . . . . . . . . . . . 1.3 Even perfect numbers . . . . . . . . . . . . . . . . . . . 1.4 Odd perfect numbers . . . . . . . . . . . . . . . . . . . 1.5 Perfect, multiperfect and multiply perfect numbers . . . 1.6 Quasiperfect, almost perfect, and pseudoperfect numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Superperfect and related numbers . . . . . . . . . . . . 1.8 Pseudoperfect, weird and harmonic numbers . . . . . . . 1.9 Unitary, bi-unitary, infinitary-perfect and related numbers 1.10 Hyperperfect, exponentially perfect, integer-perfect and γ -perfect numbers . . . . . . . . . . . . . . . . . . 1.11 Multiplicatively perfect numbers . . . . . . . . . . . . . 1.12 Practical numbers . . . . . . . . . . . . . . . . . . . . . 1.13 Amicable numbers . . . . . . . . . . . . . . . . . . . . 1.14 Sociable numbers . . . . . . . . . . . . . . . . . . . . .

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15 15 16 20 23 32

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36 38 42 45

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50 55 58 60 72

References 2

77

GENERALIZATIONS AND EXTENSIONS OF THE ¨ MOBIUS FUNCTION 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

99 99

CONTENTS

2.2

2.3

2.4

2.5

M¨obius functions generated by arithmetical products (or convolutions) . . . . . . . . . . . . . . . . . . . . . . . . 1 M¨obius functions defined by Dirichlet products . . . . 2 Unitary M¨obius functions . . . . . . . . . . . . . . . 3 Bi-unitary M¨obius function . . . . . . . . . . . . . . 4 M¨obius functions generated by regular convolutions . 5 K -convolutions and M¨obius functions. B convolution . 6 Exponential M¨obius functions . . . . . . . . . . . . . 7 l.c.m.-product (von Sterneck-Lehmer) . . . . . . . . . 8 Golomb-Guerin convolution and M¨obius function . . . 9 max-product (Lehmer-Buschman) . . . . . . . . . . . 10 Infinitary convolution and M¨obius function . . . . . . 11 M¨obius function of generalized (Beurling) integers . . 12 Lucas-Carlitz (l-c) product and M¨obius functions . . . 13 Matrix-generated convolution . . . . . . . . . . . . . M¨obius function generalizations by other number theoretical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Apostol’s M¨obius function of order k . . . . . . . . . 2 Sastry’s M¨obius function . . . . . . . . . . . . . . . . 3 M¨obius functions of Hanumanthachari and Subrahmanyasastri . . . . . . . . . . . . . . . . . . . 4 Cohen’s M¨obius functions and totients . . . . . . . . . 5 Klee’s M¨obius function and totient . . . . . . . . . . . 6 M¨obius functions of Subbarao and Harris; Tanaka; and Venkataraman and Sivaramakrishnan . . . . . . . 7 M¨obius functions as coefficients of the cyclotomic polynomial . . . . . . . . . . . . . . . . . . . . . . . M¨obius functions of posets and lattices . . . . . . . . . . . . . 1 Introduction, basic results . . . . . . . . . . . . . . . 2 Factorable incidence functions, applications . . . . . . 3 Inversion theorems and applications . . . . . . . . . . 4 M¨obius functions on Eulerian posets . . . . . . . . . . 5 Miscellaneous results . . . . . . . . . . . . . . . . . . M¨obius functions of arithmetical semigroups, free groups, finite groups, algebraic number fields, and trace monoids . . . 1 M¨obius functions of arithmetical semigroups . . . . . 2 Fee abelian groups and M¨obius functions . . . . . . . 3 M¨obius functions of finite groups . . . . . . . . . . . 2

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106 106 110 111 112 114 117 119 121 122 124 124 125 127

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129 129 130

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132 134 135

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136

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138 139 139 143 145 146 148

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148 148 151 154

CONTENTS

4 5

M¨obius functions of algebraic number and function-fields . . . . . . . . . . . . . . . . . . . . . . . . Trace monoids and M¨obius functions . . . . . . . . . . . .

References 3

159 161 163

THE MANY FACETS OF EULER’S TOTIENT 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 The infinitude of primes . . . . . . . . . . . . . . . . 2 Exact formulae for primes in terms of ϕ . . . . . . . . 3 Infinite series and products involving ϕ, Pillai’s (Ces`aro’s) arithmetic functions . . . . . . . 4 Enumeration problems on congruences, directed graphs, magic squares . . . . . . . . . . . . . . . . . 5 Fourier coefficients of even functions (mod n) . . . . . 6 Algebraic independence of arithmetic functions . . . . 7 Algebraic and analytic application of totients . . . . . 8 ϕ-convergence of Schoenberg . . . . . . . . . . . . . 3.2 Congruence properties of Euler’s totient and related functions . 1 Euler’s divisibility theorem . . . . . . . . . . . . . . . 2 Carmichael’s function, maximal generalization of Fermat’s theorem . . . . . . . . . . . . . . . . . . . . 3 Gauss’ divisibility theorem . . . . . . . . . . . . . . . 4 Minimal, normal, and average order of Carmichael’s function . . . . . . . . . . . . . . . . . . . . . . . . . 5 Divisibility properties of iteration of ϕ . . . . . . . . . 6 Congruence properties of ϕ and related functions . . . 7 Euler’s totient in residue classes . . . . . . . . . . . . 8 Prime totatives . . . . . . . . . . . . . . . . . . . . . 9 The dual of ϕ, noncototients . . . . . . . . . . . . . . 10 Euler minimum function . . . . . . . . . . . . . . . . 11 Lehmer’s conjecture, generalizations and extensions . 3.3 Equations involving Euler’s and related totients . . . . . . . . 1 Equations of type ϕ(x + k) = ϕ(x) . . . . . . . . . . 2 ϕ(x + k) = 2ϕ(x + k) = ϕ(x) + ϕ(k) and related equations . . . . . . . . . . . . . . . . . . . . . . . . 3 Equation ϕ(x) = k, Carmichael’s conjecture . . . . . 4 Equations involving ϕ and other arithmetic functions . 5 The composition of ϕ and other arithmetic functions . 6 Perfect totient numbers and related results . . . . . . . 3

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179 179 180 180

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181

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183 184 185 186 187 188 188

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189 191

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193 195 201 204 206 208 210 212 216 216

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221 225 230 234 240

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CONTENTS

3.4

3.5

3.6

3.7

The totatives (or totitives) of a number . . . . . . . . . . . . . . 1 Historical notes, congruences . . . . . . . . . . . . . . 2 The distribution of totatives . . . . . . . . . . . . . . . 3 Adding totatives . . . . . . . . . . . . . . . . . . . . . 4 Adding units (mod n) . . . . . . . . . . . . . . . . . . 5 Distribution of inverses (mod n) . . . . . . . . . . . . Cyclotomic polynomials . . . . . . . . . . . . . . . . . . . . . 1 Introduction, irreducibility results . . . . . . . . . . . . 2 Divisibility properties . . . . . . . . . . . . . . . . . . 3 The coefficients of cyclotomic polynomials . . . . . . . 4 Miscellaneous results . . . . . . . . . . . . . . . . . . . Matrices and determinants connected with ϕ . . . . . . . . . . . 1 Smith’s determinant . . . . . . . . . . . . . . . . . . . 2 Poset-theoretic generalizations . . . . . . . . . . . . . . 3 Factor-closed, gcd-closed, lcm-closed sets, and related determinants . . . . . . . . . . . . . . . . . . . 4 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . Generalizations and extensions of Euler’s totient . . . . . . . . . 1 Jordan, Jordan-Nagell, von Sterneck, Cohen-totients . . 2 Schemmel, Schemmel-Nagell, Lucas-totients . . . . . . 3 Ramanujan’s sum . . . . . . . . . . . . . . . . . . . . . 4 Klee’s totient . . . . . . . . . . . . . . . . . . . . . . . 5 Nagell’s, Adler’s, Stevens’, Kesava Menon’s totients . . 6 Unitary, semi-unitary, bi-unitary totients . . . . . . . . . 7 Alladi’s totient . . . . . . . . . . . . . . . . . . . . . . 8 Legendre’s totient . . . . . . . . . . . . . . . . . . . . . 9 Euler totients of meet semilattices and finite fields . . . 10 Nonunitary, infinitary, exponential-totients . . . . . . . 11 Thacker’s, Leudesdorf’s, Lehmer’s, Golubev’s totients. Square totient, core-reduced totient, M-void totient, additive totient . . . . . . . . . . . . . . . . . . . . . . 12 Euler totients of arithmetical semigroups, finite groups, algebraic number fields, semigroups, finite commutative rings, finite Dedekind domains . . . . . . . . . . . . . .

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242 242 246 248 249 250 251 251 253 256 261 263 263 266

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270 273 275 275 276 277 278 278 281 282 283 285 287

. .

289

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292

References 4

295

SPECIAL ARITHMETIC FUNCTIONS CONNECTED WITH THE DIVISORS, OR WITH THE DIGITS OF A NUMBER 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

329 329

CONTENTS

4.2

4.3

Special arithmetic functions connected with the divisors of a number . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Maximum and minimum exponents . . . . . . . . . . 2 The product of exponents . . . . . . . . . . . . . . . . 3 Arithmetic functions connected with the prime power factors . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Other functions; the derived sequence of a number . . 5 The consecutive prime divisors of a number . . . . . . 6 The consecutive divisors of an integer . . . . . . . . . 7 Functional limit theorems for the consecutive divisors 8 Miscellaneous arithmetic functions connected with divisors . . . . . . . . . . . . . . . . . . . . . . . . . 9 Arithmetic functions of consecutive divisors . . . . . . 10 Hooley’s function . . . . . . . . . . . . . . . . . . 11 Extensions of the Erd¨os conjecture (theorem) . . . . . 12 The divisors in residue classes and in intervals . . . . 13 Divisor density and distribution (mod 1) on divisors . 14 The fractal structure of divisors . . . . . . . . . . . . 15 The divisor graphs . . . . . . . . . . . . . . . . . . . Arithmetic functions associated to the digits of a number . . . 1 The average order of the sum-of-digits function . . . . 2 Bounds on the sum-of-digits function . . . . . . . . . 3 The sum of digits of primes . . . . . . . . . . . . . . 4 Niven numbers . . . . . . . . . . . . . . . . . . . . . 5 Smith numbers . . . . . . . . . . . . . . . . . . . . . 6 Self numbers . . . . . . . . . . . . . . . . . . . . . . 7 The sum-of-digits function in residue classes . . . . . 8 Thue-Morse and Rudin-Shapiro sequences . . . . . . 9 q-additive and q-multiplicative functions . . . . . . . 10 Uniform - and well - distributions of αsq (n) . . . . . . 11 The G-ary digital expansion of a number . . . . . . . 12 The sum-of-digits function for negative integer bases . 13 The sum-of-digits function in algebraic number fields . 14 The symmetric signed digital expansion . . . . . . . . 15 Infinite sums and products involving the sum-of-digits function . . . . . . . . . . . . . . . . . . . . . . . . . 16 Miscellaneous results on digital expansions . . . . . .

References

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330 330 332

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334 336 337 342 343

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345 349 360 363 363 366 367 369 371 371 376 379 381 383 384 387 390 401 410 414 417 418 421

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423 427 433

5

CONTENTS

5

STIRLING, BELL, BERNOULLI, EULER AND EULERIAN NUMBERS 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Stirling and Bell numbers . . . . . . . . . . . . . . . . . . . . 1 Stirling numbers of both kinds, Lah numbers . . . . . 2 Identities for Stirling numbers . . . . . . . . . . . . . 3 Generalized Stirling numbers . . . . . . . . . . . . . 4 Congruences for Stirling and Bell numbers . . . . . . 5 Diophantine results . . . . . . . . . . . . . . . . . . . 6 Inequalities and estimates . . . . . . . . . . . . . . . 5.3 Bernoulli and Euler numbers . . . . . . . . . . . . . . . . . . 1 Definitions, basic properties of Bernoulli numbers and polynomials . . . . . . . . . . . . . . . . . . . . 2 Identities . . . . . . . . . . . . . . . . . . . . . . . . 3 Congruences for Bernoulli numbers and polynomials. Eulerian numbers and polynomials . . . . . . . . . . . 4 Estimates and inequalities . . . . . . . . . . . . . . .

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459 459 459 459 464 469 488 507 508 525

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525 534

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539 574

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References

585

Index

619

6

Preface The aim of this book is to systematize and survey in an easily accessible manner the most important results from some parts of Number Theory, which are connected with many other fields of Mathematics or Science. Each chapter can be viewed as an encyclopedia of the considered field, with many facets and interconnections with virtually almost all major topics as Discrete mathematics, Combinatorial theory, Numerical analysis, Finite difference calculus, Probability theory; and such classical fields of mathematics as Algebra, Geometry, and Mathematical analysis. Some aspects of Chapter 1 and 3 on Perfect numbers and Euler’s totient, have been considered also in our former volume ”Handbook of Number Theory” (Kluwer Academic Publishers, 1995), in cooperation with the late Professor D. S. Mitrinovi´c of Belgrade University, as well as Professor B. Crstici, formerly of Timis¸oara Technical University. However, there were included mainly estimates and inequalities, which are indeed very useful, but many important relations (e.g. congruences) were left out, giving a panoramic view of many other parts of Number Theory. This volume aims also to complement these issues, and also to bring to the attention of the readers (specialists or not) the hidden beauty of many theories outside a given field of interest. This book focuses too, as the former volume, on some important arithmetic functions of Number Theory and Discrete mathematics, such as Euler’s totient ϕ(n) and its many generalizations; the sum of divisors function σ (n) with the many old and new issues on Perfect numbers; the M¨obius function, along with its generalizations and extensions, in connection with many applications; the arithmetic functions related to the divisors, consecutive divisors, or the digits of a number. The last chapter shows perhaps most strikingly the cross-fertilization of Number theory with Combinatorics, Numerical mathematics, or Probability theory. The style of presentation of the material differs from that of our former volume, since we have opted here for a more flexible, conversational, survey-type method. Each chapter is concluded with a detailed and up-to-date list of References, while at the end of the book one can find an extensive Subject index.

7

PREFACE

We have used a wealth of literature, consisting of books, monographs, journals, separates, reviews from Mathematical Reviews and from Zentralblatt f¨ur Mathematik, etc. This volume was not possible to elaborate without the kind support of many people. The author is indebted to scientists all over the world, for providing him along the years reprints of their papers, books, letters, or personal communications. Special thanks are due to Professors A. Adelberg, G. Andrews, T. Agoh, R. Askey, H. Alzer, J.-P. Allouche, K. Atanassov, E. Bach, A. Blass, W. Borho, P. B. Borwein, D. W. Boyd, D. Berend, R. G. Buschman, A. Balog, A. Baker, B. C. Berndt, R. de la Bret`eche, B. C. Carlson, C. Cooper, G. L. Cohen, M. Deaconescu, R. Dussaud, M. Drmota, J. D´esarm´enien, K. Dilcher, P. Erd¨os, P. D. T. A. Elliott, M. Eie, S. Finch, K. Ford, J. B. Friedlander, J. Feh´er, A. A. Gioia, A. Grytczuk, K. Gy¨ory, J. Galambos, J. M. DeKoninck, P. J. Grabner, H. W. Gould, E.-U. Gekeler, P. Hagis, Jr., D. R. Heath-Brown, H. Harborth, P. Haukkanen, A. Hildebrand, A. Hoit, F. T. Howard, L. Habsieger, J. J. Holt, A. Ivi´c, H. Iwata, K.-H. Indlekofer, F. Halter-Koch, H.-J. Kanold, M. Kishore, I. K´atai, P. A. Kemp, E. Kr¨atzel, T. Kim, G. O. H. Katona, P. Leroux, A. Laforgia, A. T. Lundell, F. Luca, D. H. Lehmer, A. Makowski, M. R. Murthy, V. K. Murthy, P. Moree, H. Maier, E. Manstaviˇcius, N. S. Mendelsohn, J.-L. ˇ Porubsk´y, L. Nicolas, E. Neuman, W. G. Novak, H. Niederhausen, C. Pomerance, S. Panaitopol, J. E. Peˇcari´c, Zs. P´ales, A. Peretti, H. J. J. te Riele, B. Rizzi, D. Redmond, N. Robbins, P. Ribenboim, I. Z. Ruzsa, H. N. Shapiro, M. V. Subbarao, A. S´ark¨ozy, ˇ at, J. O. Shallit, K. B. Stolarsky, A. Schinzel, R. Sivaramakrishnan, J. Sur´anyi, T. Sal´ B. E. Sagan, I. Sh. Slavutskii, F. Schipp, V. E. S. Szab´o, L. T´oth, G. Tenenbaum, R. F. Tichy, J. M. Thuswaldner, Gh. Toader, R. Tijdeman, N. M. Temme, H. Tsumura, R. Wiegandt, S. S. Wagstaff, Jr., Ch. Wall, B. Wegner, M. W´ojtowicz. The author wishes to express his gratitude also to a number of organizations whom he received advice and support in the preparation of this material. These are the Mathematics Department of the Babes¸-Bolyai University, the Alfred R´enyi Institute of Mathematics (Budapest), the Domus Hungarica Foundation of Hungary, and the Sapientia Foundation of Cluj, Romania. The gratefulness of the author is addressed to the staff of Kluwer Academic Publishers, especially to Mr. Marlies Vlot, Ms. Lynn Brandon and Ms. Liesbeth Mol for their support while typesetting the manuscript. The camera-ready manuscript for the present book was prepared by Mrs. Georgeta Bonda (Cluj) to whom the author expresses his gratitude. The author

8

Basic Symbols f (x) = O(g(x)) or f (x) g(x)

For a range of x-values, there is a constant A such that the inequality | f (x)| ≤ Ag(x) holds over the range

f (x) g(x)

g(x) f (x) (or g(x) = O( f (x)))

f (x) = o(g(x))

f (x) ∼ g(x)

f (x) = 0 (g(x) = 0 for x large). x→∞ g(x) The same meaning is used when x → ∞ is replaced by x → α for any fixed α lim

f (x) = 1 (g(x) = 0 for x large). x→∞ g(x) The same meaning when x → ∞ is replaced with x → α lim

f (x) g(x)

There are c1 , c2 such that c1 g(x) ≤ f (x) ≤ c2 g(x) for sufficiently large x (g(x) > 0)

f (x) = (g(x))

f (x) = O(g(x)) does not hold

9

Basic Notations ϕ(n) Jk (n) Sk (n) C(n, r ) ϕ(x, n) ϕ ∗ (n), ϕ∞ (n), ϕe (n) σ (n) d(n) ω(n), (n) σk (n) σ ∗ (n), σ ∗∗ (n), σ∞ (n), σe (n), σ # (n) ψ(n) P(n) λ(n) s(n) = σ (n) − n, s ∗ (n) = σ ∗ (n) − n µ(n) µ∗ (n), µ∗∗ (n), µ(e) (n) µk (n) µ A (n) ϕ(G) µG (n), µ(G)

Euler’s totient function Jordan’s totient Schemmel’s totient Ramanujan’s sum Legendre totient unitary, infinitary, exponential totient sum of divisors function number of divisors function number of distinct, resp. total number, of prime factors of n sum of kth powers of divisors of n unitary, bi-unitary, infinitary, exponential, nonunitary sum of divisors functions Dedekind’s arithmetical function greatest prime factor of n Liouville’s function; or Carmichael’s function sum of aliquot, resp. unitary aliquot, divisors of n M¨obius function unitary, bi-unitary, exponential M¨obius functions M¨obius function of order k Narkiewicz M¨obius function Euler’s totient of a group G M¨obius function of an arithmetical semigroup G, resp. of a group G

10

BASIC NOTATIONS

µ(x, y), µ)k(P), µ M (t) µ K (a) T ∗ (n), T ∗∗ (n), Te (n)

n (x) Vϕ (n) E(n) ζ (s) =

∞

n −s (Re s > 1)

M¨obius functions of posets, resp. of a trace monoid M M¨obius function of an algebraic number field unitary, bi-unitary, exponential analogs of the product of divisors of n cyclotomic polynomial valence function of ϕ (i.e. number of solutions of ϕ(x) = n) Euler minimum function; or Erd¨os’ function Riemann’s zeta function

n=1

S(n) H (n) h(n) (n) ω(v) ρ(v) H1 (n), H1 (n) H2 (n) T (n) n

F(n) = 22 + 1 t (n) sq (n), s(n) tq,m , tn a(n) = (−1)e(n) (−1)z(n) N (xn ) D N (xn ), D d(A) EX, VX

Smarandache function; or Schinzel-Szekeres function maximum exponent of n, or harmonic numbers minimum exponent of n Hooley’s function Buchstab function Dickman function DeKoninck-Ivi´c function Tenenbaum’s function product of divisors of n; or sum of iterated totients; or tangent numbers Fermat numbers set of totatives of n sum of digits of n in base q, resp. base 10 Thue-Morse sequences Rudin-Shapiro sequence Zeckendorf sequence discrepancy, resp. uniform discrepancy of the sequence (xn ) (asymptotic) density of A expactation, resp. variance of the random variable X

11

BASIC NOTATIONS

dim X dim f X φ(u) Bn , Bn , Bn∗ B(n) S(n, k), s(n, k) s ∗ (n, k) Sr (n, k) St (n, k), st (n, k), ST (n, k), sT (n, k) S(n, k, λ|θ ) S(n, k|θ ), s(n, k|θ ) S(n, k, α), S(n, k; α, β, γ ) d(n, k), b(n, k) S[n, k], s[n, k] s ∗ [n, k] S p,q [n, k], s p,q [n, k] [n] = [n]q , [n] p,q xa,b (n) P(n, k) = k!S(n, k) S(x, y), s(x, y) Bn, Bn Bn (z) Bχn E n , E n , E n∗ , E n E n (q), E n|k (q), Hk (u, q) Gn βk (q), βk,χ (q) Hk,χ (u, q)

Hausdorff dimension fractal dimension normal distribution function Bernoulli numbers Bell numbers Stirling numbers unsigned Stirling numbers of the first kind r -Stirling numbers Stirling numbers associated to the sequence t, resp. matrix T Howard’s degenerate weighted Stirling numbers Carlitz’ degenerate Stirling numbers Dickson-Stirling numbers; resp. Hsu-Shiue-Stirling numbers associated Stirling numbers q-Stirling numbers signless q-Stirling numbers of the first kind p, q-Stirling numbers q-analogue, resp. p, q-analogue of the integer n general factorial numbers number of preferential arrangements Stirling numbers of the real numbers x, y conjugate, resp. universal-Bernoulli numbers Bernoulli numbers of higher order generalized Bernoulli numbers (χ a character) Euler numbers q-Euler numbers Genocchi numbers q-Bernoulli, resp. generalized q-Bernoulli-numbers generalized q-Euler numbers

12

BASIC NOTATIONS

q(a, p), q(a, m) wp n (q) β Bn∗ (q) ζq (s), ζq (s, x) L q (s, χ) G(x, q) A(n, k), a(n, k) Bn,k (q), Am,n (q) A(n, k, α) A− (n, k) m ak,n m E(m, k) = k a|b, a b a ≡ b (mod n) c a ≡ (mod n) b d n! = 1 · 2 . . . n !n = 0! + 1! + · · · + (n − 1)! (a, b) [a, b] (x) exp(z) [x] or [x]∗ {x} = x − [x] p q n nk n = k q k Fn , L n ν p (n) [ai, j ]m×n f ∗g f g, f g

Fermat and Euler quotients Wilson quotient modified q-Bernoulli numbers p-adic q-Bernoulli numbers q-Riemann ζ -function, resp, q-Hurwitz ζ -function q − L-series q − log −-function Eulerian numbers q-Eulerian numbers Dickson-Eulerian numbers signed Eulerian numbers generalized Eulerian numbers second order Eulerian numbers a divides b, a does not divide b n divides (a − b) for integers a, b n|(ad − bc) for (b, n) = (d, n) = 1 factorial of n left-factorial of n g.c.d. of a and b, or an ordered pair l.c.m. of a and b Euler’s gamma function ez integer part of x fractional part of x Legendre (or Jacobi) symbol binomial coefficient q-binomial coefficient Fibonacci, resp. Lucas - numbers p-adic order of n m × n matrix of components ai j Dirichlet convolution unitary, resp. bi-unitary convolution

13

BASIC NOTATIONS

f ∗ A g, f ◦ g f ∗ex g, f ∇g, ( f ∗ g)∞ f ♦g, f #g f ∗l−c g f ∗G g

regular (Narkiewicz) resp. K (Davison) - convolution (or composition of functions) exponential, Golomb resp. infinitary-convolution max-product, resp. Cauchy product Lucas-Carlitz product matrix-generated convolution

14

Chapter 1

PERFECT NUMBERS: OLD AND NEW ISSUES; PERSPECTIVES

1.1

Introduction

The aim of this chapter is to survey the most important and interesting notions, results, extensions, generalizations related to perfect numbers. Many old, as well as new open problems will be stated, which will motivate - we do hope - many further research. This is one of the oldest subjects of Mathematics, with a considerable history. Some basic historical facts will be presented, as this will underline too our strong opinion on the role of perfect numbers in the development of Mathematics. It is sufficient to only mention here Fermat’s ”little theorem” of considerable importance in Number theory, Algebra, and more recently in Criptography. This theorem states that for all primes p and all positive integers a, p divides a p − a. Fermat discovered this result by studying perfect numbers, and trying to elaborate a theory of these numbers. One more example is the theory of primes in special sequences, and generally the classical theory of primes. Even perfect numbers involve the so called ”Mersenne primes”, of great importance in many parts of Number theory. Currently, about 39 such primes are known (39 as of 14-XI-2001, see e.g. http://www.stormloader.com/ajy/perfect/html), giving 39 known perfect numbers, all even. Recently (at the end of 2003) the 40th perfect number has been discovered. No odd perfect numbers are known, but we shall see on the part containing this theme, the most important and up-to-date results obtained along the centuries. An extension 15

CHAPTER 1

of perfect numbers are the ”amicable numbers” having a same old history, with considerable interest for many mathematicians. Many results, more generalizations, connections, analogies will be pointed out. Here the theory is filled again by a lot of unsolved problems. Along with the extensions of the notion of divisibility, there appeared many new notions of perfect numbers. These are e.g. the unitary perfect-, nonunitary perfect, biunitary perfect-, exponential perfect-, infinitary perfect-, hyperperfect-, integer perfect-, etc., numbers. On the other hand, there appeared also the necessity of studying, by analogy with the classical case, such notions as: superperfect-, almost perfect-, quasiperfect-, pseudoperfect (or semi-perfect), multiplicatively perfect, etc., numbers. Some authors use different terminologies, so one aim is also to fix in the literature the exact terminologies and notations. Our aim is also to include results and references from papers published in certain little known journals (or unpublished results, obtained by personal communication to the authors).

1.2

Some historical facts

It is not exactly known when perfect numbers were first introduced, but it is quite possible that the Egypteans would have come across such numbers, given the way their methods of calculation worked (”unit fractions”, ”Egyptean fractions”). These numbers were studied by their mystical properties by Pythagoras, and his followers. For the Pythagorean school the ”parts” of a number are their divisors. A number which can be built up from their parts (i.e. summing their divisors) should be indeed wonderful, perfectly made by the God. God created the world in six days, and the number of days it takes the Moon to travel round the Earth is nothing else than 28. (For the number mysticism by Pythagoras’ school see U. Dudley [85]). These are the first two perfect numbers. The four perfect numbers 6, 28, 496 and 8128 seem to have been known from ancient times, and there is no record of these discoveries. The first recorded result concerning perfect numbers which is known occurs in Euclid’s ”Elements” (written around 300BC), namely in Proposition 36 of Book IX: ”If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes a prime, and if the sum multiplied into the last make some number, the product will be perfect.” Here ”double proportion” means that each number of the sequence is twice the preceding number. Since 1 + 2 + 4 + · · · + 2k−1 = 2k − 1, the proposition states that: If, for some k > 1, 2k − 1 is prime, then 2k−1 (2k − 1) is perfect. 16

(1)

PERFECT NUMBERS

Here we wish to mention another source for perfect numbers (usually overlooked by the Historians of mathematics) in ancient times, namely Plato’s Republic, where the so-called periodic perfect numbers were introduced. It is remarkable that 2000 years later when Euler proves the converse of (1) (published posthumously, see [97]) he makes no references to Euclid. However, Euler makes reference to Plato’s periodic perfect numbers. M. A. Popov [243] says that Euler’s proof was probably inspired by Plato. Another early reference seems to be at Euphorion (see J. L. Lightfoot [191]) a poet of the third century, B.C. The following significant study of perfect numbers was made by Nichomachus of Gerasa. Around 100 AD he wrote his famous ”Introductio Arithmetica” [227], which gives a classification of numbers into three classes: abundant numbers which have the property that the sum of their aliquot parts is greater than the number, deficient numbers which have the property that the sum of their aliquot parts is less than the number, and perfect numbers. Nicomachus used this classification also in moral terms, or biological analogies: ”... in the case of too much, is produced excess, superfluity, exaggerations and abuse; in the case of too little, is produced wanting, defaults, privations and insufficiencies...” ”... abundant numbers are like an animal with ten mouths, or nine lips, and provided with three lines of teeth; or with a hundred arms...” ”... deficient numbers are like animals with a single eye,... one armed or one of his hands, less than five fingers, or if he does not have a tongue.” In the book by Nicomachus there appear five unproved results concerning perfect numbers: (a) the n-th perfect number has n digits; (b) all perfect numbers are even; (c) all perfect numbers end in 6 and 8 alternately; (d) every perfect number is of the form 2k−1 (2k − 1), for some k > 1, with 2k − 1 = prime; (e) there are infinitely many perfect numbers. Despite the fact that Nicomachus offered no justification of his assertions, they were taken as fact for many years. The discovery of other perfect numbers disproved immediately assertions (a) and (c). On the other hand, assertions (b), (d), (e) remain unproved practically even in our days. The Arab mathematicians were also fascinated by perfect numbers and Thabit ibn Quarra wrote ”Treatise on amicable numbers” in which he examined when numbers of the form 2n p ( p prime) can be perfect. He proved also ”Thabit’s rule” (see the section with amicable numbers). Ibn al-Haytham proved a partial converse to Euclid’s proposition (1), in the unpublished work ”Treatise on analysis and synthesis” (see [247]).

17

CHAPTER 1

Among the Arab mathematicians who take up the Greek investigation of perfect numbers with great enthusiasm was Ismail ibn Ibrahim ibn Fallus (1194-1239) who wrote a treatise based on Nicomachus’ above mentioned text. He gave also a table of ten numbers claiming to be perfect. The first seven are correct, and in fact these are indeed the first seven perfect numbers. For details of this work see the papers by S. Brentjes [36], [37]. The fifth perfect number was rediscovered by Regiomontanus during his stay at the University of Vienna, which he left in 1461 (see [235]). It has also been found in a manuscript written by an anonymous author around 1458, while the fifth and sixth perfect numbers have been found in another manuscript by the some author, shortly after 1460. The fifth perfect number is 33550336, and the sixth is 8589869056. These show that Nicomachus’ first claims (a) and (c) are false, since the fifth perfect number has 8 digits, and the fifth and sixth perfect numbers both ended in 6. In 1536 Hudalrichus Regius published ”Utriusque Arithmetices” in which he found the first prime p ( p = 11) such that 2 p−1 (2 p − 1) = 2047 = 23 · 89 is not perfect. In 1603 Cataldi found the factors of all numbers up to 800 and also a table of all primes up to 750. He used his list of primes to check that 219 − 1 = 524287 was prime, so he had found the seventh perfect number 137438691328. Among the many mathematicians interested in perfect numbers one should mention Descartes, who in 1638 in a letter to Mersenne wrote ([72]): ”... Perfect numbers are very few... as few are the perfect men...” In this sense see also a Persian manuscript [331]. ”I think I am able to prove there are no even numbers which are perfect apart from those of Euclid; and that there are no odd perfect numbers, unless they are composed of a single prime number, multiplied by a square whose root is composed of several other prime number. But I can see nothing which would prevent one from finding numbers of this sort... But, whatever method one might use, it would require a great deal of time to look for these numbers...” The next major contribution was made by Fermat [225], [313]. He told Roberval in 1636 that he was working on the topic and, although the problems were very difficult, he intended to publish a treatise on the topic. The treatise would never been written, perhaps because Fermat didn’t achieve the substantial results he had hoped. In June 1640 Fermat wrote a letter to Mersenne telling him about his discoveries on perfect numbers. Shortly after writing to Mersenne, Fermat wrote to Frenicle de Bessy, by generalizing the results in the earlier letter. In his investigations Fermat used three theorems: (a) if n is composite, then 2n − 1 is composite; (b) if n is prime, a n − a is multiple of n; (c) if n is prime, p is a divisor of 2n − 1, then p − 1 is a multiple of n.

18

PERFECT NUMBERS

Using his ”Little theorem”, Fermat showed that 223 − 1 was composite and also 237 − 1 is composite, too. Mersenne was very interested in the results that Fermat sent him on perfect numbers. In 1644 he published ”Cogitata physica mathematica”, in which he claimed that 2 p−1 (2 p − 1) is perfect for p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257 and for no other value of p up to 257. It is remarkable that among the 47 primes p between 19 and 258 for which 2 p − 1 is prime, for 42 cases Mersenne was right. Primes of the form 2 p − 1 are called Mersenne primes. (For a recent table of known Mersenne primes, see the site http://www.utm.edu/research/primes/mersenne/index.html). The next mathematician who made important contributions was Euler. In 1732 he proved that the eighth perfect number was 230 (231 − 1). It was the first seen perfect number discovered for 125 years. But the major contributions by Euler were obtained in two unpublished manuscripts during his life. In one of them he proved the converse of Euclid’s statement: All even perfect number are of the form 2 p−1 (2 p − 1).

(2)

By quoting R. C. Vaughan [314]: ”we have an example of a theorem that took 2000 years to prove... But pure mathematicians are used to working on a vast time scale...” Euler’s results on odd perfect numbers and amicable numbers will be considered later. After Euler’s discovery of primality of 231 − 1, the search for perfect numbers had now become an attempt to check whether Mersenne was right with his claims. The first error in Mersenne’s list was discovered in 1876 by Lucas, who showed that 267 − 1 is composite. But 2127 − 1 is a Mersenne prime, so he obtained a new perfect number (but not the nineth, but as later will be obvious the twelfth). Lucas made also a theoretical discovery too, which later modifed by Lehmer will be the basis of a computer search for Mersenne primes. In 1883 Pervusin showed that 261 − 1 is prime, thus giving the nineth perfect number. In 1911, resp. 1914 Powers proved that 289 − 1 resp. 2101 − 1 are primes. 288 (289 − 1) was in fact the last perfect number discovered by hand calculations, all others being found using theoretical elements or a computer. In fact computers have led to a revival of interest in the discovery of Mersenne primes, and therefore of perfect numbers. The first significant results on odd perfect numbers were obtained by Sylvester. In his opinion (see e.g. [317]): ”... the existence of an odd perfect number its escape, so to say, from the complex web of conditions which hem it in on all sides - would be little short of a miracle...” 19

CHAPTER 1

In 1888 he proved that any odd perfect number must have at least 4 distinct prime factors. Later, in the same year he improved his result for five factors. The developments, as well as recent results will be studied separately. Finally, we shall include here for the sake of completeness all known perfect numbers along with the year of discovery and the discoverer (s). Let Pk be the kth perfect number. Then Pk = 2 p−1 (2 p − 1) = A p , where 2 p − 1 is a Mersenne prime. P1 = A2 = 6, P2 = A3 = 28, P3 = A5 = 496, P4 = A7 = 8128, P5 = A13 (1456, anonymous), P6 = A17 (1588 Cataldi), P7 = A19 (1588, Cataldi), P8 = A31 (1772, Euler), P9 = A61 (1883, Pervushin), P10 = A89 (1911, Powers), P11 = A107 (1914, Powers), P12 = A127 (1876, Lucas), P13 = A521 (1952, Robinson), P14 = A607 (1952, Robinson), P15 = A1279 (1952, Robinson), P16 = A2203 (1952, Robinson), P17 = A2281 (1952, Robinson), P18 = A3217 (1957, Riesel), P19 = A4253 (1961, Hurwitz), P20 = A4423 (1961, Hurwitz), P21 = A9689 (1963, Gillies), P22 = A9941 (1963, Gillies), P23 = A11213 (1963, Gillies), P24 = A19937 (1971, Tuckerman), P25 = A21701 (1978, Noll and Nickel), P26 = A23209 (1979, Noll), P27 = A44497 (1979, Nelson and Slowinski), P28 = A86243 (1982, Slowinski), P29 = A110503 (1988, Colquitt and Welsh), P30 = A132049 (1983, Slowinski), P31 = A216091 (1985, Slowinski), P32 = A756839 (1992, Slowinski and Gage), P33 = A859433 (1994, Slowinski and Gage), P34 = A1257787 (1996, Slowinski and Gage), P35 = A1398269 (1996, Armengaud, Woltman, et al. (GIMPS)), P36 = A2976221 (1997, Spence, Woltman, et al. (GIMPS)), P37 = A3021377 (1998, Clarkson, Woltman, Kurowski et al. (GIMPS, Primenet)), P38 = A6972593 (1999, Hajratwala, Woltman, Kurowki et al. (GIMPS, Primenet)), P?? = A13466917 (2001, Cameron, Woltman, Kurowski, et al.). Recently, M. Shafer (see http://mathworld.wolfram.com) has announced the discovery of the 40th Mersenne prime, giving A20996011 . The full values of the first seventeen perfect numbers are written also in the note by H. S. Uhler [312] from 1954. For a quick perfect number analyzer by Brendan McCarthy see the applet at http://ccirs.camosun.bc.ca/∼jbritton/jbperfect/htm. The 39th Mersenne prime is of course, a very large prime, having a number of 4053946 digits (it seems that it is the largest known prime). There have been discovered also very large primes of other forms. For example, in 2002, Muischnek and Gallot discovered the prime number 105747665536 + 1. For the largest known primes of various forms see the site: http://www.utm.edu/research/primes/largest.html.

1.3

Even perfect numbers

Let σ (n) denote the sum of all positive divisors of n. Then n is perfect if σ (n) = 2n 20

(1)

PERFECT NUMBERS

As we have seen in the Introduction, all known perfect numbers are even, and by the Euclid-Euler theorem n can be written as n = 2k−1 (2k − 1),

(2)

where 2k − 1 is a prime (called also as ”Mersenne prime”). Actually k must be prime. The first two perfect numbers, namely 6 and 28 are perhaps the most ”human” since are closely related to our life (number of days of a week, of a month, of a woman cycle, etc.). The famous mathematician and computer scientist D. Knuth in his interesting homepage (http://www-csfaculty.stanford.edu/∼knuth/retd.htm) on the occasion of his retirement says: ”... I’m proud of the 28 students for whom I was a dissertation advisor (see vita); and I know that 28 is a perfect number...” 28 is in fact the single even perfect number of the form x3 + 1

(3)

(x positive integer), proved by A. Makowski [205]. As corollaries of this fact Makowski deduces that the single even perfect number (4) of the form n n + 1 is 28; and that there is no even perfect number of the form nn

...n

+ 1,

(5)

where the number of n’s is ≥ 3. By generalization, A. Rotkiewicz [263] proves that 28 is the single even perfect number of the form a n + bn , where (a, b) = 1 and n > 1.

(6)

If n > 2 and (a, b) = 1, he proves also that there is no even perfect number of the form a n − bn . (7) Not being aware of Rotkiewicz’s result (6), T. N. Sinha [288] has rediscovered it in 1974. The numbers 6 and 28 are the only couple of perfect numbers of the form n − 1 n(n + 1) , where n > 1; and this is obtained for n = 7. This is a result of L. and 2 Jones [157]. (8) We note that this result applies the Euclid-Euler theorem (representation of even perfect numbers), as well as a theorem by Euler on the form of odd perfect numbers (see the next section). 21

CHAPTER 1

As for the digits of even perfect numbers, as already Nicomachus (see section 2) remarked, the last digits are always 6 or 8 (but not in alternate order, as he thought), proved rigorously by E. Lucas in 1891 (see [84], p. 27, and also [138]). (9) Let us now sum the digits of any even perfect number (except 6), then sum the digits of the resulting number,..., etc., repeating this process until we get a single digit. Then this single digit will be one. (10) See [332] for this result, with a proof. Let A(n) be the set of prime divisors of n > 1. If n is an even perfect number, then it is immediate that A(n) = A(σ (n)). (11) Now, an interesting fact, due to C. Pomerance [240] states that, reciprocally, if (11) holds true for a number n, then n must be even perfect. For the additive representation of even perfect numbers, by an interesting result by R. L. Francis [103] any even perfect number > 28 can be represented as the sum of at least two perfect numbers. (12) For the values of other arithmetic functions at even perfect numbers we quote the following result of S. M. Ruiz [264]: Let S(n) be the Smarandache function, defined by S(n) = min{k ∈ N : n|k!}.

(13)

Then if n is even perfect, then one has S(n) = M p

(14)

where M p = 2 p − 1 is the Mersenne prime in the known form of n. For a simple proof, see J. S´andor [272]. In [272] there is proved also that S(M p ) ≡ 1 (mod p), and that if 22 p + 1 is prime too, then S(24 p − 1) = 22 p + 1 ≡ 1 (mod 4 p). For the values of Euler’s function on even perfect numbers, see S. Asadulla [8]. For perfect numbers concerning a Fibonacci sequence, see [234]. F. Luca [197] proved also that there are no perfect Fibonacci or Lucas numbers. In [198] he proved that there are only finitely many multiply perfect numbers in these sequences. K. Ford [102] considered numbers n such that d(n) and σ (n) are both perfect numbers (called ”sublime numbers”). There are only two known such numbers, namely n = 12 and n = 2126 (261 − 1)(231 − 1)(219 − 1)(27 − 1)(25 − 1)(23 − 1). It is not known if any odd sublime number exists. D. Iannucci (see his electronic paper ”The Kaprekar numbers”, J. Integer Sequences, 3(2000), article 00.1.2) proved that every even perfect number is a Kaprekar number in the binary base, e.g. (28)2 = 11100 and 11002 = 1100010000 with 100 + 010000 = 11100 (703 in base 10 is Kaprekar means that 7032 = 494209 where 494 + 209 = 703). 22

PERFECT NUMBERS

We wish to mention also some new proofs of the Euler theorem on the form of an even perfect number. In standard textbooks, usually it is given Euler’s proof, in a slightly simplified form given by L. E. Dickson in 1913 ([82], [80]). An earlier proof was given by R. D. Carmichael [43]. A new proof has been published by Gy. Kisgergely [174] in a paper written in Hungarian. Another proof, due to J. S´andor ([271], [273]) is based on the simple inequality σ (ab) ≥ aσ (b)

(15)

with equality only for a = 1. This method enabled him also to obtain a new proof on the form of even superperfect numbers (see later). For even perfect numbers see a paper by S. J. Bezuska and M. J. Kenney [24] (where 36 perfect numbers are mentioned (up to 1997!)). See also G. L. Cohen [56].

1.4

Odd perfect numbers

There is a good account of results until 1957 in the paper by P. J. McCarthy [46]. The first important result on odd perfect numbers was obtained by Euler [98] when he proved that such a number n should have the representation n = p α q1 1 . . . qr2βr 2β

(1)

where p, qi (i = 1, r ) are distinct odd primes and p ≡ 1 (mod 4), α ≡ 1 (mod 4). Here p α is called the Euler factor of n. Another noteworthy result, mentioned also in the Introduction is due to J. J. Sylvester [310] who proved that we must have r ≥ 4 and that r ≥ 5 if n ≡ 0 (mod 3). (2) The modern revival of interest in the problem of odd perfect numbers seems to have been begun by R. Steurwald [296] who proved that n cannot be perfect if β1 = · · · = βr = 1.

(3)

A. Brauer [33] and H.-J. Kanold [159] proved the same if β1 = 2 and β2 = · · · = βr = 1

(4)

In [158] Kanold proved that n is not perfect if β1 = · · · = βr = 2

(5)

2βi + 1 (i = 1, r ) have as a common factor 9, 15, 21, or 33.

(6)

and also if the

23

CHAPTER 1

Similarly, he proved [165] that n cannot be perfect if β1 = β2 = 2, β3 = · · · = βr = 1; and if α = 5 and βi = 1 or 2 (i = 1, r ). (7) He obtained similar results in [161]. P. J. McCarthy [47] proved that if n is perfect and prime to 3, and β2 = · · · = βr = 1, then q1 ≡ 1

(mod 3)

(8)

In 1971 W. L. McDaniel [76] improved result (6) by proving that 3 cannot be a common divisor of 2βi + 1 (i = 1, r ). (9) A year later, P. Hagis, Jr. and W. L. McDaniel [133] showed that n cannot be perfect if β1 = · · · = βr = 3. (10) We note here that in the above papers the theory of cyclotomic polynomials, as well as diophantine equations are widely used. Some computations use modern computers, too. It is of interest to note that in the proof of (10) the following results due to U. K¨uhnel [184], resp. H.-J. Kanold [158] are used: If n is odd perfect, then 105 n; (11) If n is odd perfect, and if in (1) s is a common factor of 2βi + 1 (i = 1, r ), then (12) s 4 |n. Another refinement of Euler’s theorem is due to J. A. Ewell [99]. If the prime q1 , . . . , qr (together with their corresponding exponents) are relabeled as p1 , p2 , . . . , pk , h 1 , h 2 , . . . , h t so that pi (i = 1, k) are of the form ≡ 1 (mod 4) and each h j ( j = 1, t) are of the form ≡ −1 (mod 4), say n = pα

k

pi2αi

i=1

t

2β j

hj

(13)

j=1

then: (i) the ordered set A(k) = {α1 , . . . , αk } contains an even number of odd numbers, provided α and p belong to the same residue class (mod 8); (ii) A(k) contains an odd number of odd numbers, provided that α and p belong to different classes (mod 8). (14) Of a similar nature are also the following theorems due to G. L. Cohen and R. J. Williams [67]: If n is an odd perfect number, then in (1) if we assume that β1 = · · · = βr = β, then we must have β ∈ {6, 8, 11, 14, 18}; (15) If n is as above and β1 = · · · = βr = 1, then β1 ∈ {5, 6}; 24

(16)

PERFECT NUMBERS

They obtain also (independently) a simpler proof of result (14). The theorem of Sylvester on the lower bounds for r in (1) has been improved by several authors. I. S. Gradstein [110], U. K¨uhnel [184] and G. C. Webber [324] proved that in (2) r ≥ 5 holds without any condition. (17) Recall, that r = ω(n) − 1. The result r ≥6 (18) has been discovered by N. Robbins [259] and C. Pomerance [237]; while the strongest result known today, namely r ≥7 (19) is due to P. Hagis, Jr. [120] and J. E. Z. Chein [51]. For a new, algorithmic approach, see G. L. Cohen and R. M. Sorli [On the number of distinct prime factors of an odd perfect number, J. Discr. Algorithms 1(2003), 2135]. Result (2) of Sylvester has been sharpened to r ≥ 8 if n ≡ 0

(mod 3)

(20)

r ≥ 10 if n ≡ 0

(mod 3)

(21)

by H.-J. Kanold [160] and to

by M. Kishore [179]. P. Hagis, Jr. [123] proved the same for (n, 3) = 1. For a survey of earlier results, see D. S. Mitrinovi´c - J. S´andor [218] (see p. 100-102). E. Catalan proved in 1888 (see [84]) that if an odd perfect number is not divisible by 3, 5 or 7, it has at least 26 distinct prime factors, and thus has at least 45 digits. T. Pepin showed in 1897 (see [84]) that an odd perfect number relatively prime to 3 · 7, 3 · 5 or 3 · 5 · 7 contains at least 11, 14, or 19 distinct prime factors, respectively; and cannot have the form 5(mod6). In the above mentioned paper [160] Kanold proved also that an odd perfect number must have at least a prime divisor greater than or equal to 61. (22) J. B. Muskat [223] in 1966 proved that an odd perfect number n is divisible by a prime power > 1012 . (23) C. Pomerance [238] proved that the second largest prime divisor of n must be at least 139. (24) and in 1980 P. Hagis, Jr. [121], improved this to 1000. (25) Related to the largest prime factor of an odd perfect number n, this is greater than 100129 25

(26)

CHAPTER 1

as shown by P. Hagis, Jr. and W. J. Daniel [134]; and improved to 300000

(27)

by J. T. Condict [68]. Result (25) on the second largest prime factor has been recently improved to 10000

(28)

by D. E. Iannucci [150]. Ianucci further [151] obtained for the third largest prime factor the lower bound 100 (29) Iannucci and Sorli [153] proved that if n is odd perfect, then (n) ≥ 37, (n) being the total number of prime factors of n. Kanold’s result (22) was subsequently improved. As a corollary of a general result (which we shall state later) N. Robbins [260] proved that if an odd perfect number n is divisible by 17, then n must have a prime factor not smaller than 577. (30) The strongest result was due to P. Hagis, Jr. and G. L. Cohen [132] who obtained, without any condition the lower bound 106

(31)

Very recently, this has been improved to 107 by P. M. Jenkins [155]. See also A. Grytczuk and M. W´ojtowicz [113] on the magnitude of perfect numbers. Related to the lower bounds of odd perfect numbers, a substantial result was obtained in 1973 by P. Hagis, Jr. [119], giving the lower bound 1050 . This was subsequently improved to 10160 by R. P. Brent and G. L. Cohen [34], to 10300 by Brent, Cohen and H. J. J. te Riele [35]. In a recent paper by S. Davis [79] a rationality condition for the existence of odd perfect numbers is used to derive an upper bound for the density of odd integers such that σ (n) could be equal to 2n, where n belongs to a fixed interval with a lower limit greater than 10300. Bounds of a general nature were first considered by Kanold. In [158] he proved that if n given by (1) is odd perfect, then the largest prime divisor of n is greater then 2 · max{α + 1, 2βi + 1 : i = 1, r }

(32)

In [160] he proved that if n is odd perfect and p is a Gaussian prime, then α < a(a − 1) ≤ r (r − 1)

(33)

where a is the number of qi such that qi ≡ 1 (mod p). In [165] a similar result runs as follows: 26

PERFECT NUMBERS

Suppose n is odd perfect and that ( p − 1, 2βi + 1) = 1, i = 1, r . Let a be as 3a(a − 1) 2β , when pa−1 σ (qi i ) above. Then 1 < a < r and α ≤ min a(a − 2), 4 (a − 1)(3a − 2) for any i = 1, r ; and α ≤ , otherwise. (34) 4 A complicated lower bound has been given by N. Robbins [260]. Suppose u is an odd prime. Let a1 (u) < a2 (u) < . . . be those primes p such that (i) p ≡ 2u 2k − 1 (mod 4u 2k ) for some integer k; (ii) 105 ( p + 1); (iii) 165 ( p + 1). Let V (u) be the set of all primes v such that v ≡ 1 (mod u). For each v ∈ V (u), let v = max{v ∗ : v ∗ ∈ V (u), v ∗ |φu (v)}, where φn (t) denotes the n-th cyclotomic polynomial evaluated at t. Let w(v) = max{v, v }. Let b1 (u) < b2 (u) < . . . be all the distinct w(v) such that v ∈ V (u). Finally, let X (u) be the set of all primes x such that (i) x ≡ 2u 2k−1 (mod 4u 2k−1 ) for some integer k; (ii) 105 (x 2u − 1)/(x − 1); (iii) 165 (x 2u − 1)/(x − 1). For each x ∈ X (u), let v = max{v ∗ : v ∗ ∈ V (u), v ∗ |φu (x)}, v = max{v ∗ : ∗ v ∈ V (u), v ∗ |φu (−x)}. Let y(x) = max{x, v, v }. Let c1 (u) < c2 (u) < . . . be all the distinct y(x) such that x ∈ X (u). Put L(u) = min{a1 (u), b1 (u), c1 (u)}.

(35)

Now, if n given by (1) is an odd perfect number and u is a Fermat prime such that u ||n, then m

M(n) ≥ L(u)

(36)

M(n) = max{ p, q1 , . . . , qr }

(37)

where

For an example, the values of an (3), an (5), an (17), bn (3), bn (5), bn (17), cn (3), cn (5), cn (17), consider the tables: n 1 2 3 4 5 6 an (3) 17 89 197 233 449 521 an (5) 149 349 449 1249 1549 1949 an (17) 577 1733 8669 14449 19073 22541 27

CHAPTER 1

n 1 2 3 4 5 6 bn (3) 19 61 67 79 97 127 bn (5) 1381 2221 3221 bn (17) < 1031

n 1 2 3 4 5 6 cn (3) 271 821 919 1021 1087 1093 cn (5) 400291 cn (17) < 1031 Various bounds for the prime factors of an odd perfect number have been deduced. Let p1 be the least prime factor of n given by (1). Then Cl. Servais [283] proved first that p1 ≤ r + 1 = ω(n) = A (38) where ω(n) = A denotes the number of distinct prime factors of n. For a recent extension of the Servais result see [23]. In 1952 O. Gr¨un [112] improved this to 2 A+2 3

p1 <

(39)

The bound (39) was deduced also independently by H. Sali´e [268] in 1953, as well as M. Perisastri [233] in 1958. We note that (39), or even (38) are many times rediscovered by different authors (see e.g. [86] for a weaker result than (38)). By using some general results concerning systems of Diophantine equations, H.-J. Kanold [162] in 1952 proved that if n is odd q +1 , (40) perfect (or even multiply perfect), then n has a square factor ≥ 2 where q is the largest divisor of n (denoted also as q = P(n)). A Let an odd perfect number n be written in the prime factorization n = piai . i=1

Then i(i+1)/2

pi < (4A)2

,

i = 1, A

(41)

which is due to C. Pomerance [239]. For 2 ≤ i ≤ 6 one has the improvement i−1

pi < 22 (r − i + 1) due to M. Kishore [177]. 28

(42)

PERFECT NUMBERS

D. R. Heath-Brown [143] proved that if an odd perfect number n has A distinct prime factors, then A n < 44 (43) 1 Various bounds for the sum S = (where p is prime) for an odd perfect p p/n number have been obtained by D. Suryanarayana [304] and P. Hagis Jr. and D. Suryanarayana [137]. For example, when n ≡ 1 (mod 12) and 5|n, then 0.64738 < S <

1 + log 50/31 = 0.67804 5

(44)

If n ≡ 1 (mod 12), but 5 n, then 0.66745 < S < 0.69315

(45)

See Mitrinovi´c-S´andor [218] (pp. 103-104) for more detailed results. In D. Surya p . Namely, e.g., if n is narayana [305] similar results are obtained for P = p−1 p/n odd perfect and n ≡ 1 (mod 12), 5|n, then 2 < P < 2.031002;

(46)

If n ≡ 1 (mod 12), 5 n, then 2 < P < 2.014754,

(47)

etc. The above congruence n ≡ 1 (mod 12) is not considered at random. A theorem by J. Touchard [311] states that for an odd perfect number n one has n ≡ 1 (mod 12), or n ≡ 9

(mod 36)

(48)

The initial proof given by Touchard was quite complicated, for an elementary proof see a recent paper by J. A. Holdener [144]. An earlier proof was obtained by M. Satyanarayana [280]. A somewhat later one is by M. Raghavachari [246]. An immediate application of Touchard’s theorem (48) and the fact that 2 p−1 (2 p − 1) ≡ 4 (mod 12) for any prime p ≥ 3, is that two consecutive positive integers cannot be both perfect numbers (see F. Luca [195]). (49) Another famous result is due to L. E. Dickson [82]: There are only a finite number of odd perfect numbers with a given number of prime divisor. This may be reformulated more explicitly as follows: The set D = {n = p1a1 . . . prar : p1 , . . . , pr given odd primes; ai = 0, 1, 2, . . . } contains at most a finite set of perfect numbers. (50) 29

CHAPTER 1

New proofs of this theorem were obtained e.g. by H. N. Shapiro [284], H.-J. Kanold [167], R. W. Van der Waall [316]. In fact Shapiro, in the paper [284], and in a new paper [285] obtains a very large generalization of Dickson’s theorem. Kanold’s method works also for k-multiperfect numbers (see [168]), another proof appears also in [316]. In 1990 Kanold [173] obtains a new generalization of Dickson’s theorem. We now return to the representation (1) of an odd perfect number n, in order to consider certain result of a new type concerning the term p α (”Euler factor”) or the β other terms qi i . 2β In 1947 R. J. Levit [189] proved that n is not perfect if σ ( p α )/2 and σ (qi i ) (51) (i = 1, r ) are prime powers. Recently P. Starni [294] proved that if n is an odd perfect number with qi ≡ 3 (mod 4) (i = 1, r ), then σ ( p α )/2 is a composite number. (52) Let n be written as n = p α M 2 where (see (1)) as usual p ≡ 1 ≡ 1 (mod 4), ( p, M) = 1. In 1993 P. Starni [295] proved that if α + 2 is prime or pseudoprime in base p and (α + 2, p − 1) = 1, then M 2 ≡ 0 (mod α + 2).

(53)

p−1 , As a corollary, if n = p α M 2 is odd perfect, α + 2 = prime, and α + 2 > 4 then M ≡ 0 (mod α + 2) (54) r 2β J. Slowak [291] shows that p α < σ (qi i ), so in fact one has the following i=1

representation of an odd perfect number: n = pα

σ ( pα ) d, 2

d>1

(55)

Some words on the density of odd perfect numbers: In 1954 Kanold [166] proved that this density is 0. This follows also from a result by B. Volkmann [315]. Stronger results have been obtained in the following years by B. Hornfeck [146] who showed that for V (x) = car d{n ≤ x : n perfect} (56) one has V (x) < x 1/2

(57)

A year later [147] he showed that V (x) 1 lim sup √ ≤ √ x x→∞ 2 5 30

(58)

PERFECT NUMBERS

Kanold [169] improved these to V (x) < c

x 1/4 log x log log x

(59)

while Hornfeck and E. Wirsing [148], resp. Wirsing [328] proved that c log x log log log x V (x) < exp log log x and

c log x V (x) < exp log log x

(60)

(61)

It follows from this result that even if some odd perfect numbers do exist, at least they are less numerous than, for example, the primes. In particular, the sum of the reciprocals of the perfect numbers is a convergent series. Finally, we wish to point out some new issues on odd perfect number, along with some open problems. By writing n = p α M 2 , where p ≡ α ≡ 1 (mod 4), clearly if one has

σ (M 2 ) = p α ,

σ ( p α ) = 2M 2 ,

(62) (63)

then n is odd perfect. In 1977 D. Suryanarayana [309] raised the question, that should every odd perfect number (which necessarily has the form (62)) also necessarily satisfy the relations (63)? This was answered in the negative by G. G. Dandapat, J. L. Hunsucker and C. Pomerance [74], and also later by E. Z. Chen utilizing a deep result of Ljunggern. D. Suryanarayana [309] raised also the following problem. If n = p α M 2 is an odd perfect number so that p ≡ α ≡ 1 (mod 4) and ( p, M) = 1, does it necessarily follow that there exist a divisor d|M such that σ (d 2 ) = p α M 2 /d 2 and σ ( p α M 2 /d 2 ) = 2d 2 ?

(64)

This problem is still open. Another open problem by Suryanarayana is that: is it true that every odd perfect number is of the form mσ (m) for some odd integer m; if so, is (m, σ (m)) = 1 necessarily? (65)

31

CHAPTER 1

M. V. Subbarao [301] raises the problem: Does every odd perfect number n (if such exist) have the representation 1 n = mσ (m)? 2

(66)

Another question: whenever n given by (66) is perfect, does it follow that n is odd and (m, σ (m)) = 1? (67) C. W. Anderson (see [239]) has conjectured that the set of rational numbers that arise as σ (n)/n for some n, is a recursive set. (68)

1.5

Perfect, multiperfect and multiply perfect numbers

First we wish to mention that the notion of a perfect number has been extended to Gaussian integers (see R. Spira [292], W. L. McDaniel [77], M. Hausman [141], D. S. Mitrinovi´c – J. S´andor [218]) to real quadratic fields (see E. Bedocchi [15]), or generally to unique factorization domains (see W. L. McDaniel [78]). A number n is called multiperfect, if there exist a positive integer k ≥ 1 such that σ (n) = kn (1) In this case n is called also as k-perfect number. The union of all k-perfect numbers when k ∈ N is the set of multiply perfect numbers. Thus n is called multiply perfect when n|σ (n) (2) In the sections with perfect numbers (when k = 2 in (1)) we sometimes mentioned that some results which were true for perfect numbers do hold also for multiperfect numbers. (For the history of multiply perfect and multiperfect numbers up to 1907, see Dickson [84].) For example, the famous Dickson theorem 4.(50) holds true also for k-perfect number, as proved by H.-J. Kanold [168]. A positive integer n is called primitive if cannot be written in the form m = st, where s is an even perfect number and (s, t) = 1. Kanold proved that for any k there are only a finitely many primitive k-perfect numbers with a fixed number of distinct prime factors. (3) C. Pomerance [239] proved that for every k ≥ 1 (even rational number) and every non-negative integer K , there is an effectively computable number N (k, K ) such that if ω(n) = K and n is primitive k-perfect, then n ≤ N (k, K ). 32

(4)

PERFECT NUMBERS

Here ω(n) denotes, as usual, the number of distinct prime divisors of n. Let A(n) be the set of all prime divisors of n. In 1998 W. Carlip, E. Jacobson and L. Somer [42] proved the following theorem: For fixed integers k and t, there exist at most finitely many squarefree integers n such that car d A(N ) ≤ t and n is k − perfect.

(5)

P. J. McCarthy [48] has shown that if n is k-perfect, then ω(n) ≥ k 2 − 1

(6)

For some improvements, see W. L. McDaniel [75]. McDaniel shows also that there exists no odd k-perfect numbers with k odd of the form m α , where m is an integer, and α + 1 is a prime, the square of a prime, or the cube of a prime. (7) It is not known if there exist odd k-perfect numbers, but actually up to recently (February 2002) we have a total of 5040 known k-perfect numbers. There are 39 2-perfect (i.e. classical perfect), 6 3-perfect, 36 4-perfect, 65 5-perfect, 245 6-perfect, 516 7-perfect, 1134 8-perfect, 923 10-perfect and 1 11-perfect numbers. For details regarding the discoverers see the site http://www.homes/unibielefeld.de/achim/mpn.html by A. Flammenkamp. P. Hagis Jr. and G. L. Cohen [130] have shown that the largest prime factor of a k-perfect number is at least 100129, (8) while the second largest prime factor is ≥ 1009.

(9)

Later, Hagis Jr. [125] proved that the third largest divisor is ≥ 101

(10)

The density of k-perfect numbers is 0, as first was shown by Kanold, but in 1957 Hornfeck and Wirsing proved that P(x) = o(x ε )

(11)

for any ε > 0, where P(x) denotes the number of k-perfect numbers not exceeding x. In fact in the book by Erd¨os and Graham [92] one can read that Wirsing can show that P(x) < cx c log log x log x/ log log x (12) with c, c > 0 constants. (Here k > 0 can be even rational number.) 1 in the case of an odd perfect number have been Various bounds for S = p p/n included in section 4 (see 4.44). G. L. Cohen [55] in 1980 proved the following: 33

CHAPTER 1

Let n be k-perfect. Then

 if k is odd  log k, log k log k <S< , if k is even  3 log 3/2 3 log 4/3

(13)

for n odd; and

1 log k/2 1 log 2k/3 + <S< + 2 3 log 3/2 2 3 log 4/3 log k , due to A. Krawczyk [183]. if n is even. These improve S > 2 M. Bencze [19] proved that

r ( r 3/2 − 1) < S < r (1 − r 6/k 2 ) for n even;

√ 3r r ( k 2 − 1) < S < r (1 − r 8/(kπ 2 )) for n odd

(14)

(15) (16)

where r = ω(n). A considerable interest has been devoted to triperfect numbers, i.e. to the case k = 3 in (1). R. Steuerwald [297] proved that if 6|n and n is triperfect, then the prime divisors 2 and 3 appear at the first power, the other prime divisor at a higher power. n G. L. Cohen [55] showed that if 2||n and n is triperfect, then is an odd perfect 2 number. If 2a ||n and 3|n, then a ≡ 3 (mod 4) (except if n = 120); if 2a ||n, then a ≡ 5 (mod 6) (except if n = 672). If 3b ||n, then b ≡ 3 (mod 4) and b ≡ 5 (mod 6). An odd triperfect number must be a square. In 1970 McDaniel [75], and independently Cohen [55] (but 10 years later) proved that for odd triperfect numbers n ω(n) ≥ 9

(17)

E. A. Bugulov [39] showed in fact ω(n) ≥ 11;

(18)

this has been rediscovered by M. Kishore [180]. In 1983 H. Reidling [248] showed ω(n) ≥ 12,

(19)

and this has been again rediscovered by Kishore [181]. In 1982 W. E. Beck and R. M. Najar [14] obtained for an odd triperfect number n the lower bound (20) n > 1050 34

PERFECT NUMBERS

This was improved to n > 1060

(21)

n > 1070

(22)

by L. B. Alexander [4], and to by Cohen and Hagis, Jr. [130]. As we mentioned, an odd triperfect number must be a square: D. Iannucci [152] proves that the square root of an odd triperfect number cannot be squarefree number. (23) The important theorem by Touchard (see 4.48) can be extended in some cases also to multiperfect numbers. J. A. Holdener [145] proves that if n is odd k-perfect, where k is not divisible by 3 or 4, then n≡1

(mod 12) or n ≡ 9 (mod 36).

(24)

Now certain facts on multiply perfect numbers. (Remark that many authors confuse the words ”multiperfect” and ”multiply perfect”. Clearly if n is k-perfect, then it is multiply perfect. But the converse is not true.) R. D. Carmichael [44] found all multiply perfect numbers less than 109 . These are: 1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240 (this corrected list appears in [9]). (25) Since, as we mentioned before, there are more than 5040 multiperfect numbers, and these are also multiply perfect numbers, today (in the computer era) we know multiply perfect numbers having about 10000 digits. However, it is not known whether or not there are infinitely many such numbers. See also R. K. Guy [114]. The estimate 4.(60) for the number of perfect numbers ≤ x holds true essentially also for multiply perfect numbers, so for these numbers m(x) one has c log x log log x log x m(x) = O exp (26) log log x due to Hornfeck and Wirsing [148]. We now study some miscellaneous results related to multiply perfect numbers. H. Harboth [140] takes in place of σ (n) the value S(n) = the sum of all possible sums of distinct divisors of n (in fact one obtains S(n) = σ (n) · 2d(n)−1 , where d(n) is the number of divisors of n). She proves that n|S(n) for infinitely many n

(27)

thus the open question above is solved in the case considered here. In a difficult proposed problem, C. Pomerance [242] considers multiply perfect numbers of the form n!, and shows that this is possible only for n ∈ {1, 3, 5}. (28) 35

CHAPTER 1

L. Cheng [52] has proved that n = 23 · 3 · 5, 25 · 3 · 7 and 25 · 33 · 5 · 7 are the only integers satisfying σ (n) = ω(n) · n. (29) For multiply perfect numbers in Lucas sequences see [194]. Finally, let us stop at certain problems relating complexity theory of algorithms. Gill’s complexity class ([107]) BPP is the class of languages recognized in polynomial time by a probabilistic Turing machine, with two-sided error probability bounded by a constant away from 1/2. Now Theorem 8 of [9] states that the set of multiply perfect numbers is in BPP. (30) Bach, Miller and Shallit prove the similar result for perfect numbers, as well as amicable numbers. (31) For algorithmic number theory, see [10].

1.6

Quasiperfect, almost perfect, and pseudoperfect numbers

A number n is called quasiperfect, if σ (n) = 2n + 1

(1)

No such numbers are known, but necessary conditions have been obtained by various authors. P. Cattaneo [50] proved that if n is quasiperfect and r |σ (n), then r ≡ 1 or 3

(mod 8)

(2)

If p 2α ||n and q|(2α + 1), then (q − 1)( p + 1) ≡ 0 or 4

(mod 16)

(3)

Cattaneo proved also that an odd quasiperfect number must be a perfect square. (4) H. L. Abbott, C. E. Aull, E. Brown and D. Suryanarayana [1] as well as G. L. Cohen [57] deduced other conditions. For example, in [57] it is shown that there is no quasiperfect number of the forms m 4 or m 6 (in this last case one suppose also (m, 3) = 1) (5) It is shown also that there are no quasiperfect numbers of the form 32a m 2b where 3 m, a ≡ 2

(mod 8) and b ≡ 0

If a number of the form

r

6qi +2

pi

(mod 5) or b ≡ 0

(mod 11)

(6)

( pi primes) is quasiperfect, then

i=1

r ≥ 230876 36

(7)

PERFECT NUMBERS

Abbott, Aull, Brown and Suryanarayana [1] showed that a quasiperfect number n must have at least five distinct divisors and n > 1020

(8)

M. Kishore [175] improved this to ω(n) ≥ 6 and n > 1030

(9)

while Cohen and Hagis, Jr. [61] obtained the refinement ω(n) ≥ 7 and n > 1035

(10)

When 3 n, this has been earlier obtained by Jerrard and Temperley [156]. Kanold [172] proved that there are only finitely many quasiperfect numbers n for which ω(n) < K (11) where K is a fixed constant. 1 2M + log . For a positive integer M, let λ(M) = p σ (M) p/M 1 . If n is quasiperfect, then Let S = p p/n (i) if p1 = 5, then S < λ(56 ) (12) (ii) if p1 ≥ 7, then S < log 2 (13) (iii) if p1 = 3, p2 = 5, then S < λ(34 · 56 ) (14) (iv) if p1 = 3, p2 ≥ 7, then S < λ(34 ) (15) Here p1 < p2 are the least two prime divisors of n. These results are due to Cohen [55]. Let n be quasiperfect and let p1 < · · · < pr be its prime divisors. Kishore [178] proves that i−1 (16) pi < 22 (r − i + 1) for 2 ≤ i ≤ 6 A. Makowski [203] has investigated the solutions of the equation σ (n) = 2n + 2 If 2k − 3 = prime, then n = 2k−1 (2k − 3) is a solution. We note that 2k − 3 are primes e.g. for k = 2, 3, 4, 5, 6, 9, 10, 12, 14, 20, etc. A solution of other type is n = 650. We now introduce almost perfect numbers. A number n is called so, if one has σ (n) = 2n − 1 37

(17)

CHAPTER 1

Again, we should note that in the literature ”almost perfect” sometimes means quasiperfect in the sense of (1), or almost perfect, in the sense of (17) (see e.g. [156], [71]). S. Singh [287] calls almost perfect numbers ”slightly defective” while quasiperfect numbers ”slightly excessive”. It is easy to check that powers of 2 satisfy (17); no other almost perfect numbers are known. An infinite class of numbers which are not almost perfect is given by J. T. Cross [71] as follows: Let p denote an odd prime. If 2m+1 > p, then no multiple (18) of 2m p is almost perfect. In 1978 M. Kishore [176] proved that if n is an odd almost perfect number, then ω(n) ≥ 6

(19)

Cross [71] notes that the argument by Jerrard and Temperley [156] can be modified to show that if 3 n, then ω(n) ≥ 7 (20) for such numbers. An analogous result to (16) in case of almost perfect numbers is i−1

pi < 22 (r − i + 1), 2 ≤ i ≤ 5

(21)

p6 < 23775427335(r − 5)

(22)

and due to Kishore [178]. An odd almost perfect number n should be a perfect square, and if p1a1 ||n, where p1 = 3 is the least prime divisor of n, then a1 ≥ 12. (23) For this result, see also [176].

1.7

Superperfect and related numbers

A number n is called superperfect (after D. Suryanarayana [307]) if σ (σ (n)) = 2n

(1)

Suryanarayana and Kanold [171] have determined the general form of even superperfect numbers. These can be written as n = 2k (k ≥ 1), where 2k+1 − 1 is prime

(2)

New proofs of (2) have been obtained by G. Lord [193] and J. S´andor ([271], [273]). The odd superperfect numbers are not known, but they must be perfect squares (3) 38

PERFECT NUMBERS

as proved by Kanold [307]. According to Guy [114] (p. 65), Dandepat et al. proved that if n is odd superperfect, then n or σ (n) is divisible by at least three distinct primes. (4) Suryanarayana [308] shows that odd numbers of this type cannot have the form p 2α ( p prime). (5) Hunsucker and Pomerance [149] give the lower bound 7 · 1024

(6)

for the odd superperfect numbers. D. Bode [26] obtained that the number of even superperfect numbers less than x is o(log x/ log log x)

(7)

By iterating the composition of σ , Bode [26] showed also that there are no even m-superperfect numbers for m ≥ 3, in the sense that σ (m) (n) = 2n

(8)

where σ (m) (n) = σ (. . . σ (n) . . .). m times

For a new proof, see Lord [193]. J. McCranie [70] has shown that there are no m-superperfect numbers less than 4.29 × 109 for any m ≥ 3. S´andor [270] considered the perfect number analogue for the composition of some arithmetic functions. He reproved (2) by showing that σ (σ (n)) ≥ 2n for all even n

(9)

with equality only for n = 2k , with 2k+1 − 1 prime. Let ψ be the Dedekind arithmetic function, defined by 1 ψ(n) = n (n > 1), ψ(1) = 1. 1+ p p/n Then, an improvement of (9) is the inequality σ (σ (n)) ≥ ψ(σ (n)) ≥ 2n for even n,

(10)

where the last equality holds only for n = 2k , with 2k+1 − 1 prime. Superperfect number are called σ ◦ σ -perfect numbers, while numbers with property ψ(σ (n)) = 2n 39

(11)

CHAPTER 1

are called ψ ◦ σ -perfect numbers. All even perfect numbers of this type are determined. Equation (11) has at least an odd solution n = 3. Let n be an odd solution of (11). Then (see [270]) (a) n ≡ −1 (mod 3), (b) n ≡ 7 (mod 12) (c) n ≡ −4 (mod 21), n ≡ 10 (mod 21) (d) 2α · 3β σ (n) for all α, β ≥ 1. (12) Related to σ ◦ ψ-perfect numbers, i.e. satisfying σ (ψ(n)) = 2n

(13)

S´andor proves that the only even solution is n = 2. One has σ (ψ(n)) ≥ 2n

m+1 for n even, m

(14)

with n = 2a , where m denotes the greatest odd divisor of n. If n is an odd σ ◦ ψperfect number, and p is an odd prime having the property σ ( p + 1) > 2 p, then p n (particularly 3, 5, 7, 11, 17, 19 n). (15) It is conjectured that (13) doesn’t have odd solutions. In connection to Bode’s result, S´andor [270] proves that for any m ≥ 3, there are no ψ ◦ ψ ◦ · · · ◦ ψ −perfect numbers (16) m

The problem of ”ψ-superperfect” numbers is completely settled as followed: The only solution of equation ψ(ψ(n)) = 2n

(17)

is n = 3. But there are many ”ψ-triperfect” numbers as the even solutions of ψ(ψ(n)) = 3n

(18)

are n = 2k , where k > 1. Let ϕ be the Euler totient. S´andor [270] proves that there are no ψ ◦ ϕ-perfect numbers which are odd. (19) If n is even, and the greatest odd divisor of n is ψ ◦ ϕ-deficient, then n is also ψ ◦ ϕ-deficient (i.e. (ψ ◦ ϕ)(n) < 2n). Here it is conjectured also that ψ(ϕ(m)) ≥ m for any odd m 40

(20)

PERFECT NUMBERS

Related to ψ-multiperfect numbers the following result is true (S´andor [269]). We say that n is ω-multiple of m (see Ch. Wall [318]) if m|n and the sets of prime factors of m resp. n are identical. Let us now suppose that the least solution of equation ψ(n) = kn (21) is a squarefree number n k . Then all solutions of (21) are the ω-multiples of n. Let us call n quasi-superperfect if σ (σ (n)) = 2n + 1

(22) (23)

Clearly the Mersenne primes are such; it is not known if there are others. Subbarao [301] proves that if a prime p satisfies (23) then p is a Mersenne prime, otherwise (24) σ ( p) = 2r M 2 , where M has at leat two prime divisors. Generally, if n satisfies (23), then n must be odd, and σ (a) must be of the form σ (n) = 2r M 2 , where M is an odd integer.

(25)

Subbarao proves also that there is no odd perfect number of the form n = 1 mσ (m) for which σ (m) is a power of n and with m quasi-superperfect (i.e. m 2 satisfying (23)). (26) He raises the following open problem: if m is quasi-superperfect, is the number 1 (27) n given by n = mσ (m) a perfect number? 2 The almost superperfect numbers are defined by σ (σ (n)) = 2n − 1

(28)

We cannot decide if there exist such numbers, in fact no reference to this question we were able to find in the literature. (The analogous question, however, for unitary divisors, as we will see in a further section, is much easier). Another generalization of superperfect numbers has been considered by G. L. Cohen and H. J. J. te Riele [64]. They call a number n (m, k)-perfect if σ (m) (n) = kn

(29)

The classical perfect numbers are then the (1,2)-perfect, the k-multiperfect numbers are (1, k)-perfect, the superperfect numbers are (2,2)-perfect, the m-superperfect numbers are (m, 2)-perfect. They give a table of all (2, k)-perfect numbers up to 109 . They gave a table for all (3, k)-perfect and (4, k)-perfect numbers up to 2 · 108 in 41

CHAPTER 1

1995 [63]. It is interesting to note that no new (1,3)-perfect numbers have been found in the last 350 years, nor any new (1,4)-perfect numbers in the last 70 years. For (2, k)-perfect numbers they prove the following general result: suppose l is an odd (2, k)-perfect number. For any a such that 2a |kσ (σ (2a )) and such that σ (2a ) and σ (l) are relative prime, the number 2a l is (2, 2−a kσ (σ (2a )))-perfect. (30) For example, for p > 2, 2 p−1 · 3 · 5 · 7 · 132 · 31 is (2,12)-perfect. They conjecture that all numbers n are (m, k)-perfect for some m, k; and verified it for all n ≤ 1000. (31) There are also some theoretical results supporting conjecture (31). Here one has to do with the properties relating to the iteration of σ . P. Erd¨os [90] asked if (σ (m) (n))1/m has a limit as m → ∞. He conjectures that it is infinite for each n > 1. (32) Erd¨os, Granville, Pomerance and Spiro [93] poses five new open problems, as follows: (i) For any n > 1, σ (m+1) (n)/σ (m) (n) → 1 as m → ∞ (33) (ii) For any n > 1, σ (m+1) (n)/σ (m) (n) → ∞ as m → ∞ (34) (iii) For any n > 1, there is m with n|σ (m) (n) (35) (iv) For any n, l > 1 there is m with l|σ (m) (n) (36) (v) For any n 1 , n 2 > 1, there are m 1 , m 2 with σ (m 1 ) (n 1 ) = σ (m 2 ) (n 2 ). (37) Cohen and te Riele [64] give some computational evidence to indicate that statements (32), (34), (35), (36) are true, while (33) and (36) are false. A. Schinzel [281] asks if lim inf σ (m) (n)/n < ∞ n→∞

(38)

for each m, and observe that it follows for m = 2 from a deep theorem of A. R´enyi. A. Makowski and A. Schinzel [207] gave an elementary proof that for m = 2 the limit is 1. The result is also true for m = 3, proved by H. Maier [202]. (39) Makowski [206] showed earlier this result by assuming the infinitude of Mersenne primes. He proves also that (38) is true if one assumes a general hypothesis H due to Schinzel [282]. (40) For the iteration properties of s(n) = σ (n) − n, as well as divisibility and congruence properties of σ (m) (n), see the surveys in Mitrinovi´c-S´andor [218] (pp. 92-94). We will return later on the iteration of s(n) when studying sociable numbers.

1.8

Pseudoperfect, weird and harmonic numbers

W. Sierpinski [286] called a number pseudoperfect if it was the sum of the some its distinct divisors. E.g. 20 = 1+4+5+10 is pseudoperfect but 8 not. Some authors use the ”semiperfect” terminology (see e.g. A. and E. Zachariou [330]). 42

PERFECT NUMBERS

Numbers n with the property σ (n) > 2n are called abundant (for their properties, see Mitrinovi´c-S´andor [218], and for a recent survey, see S´andor [274]. In [218] and [274] one can find results also on deficient numbers, defined by σ (n) < 2n). A number is called primitive abundant if it is abundant, but all its proper divisors are deficient; and primitive pseudoperfect if it is pseudoperfect, but none of its proper divisors are. C. Pomerance [236] called n a harmonic number, if the harmonic mean of all the divisors of n is an integer (equivalently, σ (n) divides nd(n); where d(n) is the number of divisors of n). These numbers were considered first by O. Ore [228], and some author call them ”Ore numbers”. Finally, S. J. Benkovski [21] called a number weird, if it is abundant but not pseudoperfect. Now some results on these numbers. Erd¨os (see [114], p. 46) has shown that the density of pseudoperfect numbers exists, (1) and says that probably there are integers n which are not pseudoperfect, but for which n = ab, with a abundant and b having many prime factors. (2) H. Abbott (see [114], p. 46) lets l = l(n) (where n ≥ 3) be the least integer for which there are n integers 1 ≤ a1 < a2 < · · · < an = l such that each ai divides s= ai (so that s is pseudoperfect). Then

for all n ≥ 3; and

l(n) > n c1 log log n (c1 > 0 constant),

(3)

l(n) < n c2 log log n (c2 > 0, constant),

(4)

for infinitely many n. A. and E. Zachariou [330] note that every multiple of a pseudoperfect number is pseudoperfect. (5) All numbers 2m p with m ≥ 1 and p a prime between 2m and 2m+1 are primitive pseudoperfect, but there are such numbers not of this form, e.g. 770. (6) There are infinitely many primitive pseudoperfect numbers that are not harmonic numbers. The smallest odd primitive pseudoperfect number is 945. (7) P. Erd¨os (see [114], p. 47) can show that the number of odd primitive pseudoperfect numbers is infinite. (8) Benkovski and Erd¨os [22] showed that the density of weird numbers is positive. (9) Some very large weird numbers were found by S. Kravitz [182]. There are 24 primitive weird numbers less than a million (70, 836, 4030, 5830,...). Nonprimitive weird numbers include numbers of form 70 p, where p > 144 is a prime, or 836 p with p = 421, 487, 491 or ≥ 577; also 7192 · 31. (10) Benkovski and Erd¨os [22] posed some open problems on weird numbers: are there infinitely many primitive abundant numbers which are weird? (11) 43

CHAPTER 1

Is every odd abundant number pseudoperfect (thus not weird)? (12) Can σ (n) be arbitrarily large for weird n? (13) They conjecture ”no”. For primitive weird numbers, see also [230]. Clearly, a pseudoperfect number is a perfect one, too. The similar statement for harmonic numbers is due to Ore [228]: n perfect ⇒ n harmonic. (14) Ore proved that a harmonic number has at least two distinct prime factors, (15) and Pomerance [236] (see also D. Callan [41]) proved that the only harmonic numbers with two distinct prime factors are the even perfect numbers. (16) M. Garcia [105] extended the list of harmonic numbers to include all 45 which are < 107 , and he found more than 200 larger ones. The least one, apart from 1 and the perfect numbers, is 140. (17) 9 All 130 harmonic numbers up to 2 · 10 are listed in G. L. Cohen [59], (18) and R. M. Sorli (see Cohen and Sorli [66]) has continued the list to 1010 . (19) Kanold [170] has shown that the density of harmonic numbers is 0. (20) Ore [228] conjectured that every harmonic number is even, but this probably is very difficult. Indeed, this result, if true, would imply that there are no odd perfect numbers. See also W. H. Mills [215], who proved that if there exists an odd harmonic number n, then n has a prime-power factor greater than 107 . Related to Pomerance’s result (16) one can ask: if a harmonic number has three distinct divisors, is it even? (21) In 1998 G. L. Cohen and D. Moujie [62] have introduced a generalization of harmonic numbers. A number n > 1 is called k-harmonic if Hk =

n k d(n) σk (n)

is an integer. Here k ≥ 1 denotes a positive integer, and σk (n) =

(22)

d k denotes

d/n

the sum of kth powers of divisors of n. The number n is called power-harmonic if it is k-harmonic for some k ≥ 1. A power-harmonic number is proper, it if is kharmonic for some k ≥ 2. No k-harmonic number with k ≥ 2 is known. However, Cohen and Moujie are able to prove some necessary conditions. For example, if n is power-harmonic, then has at least two distinct prime factors. (23) This generalizes Ore’s result (15). If n is k-harmonic, then Hk (n) ≤ d(n) − 1

(24)

For even k-harmonic numbers the following is true: Let n = 2a p b ( p odd prime) be a proper k-harmonic number. If k is even, then b ≡ 7 (mod 8); if k is odd then b is odd and ( p + 1)(b + 1) ≡ 0 (mod 16) (25) 44

PERFECT NUMBERS

If n is a proper k-harmonic number, then d(n) > 1 +

k−1 k ·2 , k+1

(26)

and if n is a proper power-harmonic number, then d(n) ≥ 60

(27)

Also, if n is a proper power-harmonic number, then n > 1010

(28)

Finally, we note that W. Butske et al. [40] have considered primary pseudoperfect number n which is a product of distinct primes and which satisfies 1 1 + = 1. The first few such numbers are 2, 6, 42, 1806, 47058,... n p p/n

1.9

Unitary, bi-unitary, infinitary-perfect and related numbers

n = 1. Let σ ∗ (n) denote A divisor d of n is called a unitary divisor, if d, d the sum of unitary divisors of n. It is known that (see E. Cohen [53], and see also [279] for various divisibility notions, convolutions and arithmetic functions) σ ∗ is multiplicative and σ ∗ ( p α ) = p α + 1 for any prime power α. Also σ ∗ (1) = 1. In 1966 M. V. Subbarao and L. J. Warren [303] introduced the unitary perfect numbers n satisfying σ ∗ (n) = 2n (1) They found the first four unitary perfect numbers, namely 6, 60, 90, 87360,

(2)

and in 1975 Ch. Wall [319], [321] discovered the fifth, namely 218 · 3 · 54 · 7 · 11 · 13 · 19 · 37 · 79 · 109 · 157 · 313

(3)

Subbarao and Warren proved that there are no odd solutions of (1). This was shown independently also by K. Nageswara Rao [224]. No other unitary perfect numbers than the five given by (2) and (3) are known, but if n = 2a m (m odd ) would be a new one, then Subbarao et al. [302] showed that one must have a > 10 and ω(m) > 6 (4) 45

CHAPTER 1

Later, Wall [323] improved this to ω(m) > 8,

(5)

while in [322] he proved that any new unitary perfect number has a prime power unitary divisor larger than 215 (6) H. A. M. Frei [104] has shown that if n = 2m p1a1 . . . prar is unitary perfect with (n, 3) = 1, then m > 144, r > 144, and n > 10440 (6 ) It is not known if there are infinitely many unitary perfect numbers. Subbarao (see [300]) has conjectured that there are only finitely many. It is also not known whether there exists at least a unitary perfect number not divisible by 3. (7) (see [303]), or a k-unitary perfect number for k ≥ 3, where σ ∗ (n) = kn, (8) see [289]. P. Hagis [124] proved that if (8) holds and n contains t distinct odd divisors, then k = 4 or 6 implies n > 10110 , t ≥ 51 and 249 |n. (9) (10) If k ≥ 8, then n > 10663 and t ≥ 247, while k ≥ 5, odd, imply n > 10461 and t ≥ 166, 2166 |n. (11) a In 1989 S. W. Graham [111] proved the following result: If n = 2 m is a unitary perfect number, where m is an odd squarefree number, then either a = 1 and m = 3, a = 2 and m = 3 · 5, or a = 6 and m = 3 · 5 · 7 · 13 (thus there are only three such numbers). (12) J. De Boer [129] proved that if n = 29 · 32 m is unitary perfect, with (m, 6) = 1 and squarefree, then a = 1 and m = 5. On unitary abundant numbers, see [274]. A divisor d of n is called a bi-unitary divisor if the greatest common unitary n divisor of d and is 1. Let σ ∗∗ (n) be the sum of bi-unitary divisors of n. Wall [320] d call n bi-unitary perfect, if σ ∗∗ (n) = 2n (13) It is not difficult to verify that σ ∗∗ is multiplicative and σ ∗∗ ( p α ) = σ ( p α ) = and σ ∗∗ ( p α ) =

p α+1 − 1 if α is odd, p−1

p α+1 − 1 − p α/2 , if α is even p−1 46

(14)

PERFECT NUMBERS

Hence

σ ∗∗ (n) ≤ σ (n)

(15)

with equality only if every prime which divides n does so an odd number of times. Wall proves that there are only three bi-unitary perfect numbers, namely 6, 60, 90. (16) Hagis [126] introduced bi-unitary multiply perfect numbers by n|σ ∗∗ (n)

(13 )

and proved that there are no odd numbers of this type. There are 17 numbers n < 107 satisfying (13’) (three of them are the numbers from (16)). W. E. Beck and R. M. Najar [13] essentially have determined all unitary quasiperfect numbers satisfying σ ∗ (n) = 2n + 1

(17)

as well as all unitary almost perfect numbers such that σ ∗ (n) = 2n − 1 They proved that there are no unitary quasiperfect numbers, and that n = 1 and n = 2 are the all possible unitary almost perfect numbers.

(18) (19)

(20) A. Bege [16] has proved that there are no bi-unitary quasiperfect numbers, i.e. numbers n satisfying σ ∗∗ (n) = 2n + 1. J. S´andor [270] (see pp. 11-12) showed there are no unitary quasi superperfect numbers n satisfying σ ∗ (σ ∗ (n)) = 2n + 1 (21) and that the only unitary almost superperfect number n, i.e. such that σ ∗ (σ ∗ (n)) = 2n − 1

(22)

are n = 1 and n = 3. Sitaramaiah and Subbarao [290] rediscovered this result. They study more profoundly the unitary superperfect numbers n with σ ∗ (σ ∗ (n)) = 2n

(23)

The first ten such numbers are 2, 9, 165, 238, 1640, 4320, 10250, 10824, 13500 and 23760. There are not known odd unitary superperfect numbers other than 9 and 165, and Sitaramaiah and Subbarao conjecture that the set of such numbers is finite. (24) 47

CHAPTER 1

In fact 165 is the single such number, having three distinct prime factors. (25) 238 is the only even unitary superperfect number such that n has three prime factors, while σ ∗ (n) two. (26) ∗ If n is an odd unitary superperfect number not divisible by 3, then 3|σ (n); in such a case n must be squarefree, ω(n) even, and ω(n) ≥ 52. (27) We quote one more result: All n ≤ 108 such that n|σ ∗ (σ ∗ (n)) (28) (which could be called as ”unitary multiply superperfect numbers”) are determined; there are 29 such numbers. S. Ligh and Ch. R. Wall [190] have introduced ”nonunitary divisors” and nonunitary perfect numbers. d is a nonunitary divisor of n, if n = cd, with (c, d) > 1. Let σ # (n) be the sum of nonunitary divisors of n. It is immediate that σ # (n) = σ (n) − σ ∗ (n),

(29)

and that σ # is not multiplicative. A number n is called nonunitary perfect, if σ # (n) = n

(30)

All known nonunitary perfect numbers are even. P. Hagis, Jr. [129] prove that σ # (n) is odd iff n = 2α M 2 , where M > 1 is odd and α ≥ 0. (31) As a corollary we get that an odd nonunitary perfect number must be a perfect square. (32) Another result by Hagis says that an odd nonunitary perfect number n must have at least four distinct prime factors, and n > 1015 . If 3 n, then ω(n) ≥ 7. (33) 2 4 4 If 2 · 5 · 7|n, then for such numbers one has also 3 ||n and 5 · 7 |n. Also p n if 11 ≤ p ≤ 271. (34) G. L. Cohen [58] calls a unitary divisor d of n as 1-ary divisor, and he calls d a k-ary divisor of n (for k > 1), and writes d|k n, if the greatest (k − 1)-ary divisor of d n n = 1 for 1-ary divisors d. He also calls p x an infinitary and is 1. Note that d, d d (35) divisor of p y (y > 0, p prime) if p x | y−1 p y , and writes p x |∞ p y . r y p j j , for distinct primes p1 , . . . , pr , and Let d be a divisor of n and write n = d=

r

j=1 x

p j j (where 0 ≤ x j ≤ y j , j = 1, r ). Then d is an infinitary divisor of n if

j=1 x

y

p j j |∞ p j j for each k = 1, r . 48

(36)

PERFECT NUMBERS

It is interesting to note that, by writing the binary representations x = xj2j and y − x = z j 2 j (where the sums are finite, j ≥ 0, each x j and z j is 0 or 1, and trailing zeros are allowed where required), then xjzj = 0 (37) p x |∞ p y iff Another interesting characterization (see [58]) is the following: y x y is odd, (38) p |∞ p iff x y = C yx denotes a binomial coefficient. where x Let σ∞ (n) be the sum of all infinitary divisors of n. Then σ∞ is a multiplicative function, and j σ∞ ( p α ) = (1 + p 2 ), where y = yj2j (39) y j =1

The number n is called infinitary perfect, resp. infinitary k-perfect if σ∞ (n) = 2n,

(40)

σ∞ (n) = kn (k ≥ 2)

(41)

resp. There are no odd infinitary perfect or k-perfect numbers, and Cohen conjectures that there are no infinitary perfect or k-perfect numbers not divisible by 3. (42) Cohen gives a number of 14 infinitary perfect, 13 infinitary 3-perfect, 7 infinitary 4-perfect, 2 infinitary 5-perfect numbers. Pedersen [231] found 139 new infinitary (43) perfect numbers. Cohen proves also that the only infinitary perfect numbers not divisible by 8 are 6, 60 and 90. (44) We note that D. Suryanarayana [306] and K. Alladi [6] have given different generalizations from above for unitary divisors, thus obtaining other notions of k-ary divisors. (45) ∗ ∗ A number n is called unitary harmonic if σ (n)|nd (n) (46) i.e. if the unitary harmonic mean H ∗ (n) = nd ∗ (n)/σ ∗ (n) is an integer. The main properties of unitary harmonic numbers were studied by Hagis and Lord [135], but it was earlier introduced also by N. Nageswara Rao [224] who showed that if n is unitary perfect, then it is also unitary harmonic. (47) Ch. R. Wall [Unitary harmonic numbers, Fib. Quart. 21(1983), 18-25] has proved that there are 23 unitary harmonic numbers n with ω(n) ≤ 4, and 43 unitary harmonic numbers n ≤ 106 . 49

CHAPTER 1

Let H ∗ (x) be the counting function of unitary harmonic numbers. Then for ε > 0 and large x one has (48) H ∗ (x) < 2.2x 1/2 · 2(1+ε) log x/ log log x a result due to Hagis and Lord [135]. Hagis and Cohen [131] study infinitary harmonic numbers, i.e. those numbers n for which (49) σ∞ (n)|nd∞ (n) where d∞ (n) and σ∞ (n) denote the number, resp., sum of infinitary divisors of n. Let H∞ (n) = nd∞ (n)/σ∞ (n). They show that if n is infinitary perfect, then it is also infinitary harmonic. (50) If Sc is the set of all positive integers such that H∞ (n) = c, then Sc is finite (or empty) for every real number c. (51) For the counting function H∞ (x) of infinitary harmonic function exactly the same estimate as (48) holds: H∞ (x) < 2.2x 1/2 · 2(1+ε) log x log log x

1.10

(52)

Hyperperfect, exponentially perfect, integer-perfect and γ -perfect numbers

In 1975 D. Minoli and R. Bear [217] have generalized the concept of perfect number, by introducing the hyperperfect numbers. An integer n > 1 is k-hyperperfect if n = 1 + k[σ (n) − n − 1] (1) When k = 1, this gives the perfect numbers. It easily follows that for k ≥ 2 all k-hyperperfect numbers are odd. For example 21, 2133, 19521 are 2-hyperperfect, 325 is 3-hyperperfect, 301 and 16513 are 6hyperperfect, 697 is 12-hyperperfect (see [217]). (2) Minoli [216] gave a table for all hyperperfect numbers less than 1500000. All these numbers have two distinct prime factors. But hyperperfect numbers can have more than two prime factors, for example (see te Riele [254]) 13·269·449 = 1570153 is a 5-hyperperfect number, having three distinct prime factors. (3) Minoli and Bear prove that a hyperperfect number cannot be a power of a prime. (4) If n = pq with p, q primes is k-hyperperfect, then k < p < 2k < q. (5) m m+1 m+1 If n = 3 (3 − 2), where 3 − 2 is a prime, then n is 2-hyperperfect. (6) 50

PERFECT NUMBERS

Recently, J. C. M. Nash [226] extended this to any prime p in place of 3, as follows: If n = p m ( p m+1 − ( p − 1)) and p m+1 − ( p − 1) is prime, then n is ( p − 1)hyperperfect. (7) Minoli [216] proves that if n = p1α1 p22 is k-hyperperfect, then k + 2 ≤ p1 ≤ (8) (k + 1)2 ( p1 , p2 distinct primes); and if n = p1α1 p2α2 is k-hyperperfect, then α2 ≤

log[k( p1α1 + · · · + 1) + ( p2 − 1)(k − 1)] log p2

(9)

Finally, a conjecture by Minoli and Bear [217] states that there are k-hyperperfect numbers for every k. (10) Another conjecture by them affirms that the converse of theorem (6) is true, namely if n = 3m (3m+1 − 2) is 2-hyperperfect, then 3m+1 − 2 is prime. (11) 3k + 1 , q = 3k + 4 are prime, then it If k > 1 is an odd integer, and p = 2 is immediate that p 2 q is k-hyperperfect. J. S. McCranie [69] conjectures that all khyperperfect numbers for odd k > 1 are of this form. McCranie has tabulated all hyperperfect numbers less than 1011 . For unitary hyperperfect numbers, see [122]. Here one replaces in (1) σ (n) with σ ∗ (n). It is easy to see that if n is unitary hyperperfect, and 3 n, then n ≡ 1 (mod 3). Also, if 3 nk, then n has an odd number of distinct prime factors. n cannot be a prime power, and if n has the form n = pa q b , then pa − k = 1, 2, 5, 10, 13, 17, 25, 26, 29, 34, . . . There are in total 36 unitary hyperfect numbers n ≤ 106 . Four of these numbers are unitary perfect, found by Subbarao and Warren, twenty are hyperperfect numbers given by Minoli. From the twelve new ones we quote n = 23 · 5 (k = 3), 22 · 13 (k = 3), 25 · 32 (k = 7), 32 · 73 (k = 8), 27 · 17 (k = 15), etc. D. A. Buell [38] computed all unitary hyperperfect numbers < 108 . By developing a search procedure different from that of Buell, P. Hagis [127] found all such numbers < 109 . If n = p1α1 . . . prαr , then E. G. Straus and M. V. Subbarao [299] call d an exponential divisor if d|n and d = p1b1 . . . prbr where bi |qi (i = 1, r ).

(12)

Let σe (n) be the sum of exponential divisors of n. For various arithmetic functions and convolutions on exponential divisors, see S´andor-Bege [279]. Straus and Subbarao call n exponentially perfect (or shortly eperfect) if σe (n) = 2n (13) 51

CHAPTER 1

Some examples of e-perfect numbers are 22 · 32 , 22 · 33 · 52 , 23 · 32 · 52 , 24 · 32 · 112 , 24 · 33 · 52 · 112 , 26 · 32 · 72 · 132 , 27 · 32 · 52 · 72 · 132 , 28 · 32 · 52 · 72 · 1392 , 219 · 32 · 52 · 72 · 112 · 132 · 192 · 372 · 792 · 1092 · 1572 · 3132 . If m is squarefree, then σe (m) = m, so if n is e-perfect and m = squarefree with (m, n) = 1, then m · n is e-perfect. (14) So it suffices to consider only powerful (i.e. no prime occurs to the first power) e-perfect numbers. Straus and Subbarao proved that there are no odd e-perfect numbers, (15) and that for each r the number of e-perfect numbers with r prime factors is finite. (16) Is there an e-perfect number which is not divisible by 3? (17) Straus and Subbarao conjecture that there is only a finite number of e-perfect numbers not divisible by any given prime p. (18) J. Fabrykowski and Subbarao [100] proved that any e-perfect number not divisible by 3 must be divisible by 2117 , greater than 10664 , and having at least 118 distinct prime factors. (19) In 1988 P. Hagis, Jr. [128] showed that the density of e-perfect numbers is positive (namely, is 0.0087). (20) The number n is called e-multiperfect (see [299] if σe (n) = kn

(21)

for some k > 2. Results (15) and (16) are true also for such numbers. W. Aiello, G. E. Hardy and Subbarao [2] proved that if n is e-multiperfect, then n > 2 · 107 for k = 3;

(22)

n > 1085 for k = 4;

(23)

n > 10320 for k = 5;

(24)

n > 101210 for k = 6.

(25)

For the number of powerful solutions n ≤ x of (21), L. Lucht [201] proved that this is ≤ exp(c log x/ log log x) (26) for x ≥ 3 (even, for rational k > 1), with c a constant not depending on k. A number of interesting properties are proved in the paper by J. Hanumanthachari, V. V. Subrahmanya Sastri and V. Srinivasan [139]: If m is squarefree, then m 2 is e-perfect iff m = 6. (27) The only e-perfect number having two distinct prime factors is 36. (28) 52

PERFECT NUMBERS

b cj q j , where If n is an e-perfect number not multiple of 3, and n = pi i 2 3 p ≡ 1 (mod 3), q j ≡ 2 (mod 3), ( pi , q j primes) with bi = Pi Q i Ri (Pi , Q i being squarefree, (Pi , Q i ) = 1) and e j = L j M 2j N 3j (L j , M j being squarefree and (L j , M j ) = 1), then each Q i = 1, each L j is odd, and no M j is greater than 2. (28 ) c Further, if f = the number of q j j for which M j = 1 and ω(L j ) is even; g = the c number of q j j for which M j = 2 and ω(L j ) is odd, and h = the number of pibi for which ω(Pi ) is odd, then f + g + h is odd or even, according as n ≡ 1 or 2

(mod 3)

(29) c pibi qj j For numbers divisible by 3 the following is proved: if n = 3a (a > 1, bi > 1, c j > 1), with pi , q j primes, pi ≡ 1 (mod 3), q j ≡ 2 (mod 3) is e-perfect, then either (i) some bi has in its factorization some prime highest to the index congruent to 2 (mod 3) or (ii) some c j has 2 in its factorization highest to the index congruent to 1 (mod 3) or some c j has an odd prime occurring in its factorization to the highest index congruent to 2 (mod 3). Hanumanthachari, Subrahmanya Sastri and Srinivasan study also e-superperfect numbers given by σe (σe (n)) = 2n (30)

(31) If n = pq with p, q primes, is a solution of (30), then n = 32 . If p is a prime, n squarefree and ( p, n) = 1, ( p + 1, n) > 1, then p 2 n is esuperperfect only if p = 2. (31 ) Generally, if (σe (m), n) = 1 and n is squarefree, then mn is e-superperfect iff m is superperfect, too. (32) Related to the iteration of σe (32 ), they remark that σe(4+r ) (32 ) = 2 · 3 · 5 for any r ≥ 0 and conjecture that σe(k) (22 · 32 ) is never squarefree.

(33)

Another conjecture states that for every prime p there is a least positive integer k = k( p) such that σe(k) ( p 2 ) is squarefree. (34) In this section we shall deal also with integer-perfect numbers, which are numbers n such that n= α(d)d (35) (n) where indicates that the sum is over the proper divisors d of n and α(d) ∈ (n) {1, −1} for each d. This notion is due to L. V. Marijo [209]. 53

CHAPTER 1

Let ν(n) be the number of such representation of n, and call an integer-perfect number minimal if it has no proper divisor which is integer perfect. Marijo proves the following results: No integer of the form 2a b (b odd) is integer-perfect if b is a perfect square, but otherwise all integers of this form are interger-perfect for sufficiently large a. (36) If n is an (ordinary) even perfect number, then ν(n) = ν(2n) = 1.

(37)

For any positive integer N , there exists a minimal integer-perfect number g(N ) such that ν(g(N )) ≥ N . (38) If p is a prime not dividing n, then for any positive integer a, ν( pa n) ≥ 2a (ν(n))a+1 , with equality whenever p > σ (n)

(39)

The following asymptotic results appears in Marijo [210]: car d{m ≤ x : ν(m) ≥ m} = O(x 3/4 ).

(40)

ν(n) can be arbitrarily large. (41) nk Finally, we define γ -perfect numbers. M. D. Miller [214] defines a function γ : N → N recursively by γ (n) = γ (d) (42) For each k ∈ N∗ ,

d/n,d
and γ (1) = 1. Then it is immediate that γ ( p) = 1 for a prime p, γ (4) = 2, γ (12) = 8, and generally γ ( p n ) = 2n−1 ( p prime, n ≥ 1). If n = p1α1 p2α2 . . . prαr , then it can be proved that γ (n) = 2α1 P(α1 ) (43) where P(α1 ) is a polynomial of degree α2 + · · · + αr in α1 . An integer n > 1 will be called γ -perfect if γ (n) = n

(44)

Miller (who use another terminology) proves that there are no odd γ -perfect numbers. (45) n n 2 n 3 p q is γ -perfect iff p = 2 and n + 2 = 2q, while p q or p q cannot be γ -perfect. A number of the form p n qr is γ -perfect iff p = 2, and 2qr = n 2 + 6n + 6. 54

(46)

PERFECT NUMBERS

These give example of such perfect numbers, e.g. 48, 1280, 2496, 28672, 454656, 2342912,... More generally, if f is an arithmetic function, then n is called f -perfect by J. L. Pe [On a generalization of perfect numbers, J. Recr. Math. (to appear)] if f (d) n= d|n,d
Recently, Ch. Ashbacher [On numbers that are pseudo Smarandache and Smarandache Perfect, Smarandache Notion J. 14(2004), 40-42] has raised the problem of m(m + 1) Z -perfect and S-perfect numbers (where Z (n) = min m ∈ N : n| and 2 S(n) = min{m ∈ N : n|m!}). For n ≤ 106 , the only Z -perfect numbers are n = 4, 6, 471544; and the only S-perfect number is n = 12.

1.11

Multiplicatively perfect numbers

In 2001, S´andor [275] has considered the multiplicatively perfect, superperfect, multiperfect etc. numbers. Let T (n) denote the product of all divisors of n > 1. Then n is multiplicatively perfect (in short: m-perfect), if T (n) = n 2

(1)

T (T (n)) = n 2

(2)

T (n) = n k

(3)

m-superperfect, if m-multiperfect, if for some k ≥ 2. He considered also the generalization of (2) to T (T (n)) = n k

(4)

(i.e. m-multisuperperfect numbers); as well as T (k) (n) = n k

(5)

The following results are proved: All m-perfect numbers (i.e. satisfying equation (1)) have one of the following forms: n = p1 p2 or n = p13 , where p1 , p2 are distinct primes. (6) (The author discovered later that this result appears also in K. Ireland and M. Rosen [154]. However, the first source is E. Lionnet [192] who defined ”perfect number of 55

CHAPTER 1

the second kind” as number equal to the product of its aliquot parts. See also [84], p. 58.) There are no m-superperfect numbers. (7) n = 6 is the only perfect number, which is multiplicatively perfect, too. (8) The general form of m-multiperfect numbers can be determined too, e.g.: n = p1 p22 , or n = p15 for k = 3;

(9)

n = p1 p23 or n = p1 p2 p3 or n = p17 for k = 4;

(10)

n = p1 p24 or n = p1 p24 or n = p19 for k = 5;

(11)

n=

p1 p2 p32

or n =

p1 p25 ,

n=

p111

for k = 6;

(12)

n = p1 p26 or n = p113 for k = 7;

(13)

n = p1 p2 p3 p4 or n = p1 p2 p33 or n = p13 p23 or n = p115 for k = 8;

(14)

n = p1 p22 p32 , n = p1 p28 , n = p117 for k = 9;

(15)

n = p1 p2 p34 , n = p1 p29 , n = p110 for k = 10.

(16)

As corollaries, one can deduce the following assertions: n = 28 is the single perfect number which is m-triperfect, too. (17) There are no perfect and 4-m-perfect numbers; (18) n = 496 is the only perfect number which is 5-m-perfect. (19) Related to equation (4) one can say, that it has at most a finite number of solution. (20) Particularly, it is not solvable for k = 4, 5, 6. (21) For k = 3, the general solutions are n = 812 , for k = 7, n = p13 , and for k = 9, (22) n = p1 p2 ( p1 , p2 distinct primes). Equation (5) has no solutions n > 1. (23) The function T (n) has some interesting properties: For example lim inf n→∞

log T (n) log T (n) = 1, lim sup = +∞; log n n→∞ log n

(24)

the normal order of magnitude of log log T (n) is (1 + log 2) log log n − log 2

(25)

F. Luca [200] proves that T (σ (n)) > T (n) for almost all n, and that T (n) = T (σ (n)) is impossible for any n > 1. He shows also that both of the relations T (n)|T (σ (n)) and T (σ (n))|T (n) have infinitely many solutions. 56

PERFECT NUMBERS

ˇ at and J. Tomanov´a [267] have shown that T. Sal´ log log T (n) → 1 + log 2 as n → ∞ log log n on a sequence of density 1. A similar result holds true for T (n)/n. Recently Z. Weiyi [On the divisor product sequences, Smarandache Notions J. 14(2004), 144-146] has obtained the following results: 1 1 , = log log x + C1 + O log x n≤x T (n) n≤x

1 = π(x) + (log log x)2 + B log log x + C2 + O T (n)

log log x log x

where x ≥ 1, C1 , B, C2 are constants, and T (n) = T (n)/n (π(x) is the number of primes ≤ x). A. Bege [17] introduced unitary m-perfect numbers n > 1 as solutions of T ∗ (n) = n 2

(26)

where T ∗ (n) is the product of all unitary divisors of n. Similar concepts as above for unitary m-superperfect and unitary multiperfect are also introduced and studied. All unitary m-perfect numbers have two distinct prime factors; (27) there are no unitary m-superperfect numbers. (28) If T ∗(k) (n) = n l (29) (where T ∗(k) denotes the k-times iteraton of T ∗ ) then n > 1 is called m-unitary (k, l)-perfect number. If l = 2mk , then there are no m-unitary (k, l)-perfect numbers, (30) while if l = 2mk , then every n with ω(n) = m + 1 is an m-unitary (k, l)-perfect number. (31) In a forthcoming paper, A. Bege [18] considered the similar problems for biunitary divisors. n > 1 is m-bi-unitary perfect, if T ∗∗ (n) = n 2 ,

(32)

where T ∗∗ (n) is the product of bi-unitary divisors of n etc. All m-bi-unitary perfect numbers have one of the following forms: n = p14 , n = p13 , n = p12 p22 , n = p12 p2 , n = p1 p2 , where p1 = p2 are primes. (33) 57

CHAPTER 1

All m-bi-unitary superperfect numbers have one of the following forms: n = p, n = p 2 ( p prime). (34) Let k ≥ 2. All (l, k) m-bi-unitary perfect numbers have the form k n= pi i piki −1 (35) ki =2k

where k =

l

ki =2k+1

ki and pi (i = 1, l) are distinct primes.

i=1

In a recent paper, J. S´andor [276] has introduced the multiplicatively e-perfect numbers. Let Te (n) denote the product of all e-divisors of n > 1. Then n is called m − e-perfect if Te (n) = n 2 (36) He proved the following theorem: n > 1 is m − e-perfect iff n = p k , where p is an arbitrary prime, and k is an arbitrary (ordinary) perfect number (i.e. σ (k) = 2k). (37) 6 28 496 For example, n = p , p , p , . . . are all m − e-perfect numbers. Analogously, the m − e-superperfect numbers with the property Te (Te (n)) = n 2

(38)

have the form n = p s , where s is an ordinary superperfect number (i.e. σ (σ (s)) = 2s). (39) According to the open problems relating to the ordinary perfect, or superperfect numbers, the complete determination of m − e-perfect numbers is also left to the future.

1.12

Practical numbers

A positive integer m is said to be practical (see A. K. Srinivasan [293]) if every integer n, with 1 ≤ n ≤ σ (m) is a sum of distinct divisors of m. (1) B. M. Stewart [298] showed that m = p1α1 . . . prαr with p1 < p2 < · · · < pr primes and integers αi ≥ 1 (i = 1, r ) is practical iff either m = 1 or p1 = 2 and for αi−1 every i = 2, . . . , r , pi ≤ σ ( p1α1 p2α2 . . . pi−1 ) + 1. (2) In 1950 P. Erd¨os [87] announced that practical numbers have asymptotic density. (3) Let 1 = d1 < d2 < · · · < dr = n be the divisors of n and for k ≤ r put tk = d1 + · · · + dk with t0 = 0. Then n is practical iff dk+1 ≤ tk + 1 for all k = 0, r − 1. 58

(4)

PERFECT NUMBERS

As a corollary, if n is practical, then σ (n) ≥ 2n − 1.

(5)

Results (4) and (5) are due to D. F. Robinson [261]. Let P(x) be the cardinality of practical numbers n ≤ x. Then 1 P(x) = O(x/(log x)β ) for every β < (1/ log 2 − 1)2 = 0.0979 . . . 2 and

1 P(x) ≥ Ax exp (log log x)2 + 3 log log x 2 log 2

(6)

(7)

for sufficiently large x (where A = 25/2 /5). Here (6) is due to M. Hausman and H. N. Shapiro [142], and (7) to M. Margenstern [208]. Margenstern proved also that if n is practical, distinct from a power of 2, then σ (n) ≥ 2n,

(8)

which is related to (5). As a corollary of the proof of (8) one gets: all even perfect numbers are practical (see [209]). (9) Hausman and Shapiro proved also the fact that for all x ≥ 1/3, the interval √ (x, x + 3 x) contains a practical number. (10) Margenstern proved also that lim inf σ (m)/m = 2, lim sup σ (m)/m = ∞, m→∞

n→∞

(11)

where m is practical, distinct from powers of 2. He conjectured that every even positive integer is a sum of two practical numbers, (12) and that there exist infinitely many practical numbers m such that m − 2 and m + 2 are also practical. (13) Conjectures (12) and (13) have been proved affirmatively by G. Melfi [213] (who used also a table by Margenstern for all even number < 100000 for (12)). Finally we state another conjecture (which is still open) by Margenstern, namely that x P(x) ∼ λ (x → ∞) (14) log x where λ ≈ 1.341. E. Saias [266] proved that x x c1 · ≤ P(x) ≤ c2 · for x ≥ 2 (15) log x log x for all x ≥ 2; while G. Melfi [212] showed that there exist infinitely many practical numbers in the sequences of Fibonacci, Pell, and Lucas. (16) 59

CHAPTER 1

1.13

Amicable numbers

Two numbers a and b are called amicable if σ (a) = σ (b) = a + b Clearly for a = b we reobtain the perfect numbers, so one may assume in what follows that a = b. According to Iamblicus (about 283-330), ”certain men steeped in mistaken opinion thought that the perfect numbers was called love by Pythagoreans on account of the union of different elements and affinity which exists in it; for they call certain other numbers, on the contrary, amicable numbers, adopting virtues and social qualities to numbers, as 284 and 220, for the parts of each have the power to generate the other, according to the rule of friendship, as Pythagoras affirmed. When asked what is a friend, he replied ”another I”, which is shown in these numbers. Aristotle so defined a friend in his Ethics.” Among Jacob’s presents to Esau were 200 she-goats and 20 he-goats, 200 ewes and 20 rams (Genesis, XXXII, 14). Abraham Azulai (1570-1643), in commenting on this passage from the Bible, remarked that he had found written in the name of Rau Nachson (ninth centure A.D.): ”Our ancestor Jacob prepared his present in a wise way. This number 220 (of goats) is a hidden secret, being one of a pair of numbers such that the parts of it are equal to the other one 284, and conversely. And Jacob had this in mind; this has been tried by the ancients in securing the love of kings and dignatories.” The Arab El Madschrˆıtˆı (1007) of Madrid related that he had himself put to the test the erotic effect of ”giving any one the smaller number 220 to eat, and himself eating the larger number 284”. Ben Kalonymos discussed amicable numbers in 1320 in a work written for Robert of Anjou. A knowledge of amicable numbers was considered necessary by Jochanan Allemanno (fifteenth century) to determine whether an aspect of the planets was friendly or not. In 1634 Mersenne remarked that ”220 and 284 can signify the perfect friendship of two persons since the sum of the aliquot parts of 220 is 284 and conversely, as is these two numbers were only the same thing”. For the early history of amicable numbers (up to 1919), we cite L. E. Dickson [84]. E. B. Escott [95] made a review of the subject up to 1941, while that of E. J. Lee and J. S. Madachy [187] to 1972. More recent surveys can be found in Guy [114], Mitrinovi´c-S´andor [218], Eric Weisstein [325]. Here we will study the subject in a such a manner which shows that certain forgotten or new aspects could be important for the further considerations. Only the most important results will be pointed out. 60

PERFECT NUMBERS

The first method described for finding amicable pairs is due to Thabit ibn Qurra (836-901) (see [84]): Let p = 3 · 2n−1 − 1, q = 3 · 2n − 1, r = 9 · 22n−1 − 1 be all primes (where n is a positive integer). Then (1) a = 2n pq and b = 2n r are amicable. For n = 2 one has p = 5, q = 11, r = 71, so one gets the Pythagorean pair. By rediscovering Thabit’s rule, Fermat in 1636 (in a letter for Mersenne) discovered for n = 4 (when p = 23, q = 47, r = 1151) a new pair a = 17296, b = 18416. In 1838 Descartes (again in a letter to Mersenne) found for n = 7 (when p = 191, q = 383, r = 73727) the third pair a = 9363584, b = 9437056. It was discovered later that these pairs were known to Ibn-al-Bann´a (1256-1321) of Marakesh and to K. Farisi of Bagdad (see e.g. W. Borho [29]). Despite later searches to n ≤ 8 by Euler, to n ≤ 15 by Legendre, to n ≤ 34 by Le Lasseur, to n ≤ 200 by G´erardin, to n ≤ 1000 by Riesel (see [187]), no additional Thabitean pair have been discovered. L. Euler (1747) [96] was the first to treat extensively the subject of amicable numbers and to show for the first time their great diversity. To exhaust all amicable numbers of the form (1) Euler generalized Thabit’s formulae as follows: If for g = 2n−β + 1 with some β, 0 < β < n, the numbers r1 = 2β g − 1, r2 = 2n g − 1 and s = (r1 + 1)(r2 + 1) − 1 = 2n+β g 2 − 1

(2)

are prime, then the numbers (1) are amicable, and this gives all amicable pairs of the form (1). For example, β = n − 1 gives Thabit’s rule. Euler’s generalization gives further amicable pairs for n = 8, β = 1, resp. n = 40, β = 29, as discovered by A. M. Legendre and P. Chebyshev, resp. by te Riele [252]. For various other algebraic methods due to Euler, see [187]. We note that a simple method, by the use of Diophantine equations was discovered earlier by F. van Schooten in 1657, but his method gives only the first three known pairs (see [84]). Euler’s methods enabled him to give a list of 30 new pairs. One of the Euler’s method (the third algebraic method) was discovered independently also by G. W. Kraft (1751), see [84], [187]. In 1968 E. J. Lee [186] discovered the so called ”BDE method” (i.e. Bilinear Diophantine Equation method). Given natural numbers a1 , a2 one may determine all amicable pairs of the form (3) a = a1 q, b = a2 s1 s2 where q, resp. s1 , s2 (s1 = s2 ) are primes not dividing a1 , resp. a2 , by solving a 61

CHAPTER 1

bilinear Diophantine equation on s1 , s2 as follows: Take any factorization of the number (4) ( f + d) f + dg = d1 d2 into two different natural factors d1 , d2 , where f := (σ (a1 ) − a1 )σ (a2 ), g := a1 σ (a1 ), d := a2 σ (a1 ) + a1 σ (a2 ) − σ (a2 )σ (a1 ) If then, for i = 1, 2,

si = (di + f )/d

(5)

(6)

are integers, prime, and prime to a2 , and if also q = σ (a2 )σ (a1 )−1 (s1 + 1)(s2 + 1) − 1

(7)

is prime, and prime to a1 , then we have an amicable pair (3), and this gives all such pairs. The idea of this method goes essentially back to Euler, who formulated, and extensively used it in several special cases, as did many authors later on. A new analogue of Thabit’s formula has been discovered in 1972 by W. Borho [28], and later by E. Borho and H. Hoffman [31]. We cite here the later result: Let n be a positive integer, and choose β, 0 < β < n, such that with g = 2n−β + 1, the number (8) r1 = 2β g − 1 is prime. Now choose α, 0 < α < n, such that p = 2α + (2n+1 − 1)g, r2 = 2n−α gp − 1 and s = (r1 + 1)(r2 + 1) − 1 = 2n−α+β g 2 p − 1

(9)

are also prime. Then a = 2n pr1r2 , b = 2n ps

(10)

are amicable pairs. For example, n = 2 and α = β = 1 give the amicable numbers a = 22 ·23·5·137, b = 22 · 23 · 827, discovered by other methods by Euler. A new analogue of Euler’s formula is contained in the following result ([31]): Let n, γ be positive integers with 0 < γ < n. Put either (i) C = 2n + 2γ , or (ii) C = 2n (2n+1 − 1) + 2γ (11) Take any factorization of C = f D into two positive factors f, D. Let p = D + 2n+1 − 1, r1 = p f − 1, r2 = (r1 + 1)2n−γ − 1 in case (i) 62

(12)

PERFECT NUMBERS

resp. r2 = 2n + f − 1, r2 = p(2n + f )2n−γ − 1 in case (ii)

(13)

s = (r1 + 1)(r2 + 1) − 1

(14)

Whenever the four numbers p1 , r1 , r2 , s are primes, then a = 2n pr1r2 , b = 2n ps

(15)

are amicable, and all amicable pairs (15) with p, r1 , r2 , s odd primes, are obtained in this way. For example, n = 8, γ = 7, f = 23 · 31 give Lee’s pair [186] a = 28 · 1039 · 503 · 1047311, b = 28 · 1039 · 527845247. In 1866 a 16-year old Italian school boy, N. I. Paganini found the small amicable pair (1184, 1210), which is not in the list by Euler (but he gave no indication of the method of discovery). (16) E. B. Escott was the most prolific discoverer of amicable numbers prior to the computer era. There were 390 known amicable pairs as of 1946, (see Escott [95]) and 219 of them were due to Escott. (17) H. L. Rolf [262] was the first (by using an exhaustive numerical method) to discover an amicable pair on a computer. He investigated all numbers less than 105

(18)

J. Alanen, O. Ore and J. Stemple [5] extended the numerical search to 106

(19)

discovering 8 new pairs. In 1970 H. Cohen [54] extended the range to 108 .

(20)

In 1984 te Riele [256] discovered a trick named ”daughter pair from mother pairs”, with a remarkable efficiency. This can be summarized as follows: take the inputs a1 , a2 from the list of already known amicable numbers (see the BDE-method (3)-(7)) (a, b) by splitting both numbers a, b into a = a1 v1 , b = a2 v2 , where vi (i = 1, 2) is either 1, or a large simple prime factor of a, b (v1 for a, v2 for b). From a list of known (”mother”)-pairs new (”daughter”) pairs can be computed. (21) For an improvement of te Riele’s trick, see the ”breeding” method by Borho and Hoffman [31]. 63

CHAPTER 1

te Riele [257] has found all 1427 amicable pair whose lesser members are less than 1010 (22) Moews and Moews found 3340 numbers less than 1011 (D. Moews, P. C. Moews [221]) and 5001 numbers less than 3 · 06 · 1011 (D. Moews, P. C. Moews [222]). In 1993 [258] te Riele discovered a new method similar to the Thabit-Euler-LeeBorho-Hoffman methods. In 1997 M. Garcia (see [325]) discovered a new amicable pair, each of whose members has 4829 digits. Probably this is momentary the largest known amicable pair. (23) The new pair has the form a = C M[(P + Q)P 89 − 1] b = C Q[(P − M)P 89 − 1]

(24)

where C = 211 · P 89 , M = 287155430510003638403359267, P = 574451143340278962374313859, Q = 136272576607912041393307632916794623. Note that P, Q, (P + Q)P 89 − 1 and (P − M)P 89 − 1 are all primes. During the first three months of the year 2001, M. Garcia has found over one million new amicable pairs (see his article ”A million new amicable pairs” from J. Integer Sequences (electronic) 4(2001), article 01.2.6). The earliest known amicable numbers all were divisible by 3. So P. Bratley and J. McKay [32] conjectured that there are no amicable pairs coprime to 6. However, S. Battiato and W. Borho [11] found a counterexample. (25) Today there are many known amicable pairs not divisible by 6 (see J. M. Pedersen [232]). (26) The first example of a pair coprime to 30 has been found by Y. Kohmoto (see [232]) in 1997, consisting of a pair of numbers each having 193 digits. (27) No amicable pairs which are coprime to 210 are currently known. (28) It is also not known, if there exist odd amicable pairs with one member, but not both divisible by 3. (29) A conjecture by Ch. Wall says that odd amicable pairs must be incongruent (modulo 4) (30) (see Guy [114], p. 57). It is not known whether or not there are an infinite number of amicable pairs. (31) 64

PERFECT NUMBERS

The first result in this direction is Kanold’s proof [166] in 1954 that the density of amicable pairs is ≤ 0, 204 (32) The fact that this density is zero, is due to P. Erd¨os [88]. Formulated more precisely, let B(x) = car d{(a, b) : a < b, a < x, (a, b) amicable pair} Then B(x) = o(x)

(33)

Erd¨os and G. J. Rieger [94] improved this to B(x) = O(x/ log log log x)

(34)

Finally, in 1981 C. Pomerance [241] deduced the strongest known estimates, namely B(x) ≤ x exp(−(log x)1/3 ) (35) resp. B(x) x exp(−c(log x log log x)1/3 )

(36)

Note that (35) has an interesting Corollary, not known before, namely that the sum of the reciprocals of the amicable numbers is finite. (37) Relatively prime amicable pairs are not known, (38) but conditions on their existence have been studied by O. Gmelin [109], H.-J. Kanold [163] and P. Hagis, Jr. [115], [116]. Hagis showed that there are no relatively prime amicable numbers with lesser member less than 1060

(39)

Gmelin proved that an even-odd amicable pair (a, b) must be of the form a = 2α A2 , b = B 2 , (A, B odd)

(40)

A. A. Gioia and A. M. Vaidya [108] show that A in (40) cannot be a square. (41) If in (40) α > 1, then A cannot be a prime ≡ 4 (mod 3)

(42)

B must be composite, and if α is odd, then (A, 3) = (B, 3) and there exists a prime q and a positive integer γ such that q γ ||B and q ≡ γ ≡ 1 (mod 3). 65

(43)

CHAPTER 1

If α ≡ 3 (mod 4), then there exists a prime p and a positive integer δ such that p ||B and δ

2p ≡ δ ≡ 2

(mod 5), and A ≡ B ≡

α+1 σ (A2 ) ≡ 0 4

(mod 5).

(44)

(42)-(44) are due to Gioia and Viadya, too. Kanold [164] proves that if α > 1 in (40), then a < b. (45) If α = 1 and a > b in (40), then b must contain at least five distinct prime factors. (46) Also, if a, b in (40) are relatively prime, then ab ≡ 2

(mod 24)

(47)

A corollary of (47) is the interesting fact that there are no relatively prime evenodd amicable pairs in which both members are perfect squares. (48) k If A in (40) is a pure prime power, then α = 1, and if we put M = p , then k > 6, ω(B) > 24, p ≡ 1 (mod 12), and a > b > 1075 . (49) Another result by Kanold states that if ω(a) = ω(b) = 2, then a and b cannot be an amicable pair. (50) Lee and Madachy [187] prove that if a (51) b 2 Borho [30] proved that the set of relatively prime amicable pairs a, b with ω(a)+ ω(b) ≤ K (where K is a given bound), is finite. (52) P. Hagis, Jr. [117] obtained the following results: Let a and b be relatively prime amicable pairs of opposite parity such that a < b and b even. Put 1 ( p prime) S= p p|ab If 5|ab, then S < 1.57549, while if 5 ab, then S < 1.59862. These bound are true also, if b < a, but a < 2b. If a, b are relatively prime amicables with opposite parity, then

(53) (54) (55)

S > 1.43151 if 5|ab;

(56)

S > 1.45382 if 5 ab.

(57)

In 1986 te Riele [257] found 37 pairs of amicable pairs (a, b) and (c, d) such that a+b =c+d 66

(58)

PERFECT NUMBERS

The first such pair is (609928,686072) and (643336,652664). Three such pairs were obtained in 1993 by Moews and Moews, while te Riele in 1995 obtained four pairs. In 1997 five, resp. six pairs have been discovered. Related to amicable numbers in special sequences or functions, we quote the following results: For any a, the numbers a and ϕ(a) (ϕ is Euler’s totient) cannot be amicable. (59) The same for a and (2b − 1)a + 1, where a, b > 1. (60) Results (59) and (60) are due to F. Luca [196]. Here an open question is stated too: for what value of a are a and a + 1 amicable? (they cannot be both perfect; see 4.(49)). (61) The Pell sequence (Pn ) is given by P0 = 0, P1 = 1, Pn+2 = 2Pn+1 + Pn for n ≥ 0. In [199] it is proved that there are no amicable Pell pairs. (62) The notion of amicable numbers (or pairs) has been studied also in certain other contexts. Unitary amicable numbers have been introduced by Hagis [118], and studied by him, and later by Garcia [106]. It seems that there are more than 100 such pairs. The first few are (114,126), (1140,1260), (18018,22302),... The largest known unitary amicable pair has its members with 192 digits, discovered by Y. Kohmoto [327]. We quote some theoretical results on unitary amicable pairs (a, b), i.e. satisfying ∗ σ (a) = σ ∗ (b) = a + b. For example, neither a nor b is of the form p α (prime power); they are of the same parity; if a = 2α M, b = 2β N , where β > α > 0 and M, N are odd integers having s, resp. t prime divisors, then s ≤ α and t ≤ α. (63) If a = 2M, b = 2N , where (10, M N ) = 1, then 10a and 10b also is a pair of unitary amicable numbers. If a and b are both squarefree, then (a, b) is unitary amicable pair iff it is an amicable pair. (63’) It is not known if there are an infinite number of unitary amicable pairs, or if there exists a such pair with relatively prime components. For the bi-unitary amicable numbers see another paper by Hagis [126]. Let (a, b) be a bi-unitary amicable pair, i.e. σ ∗∗ (a) = σ ∗∗ (b) = a + b. Then a and b must have the same parity. If a = 2α M, b = 2β N with M ≡ N ≡ 1 (mod 2), α < β, then ω(M) ≤ α, ω(N ) ≤ β. (64) As a corollary one gets that if a = 2M, b = 2β N with β > 1 and M, N odd, then M = p c , N = q d (prime powers). Suppose that a = m M, b = m N with (m, M) = (m, N ) = 1 and where every exponent in the prime factorization of M and N is odd. If σ ∗∗ (n)/n = σ ∗ (m)/m and (n, M) = (n, N ) = 1, then (n M, n N ) is a new bi-unitary amicable pair. (64’) There are sixty pairs of bi-unitary amicable pairs (a, b) with a < b and a ≤ 106 . The first examples are (114,126), (594,846), (1140,1260), (3608,3952), (4698,5382),...

67

CHAPTER 1

Let σ # (n) be the sum of nonunitary divisors of n. The numbers a and b are called nonunitary amicable (see [190]) if σ # (a) = b and σ # (b) = a.

(65)

All known nonunitary amicable pairs are even. In [129] Hagis proves that if (a, b) is a pair of odd nonunitary numbers, then a = M 2, b = N 2,

(66)

while if such a pair is even-odd, then a = 2α M 2 with M odd and α ≥ 1

(67)

Garcia (see [114], p. 59) has called a pair of numbers (a, b) quasi-amicable if σ (a) = σ (b) = a + b + 1

(68)

For example, (48,75), (140,195), (1575,1648), (1050,1925) are quasi-amicable pairs. Some authors use the ”betrothed numbers” terminology, but remarking that for a = b one reobtains from (68) the quasi-perfect numbers (see section 6), this one is more natural terminology. Hagis and Lord [136] have found all 46 pairs of quasi-amicable numbers with a < 107 , a < b. (69) All of them are of opposite parity. No pairs are known with a, b having the same parity. (70) Hagis and Lord prove that if a 1010 .

(71)

If (a, b) = 1 is a quasi-amicable pair, then ω(ab) ≥ 4,

(72)

while if (a, b) = 1 with a, b odd, then ω(ab) ≥ 21.

(73)

Finally, if (a, b) = 1 and a, b are of the same parity, then a and b are > 1030

(74)

The numbers a and b will be called almost-amicable if σ (a) = σ (b) = a + b − 1 68

(75)

PERFECT NUMBERS

Note that for a = b one reobtains the almost-perfect numbers (see section 6). Beck and Najar [12] (who call a, b satisfying (75) ”augmented amicable”, while for (68) they use the terminology ”reduced amicable”) found 11 almost-amicable pairs. (76) They found no unitary quasi or almost amicable pairs (a, b) such that a < b, b < 105 ,

(77)

i.e. satisfying (68), resp. (75) with σ ∗ in place of σ . Cohen [58] has introduced infinitary amicable numbers (see section 9 for infinitary divisors) a, b defined by σ∞ (a) = σ∞ (b) = a + b

(78)

Such pairs are e.g. (114,126), (594,846), (1140,1260), (4320,7920), (5940,8460), (8640,11760). He lists all infinitary amicable numbers (a, b) with a < b and a < 106 . There are 62 such pairs. (79) From the list it is interesting to remark that very often a and b have similar prime factorization, and Cohen asks for a motivation of this fact. (80) In an unpublished paper, I. Z. Ruzsa [265] called the numbers a and b lovely numbers if σ (a) = 2b, σ (b) = 2a (81) All lovely pairs (a, b) with a and b even are given by a = 2k (2q+1 − 1), b = 2q (2k+1 − 1)

(82)

where 2q+1 − 1 and 2k+1 − 1 are primes. S´andor [277] obtains a new proof of this result. He also proves that if (a, b) is a super-lovely pair, defined by σ (σ (a)) = 2b, σ (σ (b)) = 2a

(83)

with a and b even, then a = b, so one reobtains the even superperfect numbers (see section 7). The unitary, etc. analogue of lovely numbers are introduced in [278]. In 1913 Dickson [81] defined amicable triples (a, b, c) by σ (a) = σ (b) = σ (c) = a + b + c

(84)

After developing a theory analogues to Euler, he obtained eight sets of amicable triplets in which two of the numbers are equal, and two triples of distinct numbers (293 · 337a, 5 · 16561a, 993371a), resp. (3 · 89b, 11 · 29b, 359b) where a = 25 · 3 · 13, b = 214 · 5 · 19 · 31 · 151. 69

(85)

CHAPTER 1

A. Makowski [204] gave by other method new amicable triples, e.g. (22 · 32 · 5 · 11, 25 · 32 · 7, 22 · 32 · 71). In 1921 T. E. Mason [211] further generalized amicable numbers by introducing amicable k-tuples a1 , a2 , . . . , ak where σ (a1 ) = σ (a2 ) = · · · = σ (ak ) = a1 + a2 + · · · + ak

(86)

It is not known if there exist infinitely many such k-tuples (for any k ≥ 2). (87) Y. Kohmoto (see [114], p. 59) found an example of amicable 4-tuples (which he calls quadri-amicable numbers) with all of ai (i = 1, 4) not multiple of 3. (88) The smallest known amicable quadruple is (842448600, 936343800, 999426600, 1110817800). B. F. Yanney [329] defined a k-tuplet of amicable numbers a1 , . . . , ak by a1 + a2 + · · · + ak = (k − 1)σ (ai ) (i = 1, k). For k = 2 this definition coincides with that of Dickson. For k > 2, however the two definitions are distinct. E.g. a1 = 308, a2 = 455, a3 = 581 are a triplet of amicable numbers in the sense of Yanney, since σ (a1 ) = σ (a2 ) = σ (a3 ) = 672 and a1 + a2 + a3 = 2 · 672. C. W. Anderson and D. Hickerson [7] call the pair (a, b) a friendly pair, if a = b and σ (b) σ (a) = (89) a b For example, (6n, 28n) for (n, 42) = 1 are such pairs. A number a is called solitary if it doesn’t exist b = a such that (a, b) is a friendly pair (90) (i.e. a is a number ”without friends”). Clearly, if a = b are perfect numbers (i.e. σ (a) = 2a, σ (b) = 2b), then (a, b) satisfies (89). An odd solution is (135,819) or (33 · 5 · 7, 34 · 72 · 112 · 192 · 127); while a solution with a even, b odd is (42, 32 · 5 · 72 · 13 · 19).

(91)

M. G. Greening [7] shows that numbers n such that (n, σ (n)) = 1, are solitary. (92) There are 53 numbers less than 100, which are solitary, but there are some numbers, as 10, 14, 15, 20,... for which we cannot decide if they are solitary or not. (93) The density of friendly pairs is positive, first shown by Erd¨os [89]: the number of solutions of (89) satisfying a 0 is a constant (in fact C ≥ 8/147, see [7]). (94) 70

PERFECT NUMBERS

If (a, b) = 1 is a friendly pair, then (89) implies that a|σ (a), b|σ (b), so a and b are multiply perfect numbers (see section 5). It is not known if (89) has infinitely many solutions satisfying (a, b) = 1. (95) If a and b are squarefree, then (a, b) cannot be a friendly pair (see [89]). (96) The density of numbers a with friends is unity, (97) see [7]. R. D. Carmichael [45] called a pair (a, b) multiply amicable if σ (a) = σ (b) = k(a + b) for some positive integer k. (98) For k = 1 we obtain the classical amicable pairs. Mason [211] gives various multiply amicable pairs for k = 2 and k = 3. Cohen, Gretton and Hagis [60] called the integers a and b < a (α, β)-multiamicable if σ (b) − b = αa, σ (a) − a = βb, with α, β positive integers. They proved that b cannot have just one distinct prime factor, and if it has precisely two distinct prime factors, then α = 1 and b is even. For small α, β various (α, β)-multiamicable numbers have been tabulated. σ (a) M + αN N + βM Suppose that (a, M) = (a, N ) = 1 and = = . Then a σ (M) σ (N ) a M, a N are (α, β)-multiamicable. By applying this proposition, the authors have found 72 more multiamicable pairs. They proved also that the density of multiamicable numbers is zero. Cohen and te Riele [65] have studied multiple ϕ-amicable pairs (where ϕ is Euler’s totient). The pair (a, b) with 1 < a ≤ b is called ϕ-amicable pair with multiplier k (denoted as (a, b; k) if ϕ(a) = ϕ(b) =

a+b , for some integer k ≥ 1. k

(99)

The pair (a, b) is ϕ-amicable if there exists k with (99). In fact, it is easy to see that one must have k > 2. A ϕ-amicable pair (a, b; k) is called primitive if gcd(a, b) is squarefree and gcd(a, b; k) = 1.

(100)

Cohen and Riele have computed all ϕ-amicable pairs with b < 109 , and found 812 pairs for which the greatest common divisor is squarefree. Among them 499 are primitive. In fact with a ϕ-pair for which the g.c.d. is squarefree, infinitely many other ϕ-amicable pairs can be associated. (101) They proved the following general theorems: There are only finitely many primitive ϕ-amicable pairs with a given number of different prime factors. (102) 71

CHAPTER 1

Let (a, b; k) be ϕ-amicable, where ω(a) = r , ω(b) = s. Let p be the smallest prime divisor of ab, and put m = max{r, s}. Then p≤

k + 4m − 8 k−4

k + 2m − 2 p≤ k−2

with k ≥ 5, if the pair is even, (103) with k ≥ 3, if the pair is odd.

If (a, b; k) is a ϕ-amicable pair with (a, b) = 1, then k−1<

ab k2 < ϕ(ab) 4

(104)

As a corollary of (104) one obtains that if q is a prime, q < a, and (q, a) is ϕ-amicable, then ω(a) ≥ 12. (105) Finally, except for the pair (4,6) if one member of a ϕ-amicable pair has exactly two distinct prime factors, then the other member has at least four distinct prime factors.

1.14

Sociable numbers

In this final section we shall treat the iteration properties of special arithmetic functions like s(n) = σ (n) − n (1) By considering the aliquot sequence {s (k) (n)} for k = 0, 1, 2, . . .

(2)

where s (0) (n) = n, in 1902 A. Cunningham [73] said that the chain n, s(n), s (2) (n), . . . is of period k if s (k) (n) = n. The numbers in a chain of period k form a cycle (or period k). For k = 1, when s(n) = n one reobtains the perfect numbers, while for k = 2, when s(s(n)) = n, the numbers n and s(n) will be amicable. Even in 1887 E. Catalan [49] stated empirical results (in corrected versions by Dickson [83]) on the aliquot sequence. For example, that every non-periodic chain contains a prime (3) (so the next term is 1, since s( p) = 1 for a prime p), For k odd, k > 1, there is no chain an 1 , an 2 , . . . , an k of period k in which (n 1 , . . . , n k ) = 1 and (n j , a) = 1, j = 1, k, a > 1. (4) 72

PERFECT NUMBERS

In 1918 P. Poulet (see [244]) discovered a chain of period 5, namely n = 12496 = 24 · 11 · 71, s(n) = 24 · 19 · 47, s (2) (n) = 24 · 967, s (3) (n) = 23 · 23 · 79, s (4) (n) = 23 · 1783, s (5) (n) = n

(5)

and another one of period 28 for n = 14316. (6) After a gap of 50 years, and the advent of high-speed computers, H. Cohen (see [114]) discovered nine cycles of period 4, and Borho [27], as well as David and Root (see [114]) also discovered some. (7) Moews and Moews [220] have made an exhaustive search for such cycles with greatest member less than 1010 . There are 24 such cycles. (8) They found also an 8-cycle. A. Flammenkamp [101] discovered a new 8-cycle, and one 9-cycle. (9) Moews and Moews [221] have continued to uncover all cycles, of any length, whose member preceding the largest member is less than 3.6 · 1010 . They found three more 4-cycles, and one 6-cycle, all of whose members are odd (for n = 21548919483 = 35 · 72 · 13 · 17 · 19 · 431). (10) No one found a 3-cycle, and it has been conjectured that such cycles do not exist. (11) On the other hand it has been conjectured that for each k there are infinitely many k-cycles (see [114]). (12) For new four-cycles, see [25] (where fifty new such cycles are found, bringing the total number of 4-cycles to 110). Related to Catalan’s conjecture (3), today we have experimental (and heuristic) arguments that some sequences go to infinity (see e.g. [251]). (13) Perhaps (13) holds true for almost all even n. D. H. Lehmer (see [114]) showed that the aliquot sequence {s (k) (138)} after reaching a maximum s (117) (138) = 2 · 61 · 929 · 1587569, terminated at s (177) (138) = 1. (14) Guy et al. (see [114]) used a program to discover that s (746) (840) = 601

(15)

and establishing a new record s (287) (840) for the maximum of a terminating sequence. Recently, M. Dickerman found a new record showing that s (583) (1248) reaches a 73

CHAPTER 1

maximum of a number of 58 digits, the period being 1075 for this number n = 1248. (16) For recent advances in aliquot sequences (of computational nature), see M. Benito and J. L. Varona [20]. In 1977 H. W. Lenstra [188] has proved that it is possible to construct arbitrarily long monotonic increasing aliquot sequences, i.e. for each k there is an n with n < s(n) < s (2) (n) < · · · < s (k) (n)

(17)

P. Erd¨os [91] proved that for all fixed k and δ > 0 and for all n, except a sequence of density zero one has

s(n) (1 − δ)n n

i

s(n) < s (n) < (1 + δ)n n (i)

i (18)

for i = 1, k.

s(n) s (i+1) (n) > − ε for i = 1, k (for each ε > 0 and k) has (i) s (n) n asymptotic density 1. (19) Erd¨os, Granville, Pomerance and Spiro [93] proved a similar result: for each ε > 0, the set of n with s (2) (n) s(n) < +ε s(n) n The set of n with

has asymptotic density 1. (20) Let S (k) (x) denote the number of odd integers m ≤ x, not in the range of the function s (k) . Then there is a δ > 0 such that S (k) (x) x 1−δ

(21)

uniformly for all positive integers k, and x > 0. The unitary aliquot sequences and unitary sociable numbers are defined similarly, by considering s ∗ (n) = σ ∗ (n) − n (22) where σ ∗ (n) denotes the sum of unitary divisors of n. The analogue of (17) holds true here, too, proved by te Riele [249]: for each k there is an n such that n < s ∗ (n) < s (2)∗ (n) < · · · < s (k)∗ (n)

(23)

but we do not know if there are unbounded unitary aliquot sequences, or not. (24) 74

PERFECT NUMBERS

te Riele pursued all unitary aliquot sequences for n < 105 . The only one which did not terminate or become periodic was 89610. Later calculations [250] showed that this reached a maximum at s ∗(568) (89610), and terminated at its 1129th term. (25) Unitary sociable numbers may occur rather more frequently than their ordinary counterparts. M. Lal, G. Tiller and T. Summers [185] found cycles of periods 3, 4, 5, 6, 14, 25, 39, 65. For example, (30,42,54) is a 3-cycle, while (1482,1878,1890,2142,2178) is a 5-cycle. Beck and Najar [13] considered the iterates of L ∗− (n) = s ∗ (n) − 1, and

(26)

L ∗+ (n) = s ∗ (n) + 1.

(27)

They introduce ”reduced unitary sociable numbers” by considering the iterations of L ∗− , and ”augmented unitary sociable numbers” by the same for L ∗+ . We shall call these in a more natural way as unitary quasi-sociable, resp. unitary almostsociable numbers, in accordance with the used terminologies in the case of perfect and amicable numbers. Beck and Najar showed by a computer search that there are no unitary almostsociable numbers n for n < 105 . (28) The same is true for unitary quasi-sociable numbers less than 110000

(29)

Recently, the calculations were lifted up to 109 , and one cycle has been obtained for both cases (see [219]). (30) Cohen [58] introduces in a similar manner infinitary sociable numbers. He finds eight infinitary aliquot cycles of order 4, three of order 6 and one of order 11. (31) This last is the only cycle of order less than 17 and least member less than a million. Y. Kohmoto (see [326]) has considered a generalization of the sociable numbers as follows. Let (a(n)) be a sequence defined by a(n) =

σ (a(n − 1)) . m

If this sequence becomes cyclic after k > 1 terms, it is then called an 1/msociable number of order k. It is easy to see that e.g. 2m−1 Mn and 2n−1 Mm are 1/2-sociable numbers of order 2, with Mn and Mm being distinct Mersenne primes. In [219] one finds certain remarks and numerical results for the bi-unitary, exponential cycles, as well as their quasi and almost variants. (32) 75

CHAPTER 1

Hagis [129] considered nonunitary aliquot sequences. A k-tuple of distinct natural numbers (n 0 , n 1 , . . . , n k−1 ) with n i = σ # (n i−1 ) (i = 1, k − 1) and n 0 = σ # (n k−1 ) is called a nonunitary k-cycle. (33) A search was made for all nonunitary k-cycles with k > 2 and n 0 ≤ 106 . A 3-cycle was found: (619368, 627264, 1393551) (34) An open question is whether or not unbounded nonunitary aliquot sequences exist. For generalized aliquot sequences, see te Riele [253]. For the iteration of various arithmetic functions, see [255]. Finally we mention two open problems, one due to Alaoglu and Erd¨os [3], the other one to Poulet [245]. Alaoglu and Erd¨os propose the question of whether or not the sequences: σ (n), σ (σ (n)), ϕ(σ (σ (n))), σ (ϕ(σ (σ (n)))), . . .

(35)

ϕ(n), ϕ(ϕ(n)), σ (ϕ(ϕ(n))), . . .

(36)

or converge to a periodic state, a fixed number, or tend to infinity for all n? Poulet conjectures that the sequence ( f k (n))k defined by f 0 (n) = n, f 2k+1 (n) = σ ( f 2k (n)), f 2k (n) = ϕ( f 2k−1 (n)) is eventually periodic, for all n. (37)

76

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[12] W. E. Beck and R. M. Najar, More reduced amicable pairs, Fib. Quart. 15(1977), 331-332. [13] W. E. Beck and R. M. Najar, Fixed points of certain arithmetic functions, Fib. Quart. 15(1977), 337-342. [14] W. E. Beck and R. M. Najar, A lower bound for odd triperfect numbers, Math. Comp. 38(1982), 249-251. [15] E. Bedocchi, Perfect numbers in real quadratic fields, (Italian), Bolletino U.M.I. (5)15-A(1978), 94-103. [16] A. Bege, Fixed points of certain divisor functions, Notes Number Th. Discr. Math., 1(1995), no. 1, 43-44. [17] A. Bege, On multiplicatively unitary perfect numbers, Seminar on Fixed Point Theory, Cluj-Napoca, 2(2001), 59-63. [18] A. Bege, On multiplicatively bi-unitary perfect numbers, Notes Number Th. Discr. Math. 8(2002), 28-36. [19] M. Bencze, On perfect numbers, Studia Univ. Babes¸-Bolyai, 26(1981), 14-18. [20] M. Benito and J. L. Varona, Advances in aliquot sequences, Math. Comp. 68(1999), 389-393. [21] S. J. Benkovski, Problem E2308, Amer. Math. Monthly 79(1972), 774. [22] S. J. Benkovski and P. Erd¨os, On weird and pseudoperfect numbers, Math. Comp. 28(1974), 617-623; corrigendum. S. Krawitz, 29(1975), 673. [23] J. T. Betcher and J. H. Jaroma, An extension of the results of Servais and Cramer on odd perfect and odd multiply perfect numbers, Amer. Math. Monthly 110(2003), no. 1, 49-52. [24] S. J. Bezuska and M. J. Kenney, Even perfect numbers, The Math. Teacher 90(1997), 628-633. [25] K. Blamkenagel, W. Borho and A. vom Stein, New amicable four cycles, Math. Com., 72(2003), no. 244, 2071-2076 (electronic). ¨ [26] D. Bode, Uber eine Verallgemeinerung der vollkommenen Zahlen (Dissertation, Braunschweig 1971, 57 pp.). 78

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¨ [27] W. Borho, Uber die Fixpunkte der k-fach itarierten Teilersummenfunktion, Mitt. Math. Gesellsch. Hamburg 9(1969), 34-48. [28] W. Borho, On Thabit Ibn Kurrah’s formula for amicable numbers, Math. Comp. 26(1972), 571-578. [29] W. Borho, Befreundete Zahlen, Ein Zweitausend Jahre altes Thema der elementaren Zahlentheorie, in: Lebendige Zahlen, Math. Miniaturen, vol. 1, Birkh¨auser V., Basel, 1981. [30] W. Borho, Befreundete Zahlen mit gegebener Primteileranzahl, Math. Ann. 209(1974), 183-193. [31] W. Borho and H. Hoffman, Breeding amicable numbers in abundance, Math. Comp. 46(1986), 281-293. [32] P. Bratley and J. McKay, More amicable numbers, Math. Comp. 22(1968), 677-678. [33] A. Brauer, On the non-existence of odd perfect numbers of form 2 p α q12 . . . qt−1 qt4 , Bull. A.M.S., 49(1943), 712-718. [34] R. P. Brent and G. L. Cohen, A new lower bound for odd perfect numbers, Math. Comp. 53(1989), 431-437, S7-S24. [35] R. P. Brent, G. L. Cohen and H. J. J. te Riele, Improved techniques for lower bounds of odd perfect numbers, Math. Comp. 57(1991), 857-868. [36] S. Brentjes, Die ersten sieben volkommenen Zahlen und drei Arten befreundeter Zahlen in einem Werk zur elementaren Zahlentheorie von Ismal b. Ibrahim ibn Fallus, NTM Schr. Gesichte Natur. Tech. Medizin 24(1)(1987), 21-30. [37] S. Brentjes, Eine Tabelle mit volkommenen Zahlen in einer arabischen Handschrift aus dem 13 Jahrhundert, Nieuw Arch. Wiskunde (4)8(2)(1990), 239241. [38] D. A. Buell, On the computation of unitary hyperperfect numbers, Congr. Numer. 34(1982), 191-206. [39] E. A. Bugulov, On the question of the existence of odd multiperfect numbers (Russian), Kabardino Balkarsk Gos. Univ. Ucen. Zap. 30(1966), 9-19. 79

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[242] C. Pomerance, Problem 10331, Amer. Math. Monthly (1993), 796. Solution by U. Everling, same journal (1996), 701-702. [243] M. A. Popov, On Plato’s periodic perfect numbers, Bull. Sci. Math. 123(1999), 29-31. [244] P. Poulet, La chasse aux nombres, Fascicule I, Bruxelles, 1929. [245] P. Poulet, Nouvelles suites arithm´etiques, Sphinx, 2(1932), 53-54. [246] M. Raghavachari, On the form of odd perfect numbers, Math. Student 34(1966), 85-86. [247] R. Rashed, Ibn al-Haytham et les nombres parfaits, Historia Math. 16(1989), 343-352. ¨ ¨ [248] H. Reidling, Uber ungerade mehrfach willkommene Zahlen, Osterreichische Akad. Wiss. Math.-Natur. 192(1983), 237-266. [249] H. J. J. te Riele, Unitary aliquot sequences, MR139/72, Math. Centrum Amsterdam, 1972. [250] H. J. J. te Riele, Further results on unitary aliquot sequences, NW12/73, Math. Centrum, Amsterdam, 1973. [251] H. J. J. te Riele, A note on the Catalan-Dickson conjecture, Math. Comp. 27(1973), 189-192. [252] H. J. J. te Riele, Four large amicable pairs, Math. Comp. 28(1974), 309-312. [253] H. J. J. te Riele, A theoretical and computational study of generalized aliquot sequences, MCT72, Math. Centrum, Amsterdam, 1976. [254] H. J. J. te Riele, Hyperperfect numbers with three different prime factors, Math. Comp. 36(1981), 297-298. [255] H. J. J. te Riele, Iteration of number theoretic functions, Report NN30/83, Math. Centrum, Amsterdam, 1983. [256] H. J. J. te Riele, On generating new amicable pairs from given amicable pairs, Math. Comp. 42(1984), 219-223. [257] H. J. J. te Riele, Computation of all the amicable pairs below 1010 , Math. Comp. 47(1986), 361-368. 93

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[258] H. J. J. te Riele, A new method for finding amicable pairs, in Proc. Sympos. Appl. Math. 43, Amer. Math. Soc., Providence, RI, 1994. [259] N. Robbins, The nonexistence of odd perfect numbers with less than seven distinct prime factors, Notices Amer. Math. Soc. 19(1972), A-52. [260] N. Robbins, Lower bounds for the largest prime factor of an odd perfect number which is divisible by a Fermat prime, J. Reine Angew. Math. 278/279(1975), 14-21. [261] D. F. Robinson, Egyptean fractions via Greek number theory, New Zeeland Math. Mag. 16(1979), 47-52. [262] H. L. Rolf, Friendly numbers, Math. Teacher, 60(1967), no. 2, 157-160. [263] A. Rotkiewicz, Remarque sur les nombres parfaits pairs de la forme a n ± bn , Elem. Math. 18(1963), no. 4, 76-78. [264] S. M. Ruiz, Smarandache’s function applied to perfect numbers, Smarandache Notions J., 10(1999), no. 1-2-3, 114-115. [265] I. Z. Ruzsa, Personal communication to J. S´andor. [266] E. Saias, Entiers a` diviseurd denses, I, J. Number Theory, 62(1997), no. 1, 163-191. ˇ at and J. Tomanov´a, On the product of divisors of a positive integer, [267] T. Sal´ Math. Slovaca 52(2002), no. 3, 271-278. ¨ [268] H. Sali´e, Uber abundante Zahlen, Math. Nachrichten 9(1953), 217-220. [269] J. S´andor, On Dedekind’s arithmetic function, Seminarul de teoria structurilor, no. 51, Univ. Timis¸oara, 1988, pp. 1-15. [270] J. S´andor, On the composition of some arithmetic functions, Studia Univ. Babes¸-Bolyai 34(1989), no. 1, 7-14. [271] J. S´andor, On even perfect and superperfect numbers (Romanian), Lucr. Semin. Didact. Mat. 8(1992), 167-168. [272] J. S´andor, A note on S(n), where n is an even perfect number, Smarandache Notions J., 11(2000), no. 1-2-3, 139. [273] J. S´andor, On an even perfect and superperfect number, Notes Number Theory Discr. Math. (Sofia), 7(2001), no. 1, 4-5. 94

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[274] J. S´andor, Abundant numbers, in: M. Hazewinkel, Encyclopedia of Mathematics, Supplement III, Kluwer Acad. Publ., 2001 (see pp. 19-21). [275] J. S´andor, On multiplicatively perfect numbers, J. Ineq. Pure Appl. Math. 2(2001), no. 1 (electronic), 6 pp. [276] J. S´andor, On multiplicatively e-perfect numbers, (to appear). [277] J. S´andor, On Ruzsa’s ”lovely numbers”, (to appear). [278] J. S´andor, The unitary, etc. analogues of Ruzsa’s lovely numbers, in preparation. [279] J. S´andor and A. Bege, The M¨obius function: generalizations and extensions, Adv. Stud. Contemp. Math. 6(2003), no. 2, 77-128. [280] M. Satyanarayana, Odd perfecr numbers, Math. Student, 27(1959), no. 1-2, 17-18. [281] A. Schinzel, Ungel¨oste Probleme Nr. 30, Elem. Math. 14(1959), 60-61. [282] A. Schinzel and W. Sierpinski, Sur certaines hypoth`eses concernant les nombres premiers, Acta Arith. 4(1958), 185-208; corrigendum, ibid., 5(1959), 259. [283] Cl. Servais, Sur les nombres parfaits, Mathesis, 8(1888), 92-93. [284] H. N. Shapiro, Note on a theorem of Dickson, Bull. A.M.S. 55(1949), 450-452. [285] H. N. Shapiro, On primitive abundant numbers, Comm. Pure Appl. Math. 21(1968), 111-118. [286] W. Sierpinski, Sur les nombres pseudoparfaits, Mat. Vesnik 2(17)(1965), 212213. [287] S. Singh, Fermat’s Enigma: the epic quest to solve the world’s greatest mathematical problem, New York: Walker, 1997. [288] T. N. Sinha, Note on perfect numbers, Math. Student, XLII, no. 3(1974), 336. [289] V. Sitaramaiah and M. V. Subbarao, On unitary multiperfect numbers, Niuew Arch. Wiskunde 16(1998), no. 1-2, 57-61. [290] V. Sitaramaiah and M. V. Subbarao, On the equation σ ∗ (σ ∗ (n)) = 2n, Util. Math. 53(1998), 101-124. [291] J. Slowak, Odd perfect numbers, Math. Slovaca 49(1999), 253-254. 95

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[292] R. Spira, The complex sum of divisors, Amer. Math. Monthly 68(1961), 120124. [293] A. K. Srinivasan, Practical numbers, Current science, 1948, pp. 179-180. [294] P. Starni, On the Euler’s factor of an odd perfect number, J. Number Theory 37(1991), 366-369. [295] P. Starni, Odd perfect numbers: a divisor related to the Euler’s factor, J. Number Theory 44(1993), 58-59. [296] R. Steuerwald, Verscharfung einer notwendigen Bedingung f¨ur die Existenz einer ungeraden volkommenen Zahl, S.-B. Math.-Nat. Abt. Bayer, Akad. Wiss. (1937), 68-72. [297] R. Steuerwald, Ein Satz u¨ ber nat¨urliche Zahlen N mit σ (N ) = 3N , Archiv. der Math. 5(1954), 449-451. [298] B. M. Stewart, Sums of distinct divisors, Amer. J. Math. 76(1954), 779-785. [299] E. G. Straus and M. V. Subbarao, On exponential divisors, Duke Math. J. 41(1974), 465-471. [300] M. V. Subbarao, Are there an infinity of unitary perfect numbers? Amer. Math. Monthly 77(1970), 389-390. [301] M. V. Subbarao, Odd perfect numbers: some new issues, Periodica Math. Hungar. 38(1999), no. 1-2, 103-109. [302] M. V. Subbarao, T. J. Cook, R. S. Newberry and J. M. Weber, On unitary perfect numbers, Delta 3(1972), no. 1, 22-26. [303] M. V. Subbarao and L. J. Warren, Unitary perfect numbers, Canad. Math. Bull. 9(1966), 147-153. [304] D. Suryanarayana, On odd perfect numbers, II. Proc. A.M.S. 14(1963), 896904. [305] D. Suryanarayana, On odd perfect numbers. III, Proc. A.M.S. 18(1967), 933939. [306] D. Suryanarayana, The number of k-ary divisors of an integer, Monatsh. Math. 72(1968), 445-450. [307] D. Suryanarayana, Super perfect numbers, Elem. Math. 24(1969), 16-17. 96

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[308] D. Suryanarayana, There is no odd super perfect number of the form p 2α , Elem. Math. 28(1973), 148-150. [309] D. Suryanarayana, Research problems # 22, Periodica Math. Hungar. 8(1977), 193-196. [310] J. J. Sylvester, Collected papers, 4(1912), 611-614. [311] J. Touchard, On prime numbers and perfect numbers, Scripta Math. 19(1953), 35-39. [312] H. S. Uhler, Full values of the first seventeen perfect numbers, Scripta Math. 20(1954), 240. [313] A. M. Vaidya, Comment on: ”History of perfect numbers”, Ganita Bharati 1(3-4)(1979), 22. [314] R. C. Vaughan, Adventures in arithmetic, or: how to make good use of a Fourier transform, Math. Intellig. 9(1987), no. 2, 53-60. [315] B. Volkmann, A theorem on the set of perfect numbers, Bull. A.M.S. 62(1956), Abstract 180. [316] R. W. Van der Waall, On a theorem of Leopoldt and on perfect numbers, Nieuw Arch. Wiskunde (3) 18(1970), 159-161. [317] S. Wagon, Perfect numbers, Math. Intellig. 7(1985), no. 2, 66-68. [318] Ch. Wall, Topics related to the sum of unitary divisors of an integer, Univ. of Tennessee, 1970. [319] Ch. R. Wall, A new unitary perfect number, Notices A.M.S. 16(1969), 825. [320] Ch. R. Wall, Bi-unitary perfect numbers, Proc. A.M.S., 33(1972), 39-42. [321] Ch. R. Wall, The fifth unitary perfect number, Canad. Math. Bull. 18(1975), 115-122. [322] Ch. R. Wall, On the largest odd component of unitary perfect numbers, Fib. Quart. 25(1987), 312-316. [323] Ch. R. Wall, New unitary perfect numbers have at least nine odd components, Fib. Quart. 26(1988), 312-317. [324] G. C. Webber, Nonexistence of odd perfect numbers of the form 2β 2β 2β 32β p α s1 1 s2 2 s3 3 , Duke Math. J. 18(1951), 741-749. 97

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[325] E. Weisstein, http://mathworld.wolfram.com/AmicablePair.html. [326] E. Weisstein, http://mathworld.wolfram.com/SociableNumbers.html. [327] E. Weisstein, http://mathworld.wolfram.com/UnitaryAmicablePair.html. [328] E. Wirsing, Bemerkung zu der Arbeit u¨ ber vollkommene Zahlen, Math. Ann. 137(1959), 316-318. [329] B. F. Yanney, Another definition of amicable numbers and some of their relation to Dickson’s amicables, Amer. Math. Monthly, 30(1923), 311-315. [330] A. and E. Zachariou, Perfect, semi-perfect and Ore numbers, Bull. Soc. Math. Gr`ece (N.S.) 13(1972), 12-22. [331] Perfect numbers and imperfect mathematicians (Persian), Bull. Iranian Math. Soc. 9(1)(1981/82), 84-79. [332] http://www.utm.edu/research/primes/mersenne/index.html.

98

Chapter 2

GENERALIZATIONS AND EXTENSIONS OF THE ¨ MOBIUS FUNCTION

2.1

Introduction

In this chapter we will survey many generalizations in Number theory of the M¨obius function, as well as analogues functions which arose by the extensions of certain divisibility notions or product notions of arithmetical functions. Our study will include also M¨obius functions in Group theory, Lattice theory, Partially ordered sets or Arithmetical semigroups. Sometimes references to applications will be pointed out, too. This is a refined and extended version of [164]. The classical M¨obius function is defined by  

1, if n = 1 k (−1) , if n = p1 p2 · . . . · pk , pi distinct primes µ(n) =  0, if n is divisible by the squares of a prime

(1)

The function occurred implicitly in L. Euler ([67] paragraph 269) with the considerations of the ”Riemann zeta function” ∞ 1 −1 1 1− s = , ζ (s) = ns p p n=1 99

CHAPTER 2

and the reciprocal formula ∞ 1 1 µ(n) = 1− s = ζ (s) p ns p n=1

(2)

However, its arithmetical importance was first recognized by A. F. M¨obius [143] in 1832, with the discovery of a number of inversion formulae. M¨obius raised the following question: Given an arbitrary function f (z) and g(z) by ∞

g(z) =

an f (z n ),

(3)

n=1

express f (z) in terms of the functions g(z n ), say f (z) =

∞

bn g(z n ).

(4)

n=1

If f (z) = c1 z + c2 z 2 + . . . (i.e., a power series), this question may be treated as a problem in formal power series, and it is easily seen that (bn )n≥1 are obtained from the given coefficients (an )n≥1 by

ad b dn =

d|n

1, if n = 1 . 0, if n > 1

(5)

Equivalently, (5) can be written with the use of formal Dirichlet series as ∞ ∞ an bn n=1

ns

n=1

ns

= 1,

i.e. (an )n≥1 and (bn )n≥1 form Dirichlet inverse sequences. In particular, if an = 1 for all n, then bn = µ(n) by (1), and we have (formally) g(z) =

∞

f (z n ),

n=1

equivalent to f (z) =

∞

µ(n)g(z n ).

n=1

100

¨ THE MOBIUS FUNCTION

By setting f (e z ) = F(z) and g(e z ) = G(z), equations (3) and (4) take a more convenient form ∞ an F(nz), (6) G(z) = n=1

F(z) =

∞

bn G(nz)

(7)

n=1

respectively. The particular case an ≡ 1, bn ≡ µ(n) has a curious history, and a number of rediscoverers (e.g. by P. Tchebyschef in 1851 [203]; P. Bachman in 1894 [5] were unaware of M¨obius’ paper). The notation µ(n) was first introduced in 1874 by F. Mertens. All these considerations were formal and questions of convergence were neglected. Surprisingly, the first rigorous treatment of these matters appeared only in 1936 and 1937 in works by E. Hille and O. Sz´asz [108] and E. Hille [107]. They proved the following result: If an and bn are related by (5), and g(n) = where the double series

∞

am f (mn) (n = 1, 2, 3, . . . ),

(8)

m=1

ak bm f (kmn) is absolutely convergent, then

k,m ∞

f (n) =

bm g(mn)

(9)

m=1

This theorem leads to a symmetrical version of (6)-(7): The following two assertions are equivalent: (i)

g(n) =

∞

f (mn) (n = 1, 2, . . . )

(10)

m=1

and

∞

n ε | f (n)| < ∞ for some ε > 0;

n=1

(ii)

f (n) =

∞

µ(m)g(mn) (n = 1, 2, . . . )

m=1

and

∞

n ε |g(n)| < ∞ for some ε > 0.

n=1

101

(11)

CHAPTER 2

There is a need for caution here for the truth e.g. of (6)-(7) for general f and g 1 µ(m)g(mn) is identically (an = 1, bn = µ(n)). Put g(n) = ; then f (n) = n m f (mn). In the other direction, if f (n) = µ(n)/n, then zero, so that g(n) = m f (mn) is identically zero and again (6)-(7) fails. g(n) = m

The following remark by Andrew Lenard [2] has importance in Beurling’s approach to the Riemann hypothesis (see H. Bercovici and C. Foias¸ [12]. Let ∞ l(x) = (−1)n−1 L(nx), n=1

where L(x) =

∞

µ(n)K (nx),

n=1

equivalent (by (6)-(7)) to K (x) =

∞

L(nx) (x > 0)

n=1

Since l(x) = K (x) − 2K (2x), we get K (x) = l(x) + 2K (2x) = l(x) + 2l(2x) + 4K (4x) = · · · =

∞

2k f (2k x).

k=0

So, by the M¨obius inversion formula ∞ l(x) = (−1)n−1 L(nx)

(12)

n=1

is equivalent to L(x) =

∞ ∞

2k µ(n)l(2k nx)

(13)

k=0 n=1

An improvement of corollary (10)-(11) has been obtained by J. H. Loxton and J. W. Sanders [138]: Suppose that the coefficients an are completely multiplicative (i.e. amn = am · an for all m, n, and a1 = 1), and that the series am f (mn) and am g(mn) are m

absolutely convergent. 102

m

¨ THE MOBIUS FUNCTION

Then g(n) =

∞

am f (mn)

(14)

µ(m)am g(mn)

(15)

m=1

is equivalent to f (n) =

∞ m=1

Another application of the equivalence of (6)-(7) is related to the inversion of Fourier transforms. The following result is due to R. R. Goldberg and R. S. Varga [78]. Let k : [0, R] → R (R > 0) be of bounded variation, and suppose that ∞ |k(u)| log udu < ∞ 1

and put

∞

K (t) =

k(u) cos tudu. 0

Then 1 H (t) = t

∞ nπ K (0) n (−1) K + 2 t n=1

is finite almost everywhere (0 < t < ∞), and k(t) =

∞

µ2n−1 H ((2n − 1)t)

(16)

n=1

almost everywhere for t ∈ (0, ∞). In [78] it is noted that a similar result was obtained by R. J. Duffin in 1941. The ”finite” form of the M¨obius inversion formula n (17) f (d) ⇐⇒ f (n) = µ( d)g( ) g(n) = d d|n d|n was given simultaneously by R. Dedekind [60] and J. Liouville [136] in 1857. Certain M¨obius function identities are important also in Sieve theory (see H. Halberstam and H.-E. Richert [84]). Identity (17) is a characteristic property of the M¨obius function, see S. Swetharanyam [200] and U. V. Satyanarayana [169]. 103

CHAPTER 2

Interesting generalized M¨obius inversion formulae have been obtained in 1963 by D. E. Daykin [58]. Relation (17) can be written also as g(n) = f (x) (d x = n) ⇔ f (n) = µ(e)g(y) (ey = n) for n ≥ 1. Let D, E, X, Y be non-empty sets of positive integers and let k : N → R be an arithmetic function. Given a pair ( f, g) of arithmetic functions, we define another pair (F, G) by the relations G(n) = f (x) (d x = n; d ∈ D, x ∈ X ), (18) F(n) = k(e)g(y) (ey = n; e ∈ E, y ∈ Y ) (19) Empty sums take the value 0. We say that (D, E, X, Y, k) form a M¨obius system if, for all pairs ( f, g) of arithmetic functions satisfying f (n) = 0 (n ∈ X ) and g(n) = 0 (n ∈ Y )

(20)

F(n) = f (n) ⇔ G(n) = g(n)

(21)

one has

If A, B are sets of integers, let A · B denote the set of all products a ·b with a ∈ A, b ∈ B (i.e. Minkowski product). The following result is true ([58]): (D, E, X, Y, k) is a M¨obius system iff both H (n) = δ(n), where H (n) = k(e) (de = n; d ∈ D, e ∈ E), (22) and X = Y = X · D = X · E.

(23)

A set C of integers is called a product set if C can be generated multiplicatively from a sequence (which could be finite, infinite, or empty) ( pi ) of pairly coprime positive integers, such that µ( pi ) = −1 for all i. We assume that empty products take the value 1, and that 1 is in every product set. Moreover, the trivial set {1} is a product set. The following generalization of the M¨obius inversion formula holds true: Suppose that n ∈ D ⇔ n ∈ E whenever k(n) = 0

104

(24)

¨ THE MOBIUS FUNCTION

and that k(n) = 0 whenever n has a square factor. Then (22) holds iff D is a product set, D = E and k(n) = µ(n) for n ∈ D. Let D = {1, 2, 8}, E = {2α : α ≥ 0}, X = Y = N, k(1) = k(4) = 1, k(2) = −1, k(2α )+k(2α−1 )+k(2α−3 ) = 0 (α ≥ 3). Then (D, E, X, Y, k) is a M¨obius system. If D is an isomorphic image of N = {1, 2, . . . } (which is a multiplicative semigroup) and k is the function induced by µ through the isomorphism, then (D, D, N, N, k) is a M¨obius system. The particular case of the isomorphism i → i k (k fixed positive integer) has been considered by P. J. McCarthy [34]. For a generalization of an inversion formula due to Vinogradov, see R. T. Hansen [89]. A unified treatment of the M¨obius inversion formula appears in H. Breitenfellner [21]. A difference-operatorial approach to the inversion formulae is given in L. C. Hsu [114]. We note that the M¨obius function of a set of primes P was considered by A. Wintner in 1943 [220]. This is a multiplicative function µ P such that µ P ( p k ) = −1 for p ∈ P and k = 1 and 0, otherwise. Wintner proved that if 1/ p is divergent, then

∞

p∈P

µ P (n)/n = 0.

n=1

The formula (17) exemplifies the combinatorial nature of the M¨obius function. In 1964 G. C. Rota [161] and his cowerkers have developed a general theory of M¨obius function on partially ordered sets, which includes the principles of inclusionexclusion and M¨obius inversion as special cases (but, as we shall see in the following sections, many ideas and results were anticipated by S. Delsarte [61] or R. Wiegandt [219]). For many results on various estimates, inequalities, as well as many related notions and functions (as the Mertens function M(x) = µ(n)) on the classical n≤x

M¨obius function, see our Handbook I ([142]). This Chapter has, among many other things, also the aim to call the attention to many rediscoveries in the subject. But the main goal will be to survey the most important notions and results which involved the implications of M¨obius type functions (1). We will survey many generalizations in Number theory of the M¨obius function, as well as analogues functions which arose by the extensions of certain divisibility notions or product notions of arithmetical functions. Our survey will include also M¨obius functions in Group theory, Lattice theory, Partially ordered sets or Arithmetical semigroups. Sometimes references to applications will be pointed out, too.

105

CHAPTER 2

2.2 1

M¨obius functions generated by arithmetical products (or convolutions) M¨obius functions defined by Dirichlet products

The Dirichlet product ”*” of two arithmetic functions f and g is defined by ( f ∗ g)(n) =

d|n

n f (d)g( ) d

(1)

where d | n denotes that d runs through all divisors d of n. The operation ”*” was considered as a binary operation on the set F of all arithmetical functions only at the beginning of the 20th century. M. Cipolla and E. T. Bell [10] proved first that (F, +, ∗) is an integrity domain. E. D. Cashwell and C. J. Everett [38] proved that this is in fact a factorial ring. Let F1 = { f ∈ F | f (1) = 0}, MF = { f ∈ F | f multiplicative and f ≡ 0}. It can be proved that (F1 , ∗) is a commutative group, while (MF, ∗) is a subgroup of this Abelian group. In what follows, we shall use more times the arithmetical functions defined by E k (n) = n k , E(n) = n, e(n) = 1

(n ∈ N)

(2)

and δ(n) =

1, n = 1 . 0, n > 1

(3)

Clearly e(n) = E 0 (n). By using the above group terminolgy, the inverse of the function e is exactly the M¨obius function µ: (as δ is the unity element of this group: f ∗ δ = δ ∗ f = f ), e ∗ µ = µ ∗ e = δ. We note that for the classical arithmetical functions ϕ, σ, d (representing Euler’s totient, the sum of the divisors and the number of the divisors, respectively) we have ϕ = µ ∗ E, σ = E ∗ e, d = e ∗ e

(4)

or more generally Jk = µ ∗ E k , σk = E k ∗ e, dk = dk−1 ∗ e

(k ≥ 2)

(5)

where Jk is Jordan’s totient; σk is the sum of k-th powers of divisors; and dk is the Piltz divisor function (for more details, see e.g. [3], [90], [36], [177], [162]). 106

¨ THE MOBIUS FUNCTION

In 1963 in a little known paper, C. Popovici [151] introduced and studied a generalized M¨obius function, defined by µk = µ ∗ µ ∗ . . . ∗ µ

(6)

k

where k ≥ 1. It is immediate, by induction that k (7) µk ( p ) = (−1) r k where the binomial coefficient = 0 for k < r . Popovici proved more properties, r e.g. that µk is multiplicative; µk−1 = µk ∗ e, etc. Certain combinatorial applications, group theoretical considerations, as well as extensions of (6) to other functions in place of µ, can be found in a paper by E. Schwab and E. D. Schwab [174]. In 1998 L. C. Hsuand J. Wang [115] rediscovered this function. By a generalized α binomial coefficient for complex number α, via (7) this M¨obius function can be r extended, by r r α (8) µα ( p ) = (−1) r r

r

and the assumption that µα is multiplicative, i.e. µα (n) =

ν p (n)

(−1)

p|n

α ν p (n)

(9)

where ν p (n) denotes the power of the prime number p in the factorization of n. This definition, as well as the repetition of some known properties are included in a paper by T. C. Brown, L. C. Hsu, J. Wang and P. J-S. Shiue [23]. We note that in 1996 R. G. Buschman [25] has introduced also µk for a positive integer k, by another definition, namely as the Dirichlet inverse of the Piltz divisor function dk (in fact dk = µ−k ). He has remarked that M¨obius generalized functions in this way were introduced also by E. D. Cashwell and C. J. Everett [38], and even by R Vaidyanathaswamy [210]. See also H. Scheid [170] and P. J. McCarthy [36]. More recent investigations by us reveal that µk was in fact first introduced in the literature in 1915 by A. Fleck [69]. For applications of µk in the theory of Diophantine Equations, see R. Tschiersch [208]. 107

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Extensions of µk by regular convolutions will be studied later. In 1968 D. Rearick [155] defined the k-th power of an arithmetic function by fk = f ∗ f ∗ ... ∗ f .

(10)

k

Clearly, for f = µ, by (6) one reobtains µk . Let A denote the set of all real valued arithmetic functions, and let P denote the subset of A consisting of arithmetical functions f with f (1) > 0. Rearick [155] defines the logarithm operator Log : P −→ A by Log f (1) = log f (1), n 1 Log f (n) = log d, f (d) f −1 f (n) d|n d

(n > 1)

(11)

where f −1 is the Dirichlet inverse of f . It can be proved that Log is a bijection of P onto A; therefore it is possible to define the exponential operator Exp : A −→ P,

Exp = Log−1 .

(12)

Let f ∈ P. The real power of f , i.e. f α (α ∈ R) is defined by f α = Exp(αLog f ).

(13)

Then, it can easily be shown that f −k = f −1 ∗ f −1 ∗ . . . ∗ f −1 k

where k is a positive integer. T. Caroll and A. A. Gioia [33] determined explicitely the powers f α of com1 1 pletely multiplicative functions for α = − and α = , and for positive rational 2 2 numbers α. In 1997 P. Haukkanen [97] proved the following surprising fact: If f is a completely multiplicative function, then f α = µ−α · f

(14)

for all real numbers α, where µ−α is the generalized M¨obius function of (8), (9) (put ”−α” in place of ”α”).

108

¨ THE MOBIUS FUNCTION

For a recent application of the function µα , see V. Laohakosol, N. Pabhapote and N. Wechwiriyakul [130]. By applying (14), in [130] the following is proved: Let f be a nonzero multiplicative function and α a nonzero real number. Then f is completely multiplicative iff (µα f )−1 = µ−α f

(15)

For α = 1 this gives a result of T. M. Apostol (see [2], p. 49). Let us assume now that if α is a negative even integer, then f ( p −α−1 ) = f ( p)−α−1 for each prime p.

(16)

Let f be a nonzero multiplicative function and α ∈ R \ {0, 1}. Assuming condition (16), if f α = µ−α f, then f is completely multiplicative. (17) For details, see [130]. Finally note that the Dirichlet product of arithmetic functions can be extended to functions f (n 1 , . . . , n k ) of k variables: f (a1 , . . . , ak )g(b1 , . . . , bk ) ( f ∗ g)(n 1 , . . . , n k ) = (18) ajbj = n j j = 1, . . . , k The arithmetic function δk (n 1 , . . . , n k ) =

1, n 1 = . . . n k = 1 0, otherwise

is the identity element for this operation. In 1975 Paula A. Kemp [123] defined a M¨obius function of n arguments, ∼

µ (n 1 , . . . , n k ) = µ(n 1 ) · · · µ(n k )

(19)

and proved an inversion theorem. The theory of such functions, however (including (19)) was first extensively studied by R. Vaidyanathaswamy [210] (see sec. 4 for (19)). See also a paper by M. V. Subbarao [191] (pp. 261-264).

109

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2

Unitary M¨obius functions The unitary product of two arithmetical functions f, g ∈ F is defined by n ( f g)(n) = f (d)g d d|n

(20)

(d, dn )=1

n i.e. one consider a sum on divisors d of n with d, = 1. The idea of a such d convolution is due to R. Vaidyanathaswamy [210] again, but its extensive study was initiated by E. Cohen [47], [48]. It can be shown that (F, +, ) is a commutative ring with unity element, (F, ) is an Abelian group having as (MF, ) as a subgroup. The function δ is also the unity element, while the inverse of e will be denoted, by analogy, by µ∗ , as being the M¨obius function. In fact µ∗ (n) = (−1)ω(n) ,

(21)

where ω(n) denotes the number of distinct prime divisors of n. By analogy with (4), one has ϕ ∗ = µ∗ e, σ ∗ = E e, d ∗ = e e, dk∗

=

∗ dk−1

(22)

e,

(k ≥ 2).

In [173] A. Schinzel proved that if f (1) = 1, the inverse function of f under the unitary convolution exists and is given by the formulae g(1) = 1 and g(n) =

ω(n) k=1

(−1)k

k

f (di ),

d1 d2 ···dk =n i=1

for n > 1, where ω(n) is the number of distinct prime factors of n. For many properties of these functions, we quote only E. Cohen [47], [48], M. V. Subbarao [191], D. Suryanarayana [194] [195], J. Chidambaraswamy [43], J. S´andor [162], J. S´andor and L. T´oth [166] (see also the monograph Mitrinovi´c-S´andor-Crstici [142]). In 1971 E. M. Horadam [110] has extended Daykin’s M¨obius inversion formula for unitary divisors. Let D, E be non-empty sets of positive integers. Let k be an arithmetic function such that k(n) = ±1 whenever n ∈ E. 110

(23)

¨ THE MOBIUS FUNCTION

Consider the statement H (n) = δ(n), where H (n) =

k(e)

(de = n, (d, e) = 1, d ∈ D, e ∈ E). (24)

A set C of positive integers is called a unitary product set if it can be generated multiplicatively from a finite or infinite set of positive integers ( pi ), mutually coprime. A general member of the set of generators will be denoted by pa(i) . If n ∈ C, αk α1 then n has the form n = pa(1) . . . pa(k) , but as we shall mainly be concerned with the evaluation of H (n), we shall take n = pa(1) . . . pa(k) . The unitary M¨obius function of the set C is defined by µC∗ (1) = 1, µC∗ ( pa(i) ) = −1, and if n = pa(1) . . . pa(k) , then µC∗ (n) = (−1)k .

(25)

Now, related to the statement (24), the following is true: Suppose that (23) holds true. Then (24) is true if and only if D is a unitary product set, D = E, and k(n) = µ∗D (n) for all n ≥ 1.

3

(26)

Bi-unitary M¨obius function

We introduce the bi-unitary convolution formula (see [100]). A divisor d > 0 of the positive integer n is called bi-unitary if the greatest common unitary divisor of n d and is 1. We denote by (a, b)1 the greatest unitary divisor of both a and b. The d bi-unitary product of two arithmetical function f, g ∈ F is defined by n . (27) ( f g)(n) = f (d)g d d|n (d, dn )1 =1

The properties of these convolutions appear in [100]. If we consider the greatest common bi-unitary divisor of a and b and denote by n (a, b)2 , we call d triunitary divisor of n if (d, )2 = 1. A. Bege introduced in [8] the d following generalization of M¨obius’ function:  1, if n = 1  ∗∗ ∞ α p (28) µ (n) = pα − k  k=1 2 , if n > 1 (−1) pα n and proved the following inversion formula. If f, g ∈ F we have n g(n) = . f (d) ⇐⇒ f (n) = µ∗∗ (d)g d d|n d|n (d, dn )1 =1

(d, dn )1 =1

111

(29)

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4

M¨obius functions generated by regular convolutions

The regular convolution will be a common generalization of the Dirichlet and the unitary product. Let A(n) be a subset of the set of positive divisors of n. The A-convolution of two arithmetic functions f and g is defined by

( f ∗ A g)(n) =

f (d)g

n

d∈A(n)

d

.

(30)

In 1963 W. Narkiewicz [145] defined an A-convolution to be regular if (a) the set of arithmetical functions forms a commutative ring with unity with respect to the ordinary addition and A-convolution; (b) the A-convolution of multiplicative functions is multiplicative; (c) the function e has an inverse µ A with respect to the A-convolution, and µ A (n) = 0 or −1 whenever n is a prime power. It is known that ([145]) an A-convolution is regular iff: (i) The conditions d ∈ A(n), m ∈ A(n) and d ∈ A(n),

n m ∈A d d

are equivalent; n (ii) d ∈ A(n) implies ∈ A(n); d (iii) {1, n} ⊆ A(n); (iv) A(mn) = {de | d ∈ A(m), e ∈ A(n)}, whenever (m, n) = 1; (v) for each prime power pa (a ∈ N) there exists a positive integer t = t A ( pa ) such that A( pa ) = {1, p t , p 2t , . . . , p st }, where st = a and p t ∈ A( p 2t ), p 2t ∈ A( p 3t ), . . . , p (s−1)t ∈ A( pa ). For example, the Dirichlet convolution D = ” ∗ ” will be obtained by D(n) = the set of all positive divisors ofn, while the unitary convolution U = ” ” follows n = 1}. It is immediate that these convolutions from U (n) = {d > 0 | d | n, d, d are regular. In what follows we shall assume that A is a regular convolution. The generalized M¨obius function µ A is the multiplicative function (introduced by W. Narkiewiecz 112

¨ THE MOBIUS FUNCTION

[145]) given by µ A ( pa ) =

 

1, if a = 0 −1, if pa > 1 is A-primitive .  0, if pa is non A-primitive

(31)

We note that a positive integer n is said to be A-primitive, if A(n) = {1, n}. Let Ak (n) = {d ∈ N | d k ∈ A(n k )}. In 1978 V. Sita Ramaiah [178] considered Ak -convolutions. The corresponding M¨obius function µ Ak therefore may be named as ”Sita Ramaiah’s M¨obius function”. He proved that the Ak -convolution is regular whenever the A-convolution is regular. In 1995 L. T´oth [205] defined the notion of a ”cross-convolution”. Let A be a regular convolution. Then A is named as a cross convolution if for every prime p we have either t A ( pa ) = 1 ( i.e. A( pa ) = {1, p, p 2 , . . . , pa } ≡ D( pa ), for every a ∈ N), or t A ( pa ) = a ( i.e. A( pa ) = {1, pa } ≡ U ( pa ), for every a ∈ N). It follows by the work of Sita Ramaiah [178] that if A is a cross convolution, then Ak = A for every k. For arithmetical functions involving cross convolutions, see L. T´oth [206] and L. T´oth and J. S´andor [207]. Let now S be an arbitrary subset of N and A a regular convolution. Let S be the characteristic function of the set S, i.e. 1, if n ∈ S

S (n) = . 0, if n ∈ S P. Haukkanen [93] defined the M¨obius function µ S,A by µ S,A ∗ A e = S ,

(32)

i.e.

µ S,A (d) = S (n).

d∈A(n)

If S = {1}, then µ S,A ≡ µ A -the Narkiewiecz M¨obius function. We note that the function µ S,D (where D is the Dirichlet convolution) was introduced by E. Cohen in 1959 [46], and considered also by M. V. Subbarao and V. C. Harris [192]. 113

CHAPTER 2

Other interesting particular cases of the Haukkanen M¨obius function appear in [93]. For example, let us define n to be a square with respect to the A-convolution if for each pa n (i.e. pa = 1, pa | n, pa+1 n), we have a = 2st, t = t A ( pa ). Now, if S is the set of all squares with respect to the A-convolution, then µ S,A = λ A , where 1, if n = 1 , λ A (n) = a1 +...+ar (−1) , if n = p1a1 t1 · · · prar tr with ti = t A ( piai ti ). We note that λ A is a generalized Liouville function. In [93] a generalized Ramanujan sum is considered, too. Following the paper [98] we say that an integer n > 1 is (A, r )-powerful if for each A-primitive prime power p t ∈ A(n) we have θ ( p t ) ≥ r and pr t ∈ A(n), where θ ( p t ) = sup{s ∈ N | t A ( p st ) = t}. Let A be a regular convolution. An arithmetical function f is said to be An whenever multiplicative (see K. L. Yocom [221]) if f ≡ 0 and f (n) = f (d) f d d ∈ A(n). An arithmetical function f is said to be A-rational function of order (s, r ) if there exist A-multiplicative functions f 1 , . . . , f s and g1 , . . . , gr such that f = f 1 ∗ A f 2 ∗ A . . . ∗ A f s ∗ A (g1 )−1 ∗ A (g2 )−1 ∗ A . . . ∗ A (gr )−1

(33)

This notion in case of A = D was introduced by R. Vaidyanathaswamy [210]. A combinatorial interpretation of a generalized Euler function, involving Arational arithmetic function, and the A-extensions of the Fleck-Popoviciu-BuschmanHsu-Wang M¨obius function is contained in [99]: µ A,k = (µ A ) ∗ A (µ A ) ∗ A . . . ∗ A (µ A ) .

(34)

k

Another M¨obius function µ B , which is an analogue of µ A appears in [99].

5

K -convolutions and M¨obius functions. B convolution

The K -convolution or Davison-convolution of the arithmetical functions f and g is defined by n K (n, d), (35) ( f ◦ g)(n) = f (d)g d d|n where K is a complex valued function on the set of all ordered pairs (n, d) such that n is a positive integer and d is a positive divisor of n. The concept of K -convolution 114

¨ THE MOBIUS FUNCTION

originates to T. M. K. Davison [56] from 1966. In the case in which K (n, d) depends n only on the g.c.d. (d, ), the concept is due to A. A. Gioia [75] and A. A. Gioia and d M. V. Subbarao [76]. Further study of K-convolutions is included in M. Ferrero [68], I. P. Fotino [72], M. D. Gessley [74], P. J. McCarthy [35], P. Haukkanen [92]. It is known (see the above References) that the set MF of multiplicative arithmetical functions forms an Abelian group with identity with respect to the K convolution iff (a) K (n, n) = K (n, 1) = 1 for all n; (b) K (mn, de) = K (m, d)K (n, e) for all m, n, d, e such that d | m, e | n, (m, n) = 1; (c) K (n, d)K (d, e) = K (n, e)K

n d , e e

for all n, d, e such that d | n, e | d; (d) n K (n, d) = K n, d for all n, d with d | n. For example, the regular convolution by Narkiewcz satisfies (a)-(d). The k-th K -iterate of an arithmetical function f is defined by f (k) = f ◦ f ◦ . . . ◦ f

(36)

k

Clearly, this is a generalization of M¨obius’ function considered in former paragraphs (see e. g. (10), (35)). Clearly f (k) (n) = f (a1 ) · · · f (ak )K (n, a1 )K (a2 · · · ak , a2 ) · · · K (ak−1 ak , ak−1 ) a1 a2 ···ak =n

(37) The inverse of an arithmetic function f with respect to the K -convolution is defined by f ◦ f (−1) = f (−1) = δ. The inverse exists and is unique iff f (1) = 0 (see [56]). 115

CHAPTER 2

In paper [94], among other results, the following is proved: Suppose f is an arithmetical function such that f (1) = 1. Then f (r ) is multiplicative iff f is multiplicative. Questions of ”quasi-multiplicativity” as well as ”semi-multiplicativity” are also considered (see also D. Rearick [154]). An interesting particular case, introduced in the book J. McCarthy ([36], p. by P. ν p (n) 168) is the so called ”binomial convolution”. For n = p|n p define the function B(n, d) as ν p (n) (38) B(n, d) = ν p (d) p|n a where ν p (n) denotes the power of the prime p in the canonical form of n, and b is a binomial coefficient. The binomial convolution follows from (35), when K ≡ B. In what follows, we shall use the notation ” ◦ ” = ” ◦ B ” and the binomial convolution will be called simply as the ”B-convolution”. We note, that the B-convolution shares many basic properties with the Dirichlet convolution. For example, the set of all arithmetic functions f with f (1) = 0 forms a group under the B-convolution, and the set of all multiplicative functions forms a subgroup of this group. On the other hand, the B-convolution has properties which differ from those of the Dirichlet convolution. A fundamental diference is that the Bconvolution preserves complete multiplicativity, while the Dirichlet product of two completely multiplicative functions generally is not completely multiplicative (but it is ”specially multiplicative” (see [36], p. 24 and [181])). The following interesting theorem is proved in [96]. We have f (n) = g(d)B(n, d) ⇐⇒ g(n) = f (d)λ(d)B(n, d). (39) d|n

d|n

where λ(n) = (−1) (n) is the Liouville function. This means that the inverse e−1 of the function e, related to a B-convolution, is the Liouville function. In other words, µ◦ B (n) = λ(n)

(40)

i.e. the Liouville function appears here as the M¨obius function generated by the B-convolution. 116

¨ THE MOBIUS FUNCTION

The B-convolution has also an interesting connection to the convolution of sequences. It is well known that the binomial convolution of sequences (an )n≥0 , (bn )n≥0 is a sequence (cn )n≥0 given as n n ak bn−k cn = k k=0

(41)

(see R. L. Graham, D. E. Knuth and O. Patashnik [80], section 7.6). If f and g are multiplicative functions, then their binomial convolution f ◦ B g is completely determined by their values at all prime powers p n . For a fixed prime p, let f ( p n ) = an and g( p n ) = bn for all n ≥ 0. Then n n n f ( p k )g( p n−k ) = ( f ◦ B g)( p ) = k k=0 n n ak bn−k = cn = k k=0 This is similar to the connection between Dirichlet convolution and Cauchy product n

ak bn−k .

k=0

For related questiones, involving e.g. ”roots of sequences” under the Dirichlet and binomial convolution, see [95].

6

Exponential M¨obius functions The exponential convolution was defined by M. V. Subbarao in 1972 [191] by  ( f ∗ex g)(1) = f (1)g(1)    nk (42) n1 d1 d d1  dk  ... f p1 · · · pk g p1 · · · pk k  ( f ∗ex g)(n) = d1 |n 1 d2 |n 2

dk |n k

where n = p1n 1 p2n 2 · · · pkn k is the prime factorization of n. It is immediate that this convolution is commutative n and associative. This is not . of Narkiewicz type, not being of the form f (d)g d d|n Subbarao proved the following results: 1) (F, ∗ex ) is a commutative semigroup with identity element |µ| (where µ is the classical M¨obius function); 117

CHAPTER 2

2) The units of (F, ∗ex ) are those functions f for which f (n) = 0 whenever n is a product of distinct primes and f (1) = 0; 3) The semigroup (F, ∗ex ) has an infinity of zero divisors. An element f of F, ∗ex ) is a non zero divisor only if, given any finite number of primes p1 , . . . , pr , there exist coresponding positive integers a1 , . . . , ar such that f p1a1 · . . . · prar = 0; 4) (F, ∗ex ) has no nontrivial nilpotent elements. He also defined the exponential analogue µ(e) of the M¨obius function as follows: µ(e) (1) = 1; µ(e) (n) = µ(a1 ) · . . . · µ(ar ),

(n > 1)

(43)

where n = p1a1 · . . . · prar . Then µ(e) is multiplicative and e-multiplicative (i.e. satisfies the functional equation f ( pab ) = f ( pa ) f ( p b ), for (a, b) = 1 and all primes p, where f is multiplicative). The exponential inverse of the function given by e(n) = 1, (n ∈ N) will be exactly µ(e) . Therefore f = g ∗ex e ⇐⇒ g = f ∗ex µ(e) .

(44)

Let now A be a regular convolution of Narkiewicz type. The exponential Aconvolution of the functions f and g has been introduced by J. Hanumanthachari [87] and K. Shindo [176]:  ( f ∗ Aex g)(1) = f (1)g(1)    nk n1 d d1  dk d1  ... f p1 · · · pk g p1 · · · pk k  ( f ∗ Aex g)(n) = d1 ∈A(n 1 ) d2 ∈A(n 2 )

dk ∈A(n k )

(45) where n = p1n 1 p2n 2 · · · pkn k is the prime factorization of n. Hanumanthachari proved that (F, ∗ Aex ) is a commutative semigroup with identity |µ|. Units are those functions f , for which f (γ (n)) = 0, where γ (n) = p1 p2 · · · pk is the greatest squarefree divisor (or the ”core”) of n. Further, it can be verified that the set of all exponentially multiplicative functions forms an Abelian 118

¨ THE MOBIUS FUNCTION

group under the exponential A-convolution. For example, the inverse of the function e is e−1 = µ(e) A , given by µ(e) A (1) = 1; µ(e) A (n) = µ A (n 1 ) · . . . · µ A (n k ),

(n > 1)

(46)

where n = p1n 1 · . . . · pkn k . Clearly, µ(e) A is multiplicative and exponentially multiplicative. For the exponential totient function, see J. S´andor [163], P. Haukkanen and P. Ruokonen [103] and for a survey of results, including properties on asymptotics, see Mitrinovi´c-S´andor [142]. Exponentially A-multiplicative functions, as well as their Apostol type characterizations, appear in [103]. For example the following result is proved: An exponentially multiplicative function f is exponentially A-multiplicative (i.e. n f (γ (n)) = 0 and f ( p n ) = f ( p d ) f ( p d ) for all primes p and all d, n with d ∈ A(n)) iff f −1 = µ(e) A f,

(47)

where µa(e) is given by (46). For a notion and results related to strong regular A-convolution, see A. C. Vasu, V. V. Subrahmaya Sastri and C. S. Venkataraman [211].

7

l.c.m.-product (von Sterneck-Lehmer)

The von Sterneck-Lehmer, or l.c.m.-product, appeared in R. D. von Sterneck [190] and studied by D. H. Lehmer [131], is defined by f (k)g(m) (48) ( f g)(n) = [k,m]=n

where [k, m] denotes the l.c.m. of k and m. von Sterneck proved the following connection between l.c.m. product and Dirichlet product: ( f g) ∗ e = ( f ∗ e) · (g ∗ e)

(49)

where e(n) = 1, ∀n ∈ N, and ”·” is the pointwise product. From this relation directly follows that the l.c.m.-product of multiplicative functions is multiplicative. It can be verified that there exist non-trivial divisors of zero for this product. Let f be any function such that f (1) = 0 and let g = µ. In fact, H. Scheid [170] pointed out that it is easy to show that if f = δ, and f f = f, 119

CHAPTER 2

then f is a zero divisor. If however, (g ∗ e)(n) is non zero for all n, then from (49) it is an easy proof to show that f g = 0 ⇒ f = 0. Consequently, we can prove that if g is a non-negative function which has a Dirichlet inverse, then we have the cancellation property: f g = h g ⇒ f = h.

(50)

In order to state an inversion formula for the l.c.m. product, Lehmer [131] introduced the arithmetical function  if n = 1  1, r r Ms (n) = (51) {(ak + 1)s − aks }, if n = pkar  k=1

k=1

where s ∈ R. Ms is a common generalization of δ and e, since M0 = δ, M1 = e. In fact k Mk = e e . . . e = µ ∗ d (k ≥ 0),

(52)

k

see R. G. Buschman [29]. Lehmer proved the following inversion theorem of M¨obius type: f = e g ⇐⇒ g = M−1 f.

(53)

This means that M−1 provides the ”M¨obius function” analogue in the case of l.c.m. products: µ ≡ M−1 .

(54)

We note that M−1 can be written also as M−1 (n) =

r k=1

1 1 − ak + 1 ak

=

r

(−1)ω(n)

k=1

120

ak (ak + 1)

¨ THE MOBIUS FUNCTION

A list of l.c.m. products appears in [29]. We quote only the following interesting relations: f δ =

f

f µ =

f (1)µ

ϕ ... ϕ =

Jk

k

ϕ Jk

=

Jk+1

Mk Mm = Mk+m Jh Jk

=

Jh+k

∗

µ (d · λ) = λ.

8

Golomb-Guerin convolution and M¨obius function

The Golomb-Guerin convolution will be defined as follows. Let F denote the set of elements of positive integers which are not perfect powers > 1 (thus 1 ∈ F). Any integer n ≥ 2, n = p1a1 · · · prar can be expressed as n = m g , where g = g.c.d.(a1 , . . . , ar ) and m ∈ F. In 1973 S. W. Golomb [79] defined the ”root function” r (n) equal the number of distinct representations n = a b , with a, b positive integers, and noted that r (n) = d(g) for n = m g , (m ∈ F, g ∈ N), with d denoting the number of distinct divisors. For two arithmetical functions f, g and n = m g , (n ∈ F, g ∈ N), E. E. Guerin [81] defined the G-convolution as  ( f ∇g)(1) = 1    (55) d g .   f m g md  ( f ∇g)(n) = d|g

Guerin [81] has proved the following result: (1) (F, +, ∇) is a commutative ring with unity δ∇ (where δ∇ (n) = 1 for n = 1 or n ∈ F; δ∇ (n) = 0 otherwise). (2) f is a unit in (F, +, ∇) iff f (1) = 0 and f (m) = 0 for all m ∈ F. (3) An arithmetic function f is a nondivisor of zero iff f (1) = 0 and for each m ∈ F there exists a positive integer g such that f (m g ) = 0. He also showed that the set of G-multiplicative functions (i.e satisfying f (1) = 1 and f (m ab ) = f (m a ) f (m b ) for m ∈ F, and (a, b) = 1) which are units in (F, +, ∇) form an Abelian group.

121

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Now, as usual, the inverse of e with respect to ∇ will be named a M¨obius function. Put e∇µ∇ = δ∇ , i.e.

µ∇ (n) =

(56)

1, if n = 1 . µ(g), if n = m g (m ∈ F, g ∈ N)

(57)

This is the Guerin analogue of the M¨obius function. Euler’s totient and the sum of divisors functions will be introduced as ϕ∇ (n) = (µ∇ ∇ E)(n) = g µ(d)m d (n = m g (m ∈ F, g ∈ N)), =

(58)

d|g

σ∇ (n) = e∇ E. It can be verified that ϕ∇ and σ∇ are not G-multiplicative functions.

9

max-product (Lehmer-Buschman) The Lehmer-Buschman or max product is defined by f (k)g(m) ( f ♦g)(n) =

(59)

max{k,n}=n

introduced by D. H. Lehmer [133]. Lehmer proved that the ♦ product is associative, commutative and has δ as the identity. There are nontrivial divisors of zero. A fundamental identity proved by Lehmer is ( f ♦g)#e = ( f #e) · (g#e),

(60)

where the product ”#” is defined by ( f #g)(x) =

k≤x

f (k)g

x k

,

(61)

which has a name ”Cauchy product”. There exists a useful identity which connects Dirichlet and # products (see T. M. Apostol [3], I. Niven-H. S. Zuckerman [148]): f #(g#h) = ( f ∗ g)#h. 122

(62)

¨ THE MOBIUS FUNCTION

As corollaries of (62) note that: (1) if f (1) = 0, then f #g = f #h ⇒ g = h; (2) if S is the greatest integer function f = e ∗ g ⇒ f #e = g#S; (3) f #(g#h) = g#( f #h). Left inversion does have the same form as the Dirichlet inversion, and it follows by the use of (62), but the order of factors is now important: f = e#g ⇐⇒ g = µ# f.

(63)

Right inversion is a different matter. We have f = g#e ⇐⇒ g = f,

(64)

where f (1) = f (1) and f (n) = f (n) − f (n − 1), (n ≥ 2). Since the inverse operator for the right #-multiplication by e is the difference operator , we can write another form for (60): ( f )♦(g) = ( f · g).

(65)

By manipulating with the operator and equation (65), we can derive the right inversion formula of M¨obius type ([28]): f = e♦g ⇒ g = (E −1 )♦ f,

(66)

where (E −1 )(n) =

1 1 − , n n−1

(n > 1).

More generally, it can be proved [28]: f = e♦ . . ♦e ♦g ⇐⇒ g = (E −k ) ♦ f. .

(67)

k

Remark that letting in (66) f ≡ δ, we can deduce a ”right-M¨obius function” µ♦ by µ♦ = (E −1 ) ♦δ. 123

(68)

CHAPTER 2

10

Infinitary convolution and M¨obius function

The infinitary divisors have been introduced also in Chapter I (see paragraph 9). A slightly distinct introduction can be done via the I -components of a number. Let α I = { p 2 : p prime, α ≥ 0}. It is easy to see that n > 1 can be written in exactly one way (except the order of factors) as a product of distinct elements of I . We shall call each element of I in this product an I -component of n, and we shall say that d is an I -divisor of n if every I -component of d is also an I -component of n. We write d|∞ n for an I -divisor d of n. Suppose that n = P1 P2 . . . Pt , where P1 < P2 < · · · < Pt are the I -components of n. Put t = J (n) (let (J (1) = 0, by definition). The infinitary M¨obius function µ∞ is given by (see G. L. Cohen and P. Hagis, Jr. [53]) µ∞ (n) = (−1) J (n) .

(69)

It is immediate that

µ∞ (d) = δ(n).

(70)

d|∞ n

The infinitary convolution of two arithmetic functions f and g is defined by n . (71) f (d)g ( f ∗ g)∞ (n) = d d|∞ n Then it follows that the set of arithmetic functions f with f (1) = 0 forms an abelian group with respect to the operation of infinitary convolution, with identity element δ, i.e. δ(n) = 1, n = 1, δ(n) = 0, n > 1. The following M¨obius inversion formula holds true: n F(n) = µ∞ (d). f (d) ⇔ f (n) = F (72) d d|∞ n d|∞ n Another infinitary version of the M¨obius functions appears in sequence A064179 of N. J. A. Sloane’s Encyclopedia [183].

11

M¨obius function of generalized (Beurling) integers

A. Beurling [13] defined the so-called ”generalized integers” as follows: Suppose there is given a finite or infinite sequence of real numbers (”generalized primes”) such that 1 < p1 < p2 < . . . . From the set {l} of all possible p-products, i.e. products p1α1 . . . prαr with α1 , . . . , αr ≥ 0 integers of which all but a finite number are 0. Call 124

¨ THE MOBIUS FUNCTION

these numbers generalized integers and suppose that no two generalized integers are equal if there α’s are different. Suppose also that {l} may be arranged as an increasing sequence 0 < 1 = l1 < l2 < l3 < · · · < ln < . . .

(73)

Denote for simplicity ln = N , lr = R, ld = D etc. Then a notion of divisibility will follow by D|N iff ∃ = lδ : N = D

(74)

Let (R, S) denote the greatest common l-divisor of R = lr and S = ls . Similarly, (R, S)k denotes the greatest of the generalized integers, which are kth powers of generalized integers and which divide R and S. Therefore, if (R, S)k = 1, then R and S are k-prime to each other. E. M. Horadam [111], [112] defined the M¨obius function of a generalized integer N as (−1)k , if N is squarefree among {l} (75) µ(N ) = 0, otherwise V. V. Subrahmanyasastri [193] has introduced the generalization µ(N 1/k ), if N is a kth power of a generalized integer, µk (N ) = 0, otherwise

(76)

By defining the convolution of two arithmetic functions defined on generalized integers by N ( f · g)(N ) = (77) f (D)g d D|N and using (75), as well as (76), M¨obius-type inversion formulae can be deduced. Similarly, various totient functions or Ramanujan sums can be introduced and studied (see e.g. [111], [112], [193]). For a survey of solved and unsolved problems on generalized integers see E. M. Horadam [113].

12

Lucas-Carlitz (l-c) product and M¨obius functions Let p be a prime and for the nonnegative integers n and r put n = n 0 + n 1 p + n 2 p2 + . . .

(0 ≤ n j < p),

r = r0 + r1 p + r2 p 2 + . . .

(0 ≤ r j < p).

125

CHAPTER 2

A famous theorem of E. Lucas [139] states that n0 n1 n2 n = . . . (mod p) r r0 r1 r2

(78)

n = Cnr is prime In particular, it follows from (78) that the binomial coefficient r to p if and only if 0 ≤ rj ≤ nj

( j = 0, 1, 2, . . . )

(79)

We accordingly define the Lucas-Carlitz product [31] by ( f ∗l−c g)(n) =

n ∗

f (r )g(n − r )

(80)

r =0

where the summation is restricted to those r that satisfy (79). Here f and g are arithmetic functions defined on the set of nonnegative integers. It is immediate from the definition that the l-c product is both associative and commutative; also it is distributive with respect to the usual addition of function. The function z(n) = 0 (n = 0, 1, . . . ) is the identity for addition, while the function δ modified to δ0 (n) = 1, n = 0; δ0 (n) = 0 if n > 0 becomes the multiplicative identity for (80). If f ∗l−c g = δ0 , we call g the inverse of f . The following theorem holds (see [31]): An arithmetic function f possesses an inverse if and only if f (0) = 0.

(81)

Particularly, the function e0 given by e0 (n) = 1 (n = 0, 1, 2, . . . ) satisfies the conditions, so has an inverse µl−c : e0 ∗l−c µl−c = δ0

(82)

If n = a0 + a1 p + · · · + ar pr (0 ≤ a j < p), it is easily verified that  if a = 0 r  1 r −1 if a = 1 and µl−c (n) = µl−c (a j p j ) µl−c (ap ) =  j=0 0 if a > 1

(83)

where 0 ≤ a < p. The following inversion theorem is valid: g(n) =

n ∗

f (r ) ⇔ f (n) =

r =0

n ∗ r =0

126

µl−c (r ) f (n − r ).

(84)

¨ THE MOBIUS FUNCTION

If m, n are defined by n = a0 + a1 p + · · · + ar pr (0 ≤ a j < p) and m = b0 + b1 p + · · · + br pr (0 ≤ b j < p) we define (m, n) p =

r

p j min{a j , b j }

(85)

j=0

An arithmetic function f is called factorable if f (0) = 1 and f (m + n) = f (m) f (n) for (m, n) p = 0.

(86)

For example, e0 and µl−c are factorable functions. Related to factorable functions the following general theorem holds true: Let h = f ∗l−c g. Then if any two of the functions f, g, h are factorable the third is also.

As a corollary one deduces that if g(n) =

n ∗

(87)

f (r ), then

r =0

f is factorable ⇔ g is factorable.

13

(88)

Matrix-generated convolution

In what follows we shall consider matrix-generated convolutions and corresponding M¨obius functions. Suppose that G = (gi j ) is an infinite dimensional matrix having elements 0 and 1: gi j = 1 if i = j and gi j = 0 if i > j, and that the 1’s in column n of G appear in rows n 1 , n 2 , . . . , n k (n 1 < n 2 < · · · < n k = n). Let f, g be two arithmetic functions and define the convolution (see E. E. Guerin [82]) ( f ∗G g)(n) =

k

f (n i )g(n k+1−i ),

n = 1, 2, 3, . . .

(89)

i=1

generated by the matrix G. Clearly, ∗G is a commutative operation on the set of arithmetic functions. Let G n be the n × n submatrix of G = (gi j ) with 1 ≤ i ≤ n, 1 ≤ j ≤ n. Examples. 1) The matrix D = (di j ) with di j = 1 if i| j, di j = 0 otherwise, generates the Dirichlet convolution ∗ D . Here Dn is the n × n divisor matrix, and the set {n 1 , n 2 , . . . , n k } is the set of positive divisors of n. 2) The unitary convolution is generated by the matrix U = (u i j ) with u i j = 1 if i ≤ j, and i| j and i and j|i are relatively prime; u i j = 0 otherwise. 127

CHAPTER 2

3) The matrix C = (ci j ) defined by ci j = 1 if i ≤ j, ci j = 0 otherwise, generates a convolution ∗C related to the Cauchy product. Since {n 1 , n 2 , . . . , n k } = {1, 2, . . . , n}, we have ( f ∗C g)(n) = f (1)g(n) + f (2)g(n − 1) + · · · + f (n)g(1). 4) For j−1a fixed prime p, let the matrix L = (li j ) be defined by li j = 1 if i ≤ j , li j = 0 otherwise. The convolution ∗ L generated by L is related to the and p i−1 Lucas-Carlitz product. Define a general M¨obius function µG by e ∗G µG = δ

(90)

where, as usual e(n) = 1 for all n and δ(n) = 1 if n = 1, = 0, n > 1. It is immediate −1 from G −1 n G n = In (the n × n identity matrix) that if G n = (g i j ), then g i j = µ( j)

(91)

for j = 1, 2, . . . , n (where µ is the classical M¨obius function). For example, the elements in row one of D6−1 (see Example 1 above) are µ D (1) = µ(1), µ(2), . . . , µ(6) (in that order). Define now two arithmetic functions A and B by A(n) =

n

gin f (i),

B(n) =

i=1

n

g in g(i).

(92)

i=1

Now suppose that G and ∗G satisfy the following two properties: (i) f ∗G δ = f for all arithmetic functions f ; (ii) ∗G is an associative operation on the set of arithmetic functions (E.g. in Examples 1)-4) properties (i) and (ii) are valid). As from (92) one has A = e ∗G f and by (i), (ii) ( f ∗G g)(n) = ( f ∗G e ∗ B)(n), so g(n) = (e ∗G B)(n) ⇔ B(n) = (g ∗ µG )(n)

(93)

which is a M¨obius inversion formula. For details, see [82]. We stop here the consideration of various convolution products. We note that other convolutions are included in the references (e.g. semi-unitary product [43], Lehmer ψ-product [132], [179], [147], partial product [26], integral product [27], convolutions with unbounded unity [180], etc.). 128

¨ THE MOBIUS FUNCTION

2.3 1

M¨obius function generalizations by other number theoretical considerations Apostol’s M¨obius function of order k

M¨obius functions of order k, introduced by T. M. Apostol [2] in 1970 are defined by the formulas  1, if n = 1,     0, if p k+1 | n for some p, prime αi k r k (1) µk (n) = (−1) , if n = p1 · · · pr pi , with 0 ≤ αi < k,   i>r   1, otherwise. Among other results (e.g. µk is multiplicative) Apostol obtained an asymptotic formula like 1 µk (n) = Ak x + O(x k log x), (2) n≤x

2 1 1 − k + k+1 . Ak = p p p

where

By assuming the Riemann hypothesis, D. Suryanarayana [196] showed that, the error term in (2) can be improved to 4k (3) O x 4k 2 +1 f (x) , where

f (x) = exp{A log x(log log x)−1 }

for some positive constant A. A further generalization of the Apostol function has been provided by A. Bege [9], who considered the application  1 if n = 1,     1 if p k n for each prime p, m r m µk,m (n) = (4) (−1) if n = p · · · p piαi , with 0 ≤ αi < k, r 1   i>r   0 otherwise. We remark that if m = k, µk,k (n) = µk (n). A. Bege [9] proved for x ≥ 3 and m > k ≥ 2 the following asymptotic result: xn 2 αk,m 1 + 0 θ (n)x k δ(x) . µk,m (r ) = (5) ζ (k)ψk (n)αk,m (n) r ≤x (r,n)=1

129

CHAPTER 2

uniformly in x, n and k, where 1− αk,m = p

and αk,m (n) = n

1 p m−k+1 + p m−k+2 + · · · + p m

1−

p|n

;

1 p m−k+1 + p m−k+2 + · · · + p m

;

θ (n) = d ∗ (n) = the number of square-free divisors of n; ψk (n) = n

p|n

1 1 1 + + · · · + k−1 ; p p

where k is an integer ≥ 2, 3

1

δ(x) = exp {−A log 5 x (log log x)− 5 }, with A > 0 an absolute constant. By assuming that the Riemann hypothesis is true, the following asymptotic formula holds xn 2 αk,m 2 + 0 θ (n)x 2k+1 f (x) . µk,m (r ) = (6) ζ (k)ψk (n)αk,m (n) r ≤x (r,n)=1

uniformly in x, n and k, with f (x) = exp {A log x (log log x)−1 }, (A > 0). A main ingredient in the proof is the identity µ(d) qk (δ) µk,m (n) =

(7)

δd m =n (d,δ)=1

where k ≤ m, µ is the classical M¨obius function and qk is the characteristic function of the k-free integers.

2

Sastry’s M¨obius function

A somewhat similar M¨obius generalization has been considered by K. P. R. Sastry [167]. If n = ri=1 piαi , let  1, if n = 1,  0, if n contains a k-th power divisor (8) µk (n) =  (−1)α1 +α2 +...+αr otherwise 130

¨ THE MOBIUS FUNCTION

Clearly, µ2 (n) = µ(n). This function is multiplicative, and the sum Sk (n) = µk (d) d|n

has the property:

Sk ( p α ) =

 0,     

if (i) α < k for k odd if (ii) α ≥ k for k even (9)

  1,   

if (i) α < k for k even if (ii) α ≥ k for k odd

As a corollary one obtains that Sk (n) vanishes iff one of the following holds: (i) n > 1 and admits a kth power divisor, and k is even, (ii) n > 1, k is odd and at lest one αi < k, and k is odd, (iii) n > 1 and does not admit a kth power divisor, and one of αi is odd

(10)

In every other case one has Sk (n) = 1.

(11)

For the sum µk (n) for k = 1, 2, . . . , m one has m 0, if µm (n) = 0, µk (n) = αi (m − l + 1) if µl−1 (n) = 0, µl (n) = 0 and l ≤ m (−1) k=1 The Dirichlet series of this function is   ζ (2s) · 1 , ∞ µk (n)  ζ (s) ζ (ks) =  ns  ζ (2s) · ζ (ks) , n=1 ζ (s) ζ (2ks)

(12)

for k even .

(13)

for k odd

Another property is related to the sum r n βi i=1 µk k = (−1) , d d k |n

where αi ≡ βi (mod k) and 0 ≤ βi < k (1 ≤ i ≤ r ). Particularly, if k is even,  r   1, if αi =even    i=1 n  . µk k =  d r  d k |n    αi =odd.  −1, if i=1

131

(14)

(15)

CHAPTER 2

3

M¨obius functions of Hanumanthachari and Subrahmanyasastri

In 1978 J. Hanumanthachari and V. V. Subrahmanyasastri [88] introduced a M¨obius function µs (M, N ) as follows: Let s and M be any two positive integers; then for any positive integer n define µs (M, N ) =

0, (−(s + λ( p)),

if N has a square factor , otherwise

(16)

p|N

where, for any prime p, λ( p) = λ(M, p) is the number of k ∈ {1, 2, . . . , s} for which (M − k + j, p) = 1 for each j ∈ {1, 2, . . . , s}. In fact, this new M¨obius function arises in perceiving an extended Nagell totient function θs (M, N ) in the light of the definition of a totient function of Vaidyanathaswamy [210]: The Dirichlet product of a totally multiplicative function and the inverse of another totally multiplicative function is called a totient. When s = 1, we get the function µ1 (M, N ) ≡ µ(M, N ) =

0, N (−1)ω( m ) · (−2)ω(m) ,

if N has a square factor , otherwise (17)

where ω(m) denotes the number of distinct prime factors of m; m being the greatest divisor of N relatively prime to M. When M ≡ 0 (mod N ); (16) reduces to µs (N ) =

0, (−s)ω(N ) ,

if N has a square factor , otherwise

(18)

and this reduces to the classical M¨obius function when s = 1. Denote the Jordan-Nagell totient function by θ (s) (M, N ), equal to the number of ordered s-tuples < x1 , . . . , xs > with x j (mod N ), j ∈ {1, . . . , s} such that ((x j ), N ) = 1 and ((M j − x j ), N ) = 1, where (r j ) denotes the greatest common divisor of (r1 , . . . , rs ). This is given by θ (M, N ) = N s

s

p|(M,N )

1 1− s p

p|N

pM

132

2 1− s p

.

(19)

¨ THE MOBIUS FUNCTION

The Schemmel-Nagell totient, denoted by θs (M, N ) is equal to the number of stuples of consecutive integers < x + 1, x + 2, . . . , x + s > with x (mod N ) such that (x + j, N ) = (M − j − (x + j), N ) = 1,

j ∈ {1, . . . , s}.

One can see that θs (M, N ) = 0 if s is greater or equal than any prime factor of N ; otherwise it is given by s + λ( p) . (20) 1− θs (M, N ) = N p p|N When s = 1, θ (1) (M, N ) = θ1 (M, N ) = θ (M, N ) is Nagell’s totient [181]. When N | M, θ (s) (M, N ) and θs (M, N ) reduce to Jordan’s totient Js (N ) and Schemmel’s totient Ss (N ) [181], respectively. Obviously, µs (M, N ) is multiplicative in N . Other properties include: N = e(n), (21) µs (M, d) · M, d d|N where

(M, N ) = 2 (M) , m being the greatest divisor of N , relatively prime to M, and (m) = total number of prime factors of m; N ; (22) dµs M, θs (M, N ) = d d|N (s)

θ (M, N ) =

d µs s

d|N

µ(M, N ) =

N M, d

µ(γ (d))µ

d|N

N d

;

(23)

,

(24)

(d,M)=1

where γ (N ) denotes the largest squareefree divisor of N (”core”of N ). In the paper there appear many other interesting identities, e.g. N ϕ(N ) = θs (M, N ) θs (M, d)µ ; d N d|N d|N

N θ (M, d)µ M, d (s)

133

(25)

2 θ (s) (M, N ) = . Ns

(26)

CHAPTER 2

4

Cohen’s M¨obius functions and totients

The extension of Euler’s totient (as in the above paragraph) leads many times to generalized M¨obius functions. In 1963 E. Cohen [51] defined such an extension, involving a M¨obius generalization. Let p1 , p2 , . . . denote the increasing sequence of primes, and for a positive integer n, let νi (n) denote the multiplicity of pi as a divisor of n for each i ≥ 1. In particular, ν1 (n) = 0 if pi n. Given a positive integer k, n is called k-free if νi (n) < k for every i ≥ 1, and is k-full if νi (n) ≥ k for every i for which νi (n) = 0. Let Q k , L k denote the set of k-free and k-full integers respectively. Since n = n 1 · n 2 with (n 1 , n 2 ) = 1, n 1 ∈ Q k , n 2 ∈ L k ; the part n 1 = αk−1 (n) is called the k-free part, while n 2 = βk−1 (n) the k-full part of n. Let ϕ r (n) denote the number of integers (mod n) relatively prime to αr (n). Then ϕ r (n) = βr (n)ϕ (αr (n)) , where ϕ is the classical Euler totient. Denote by µr (n) the multiplicative function determined by  r  p , if a = r + 1 −1, if a = 1 . µr ( pa ) =  0, otherwise

(27)

(28)

The following elementary properties follow: ϕ r (n) = n

µr (d) d|n

µr (d) =

d|n

γ r (n), 0,

d

;

if n ∈ L r +1 . othwewise

(29)

(30)

where γ is the core of n. An interesting property follows from the definition: lim ϕ r (n) = ϕ(n).

r →∞

(31)

The following asymptotic result is true: n≤x

ϕ(n) =

ax 2 + O(x log2 x) 2 134

(32)

¨ THE MOBIUS FUNCTION

(ϕ(n) = ϕ 1 (n)), n≤x

ϕ r (n) =

ar x + O(x 1+ε ), 2

(33)

with a, ar constants. More generally, one can introduce M¨obius functions of arbitrary direct factor sets. Let P, Q = ∅ be subsets of N such that if n 1 , n 2 ∈ N, (n 1 , n 2 ) = 1, then n = n 1 n 2 ∈ P ⇔ n 1 , n 2 ∈ P; and similarly for Q. If every integer n ∈ N possesses a unique factorization of the form n = ab (a ∈ P, b ∈ Q), then each of the sets P and Q will be called a direct factor set of N. In what follows P will denote such a direct factor set with (conjugate) factor set Q. The M¨obius function can be generalized to arbitrary direct factor sets, by writing n , (34) µ P (n) = µ d d|n,d∈P where µ is the classical M¨obius function. For example, µ1 (n) = µ{1} (n) and µN (n) = δ(n) (where δ(n) = 1, n = 1; 0, n > 1). It is immediate that for n = n 1 n 2 , (n 1 , n 2 ) = 1, n 1 , n 2 ∈ P one has µ P (n) = µ P (n 1 )µ P (n 2 ). On the other hand: n = δ(n) (35) µP d d|n,d∈Q which leads to the following M¨obius-type inversion formula: n n f (n) = ⇔ g(n) = g f (d)µ P d d d|n,d∈Q d|n

(36)

By using these notions and properties, generalized totient functions may be introduced (see also E. Cohen [52]).

5

Klee’s M¨obius function and totient

V. L. Klee [124] in 1948 introduced the function Tk (r ) as the number of integers h among the numbers 1, 2, . . . , r for which (h, r ) is k-free. For k = 1, Tk ≡ ϕ. Klee defined a generalized M¨obius function µk by 1 µ r k , if r is a kth power . (37) µk (r ) = 0, otherwise 135

CHAPTER 2

Then the function Tk can be expressed as r µk (d) , Tk (r ) = d d|r

(38)

i.e. Tk = µk ∗ E. For fixed r , T2 (r ) represents the number of those arithmetic progressions r s + t (t ∈ {0, 1, 2, . . . , (s − 1)}) which contain infinitely many squarefree numbers. E. T2 (r ) K. Haviland [104] showed that is a Bohr almost-periodic function with an abr solutely and uniformly convergent Fourier series. A generalization of Klee’s function Tk has been introduced by D. Suryanarayana [53]. This is based on a notion of k-ary divisor of an integer. Let the symbol (a, b)k denote the greatest among the common k-th power divisors of a and b. If (a, b)k = 1 we say that an is relatively k-prime to b and vice-versa. A divisor d of n is called k-ary if d, = 1. We note that for k = 1 one reobtains the unitary divisors. Let d k Tk (x, n) denote the number of positive integers ≤ x which are relatively k-prime to n. Then Tk (n, n) = Tk . As a generalization of (38) one has x (39) µk (d) Tk (x, n) = d d|n

6

M¨obius functions of Subbarao and Harris; Tanaka; and Venkataraman and Sivaramakrishnan

In 1966 M. V. Subbarao and V. C. function µk,q (r ) by   1, −1, µk,q ( pa ) =  0,

Harris [192] introduced the multiplicative if a ≡ 0 (mod k) if a ≡ q (mod k) . otherwise

(40)

They proved the interesting relation lim µk,1 (r ) = µ(r )

k→∞

(41)

and that, if ϕk,q (r ) denotes the number of integers a (mod r ) such that (a, r ) has its kth power free part qth powerfree, then r (42) ϕk,q (r ) = µk,q (d) . d d|r 136

¨ THE MOBIUS FUNCTION

From (42) follows also that lim ϕk,1 (r ) = ϕ(r ).

k→∞

(43)

Many generalized totient functions are also surveyed in [165]. Let λ be the Liouville function, given by λ(n) = (−1) (n) (where (n) denotes the total number of prime factors of n). A common generalization of Liouville’s function and the M¨obius function has been introduced by M. Tanaka [202]. Let k ≥ 2 be positive integer or infinite and define   λ(n), if n is k − free 0, otherwise (44) µk (n) =  λ(n), if k = ∞ Clearly, µ2 (n) = µ(n), µ∞ (n) = λ(n). For 2 ≤ k ≤ ∞ let Mk (x) = µk (x). Then n≤x

√ √ Mk (x) − Bk x = ± ( x)

(45)

1 , Bk = ζ (1/2) ζ (k/2)/ζ (1/2)ζ (k), if k is odd; and Bk = 1/ζ (1/2)ζ (k/2) if k is even. Another generalization of the M¨obius function, leading to an extension of Ramanujan sums has been introduced by C. S. Venkataraman and R. Sivaramakrishnan [212]. Put 0, if n contains a squared factor > 1, µk (n) = (46) exp(πiω(n)/k), otherwise (for the symbol ± see Handbook I [142]) where B2 = 0, B∞ =

where ω(n) denotes the number of distinct prime factors of n and k ≥ 1. Clearly µ1 (n) = µ(n) by exp(πiω(n)) = (−1)ω(n) . One has 1, if n = 1 (47) µk (d) = ω(n) , if n>1 (1 + s) d|n where s = exp(πi/k). The function µk leads to a parallel generalization of Euler’s totient, as well a Ramanujan sums, by introducing n · d, (48) µk ϕk (n) = d d|n 137

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and Ck (n, r ) =

µk

d|(n,r )

n d

·d

(49)

The functions ϕk and Ck are multiplicative (the second one is both n and r ), and are connected by the following formula: Ck (n, r ) = µk (r/g)ϕk (g)

t ϕk (t)

(50)

where g = (n, r ) and t is the product of the common distinct prime factors of r/g and g.

7

M¨obius functions as coefficients of the cyclotomic polynomial

In 1983 R. Dussaud [165] introduced a generalized M¨obius function, by application of results a cyclotomic polynomial Φn (x), having as roots the primitive roots of unity of order n. It is well known that the sum of all primitive roots of unity of order n is µ(n). Therefore, write Φn (x) = 1 − µ(n)x + µ2 (n)x 2 − µ3 (n)x 3 + . . . So µ2 (n) is the sum of all zeros of Φn taken two-by-two, i.e. xi x j µ2 (n) =

(51)

(52)

i< j

where (xi )1≤i≤ϕ(n) are the roots of Φn (x) = 0. In an analogous way, Dussaud defines µk (n) by xi1 xi2 · · · xik . µk (n) =

(53)

i 1
The case k = 2 is studied, the general case is left to future considerations of other authors. For example  1, if n = p1 p2 · · · pk with pi distinct primes and k is odd     1, if n = 2 p1 p2 · · · pk with pi distinct odd primes and k is even      1, if n = 4 p1 p2 · · · pk with pi distinct odd primes and k is even 0, if n = p1 p2 · · · pk with pi distinct odd primes and k is even µ2 (n) =   0, if n = 2 p1 p2 · · · pk with pi distinct odd primes and k is odd      −1, if n = 4 p1 p2 · · · pk with pi distinct odd primes and k is odd   0, otherwise (54) 138

¨ THE MOBIUS FUNCTION

We note that in condensed form, by another general method, these can be found in a paper by M. Deaconescu and J. S´andor [59]. In fact, this generalization involves the coefficients of the cyclotomic polynomial and this question has been extensively studied (but not completely solved) by many authors (see e.g. [19], [159], [217]). Finally, we note that other generalizations of the M¨obius function in the spirit of this paragraph are due to Golubev (see [24]); H. Gupta (see [83]); C. A. Nicol (see [146]); M. H. Eggar (see [66]).

2.4 1

M¨obius functions of posets and lattices Introduction, basic results

The earliest papers on M¨obius functions and inversion theorems in settings other that of Number theory were concerned with lattices of subgroups of a group. L. Weisner [215], [216], P. Hall [85] were interested in prime-power groups, while S. Delasarte [61] and E. Cohen [49], [50] in Abelian groups. M. Ward [213] considered more abstract cases. He defined the convolution of incidence functions of a locally finite lattice, and pointed out that such functions form a ring with respect to addition and convolution. See also a paper by R. Wiegandt [219] (who mentions interesting application of Delsarte’s inversion formula in group theory, due to L. R´edei [156], [157]). For a recent book studying also the ”Wiegandt convolution”, see J. Golan [77]; and for related work on Incidence algebras, see E. Spiegel, Chr. J. O’Donnell [188]. In 1964, G.-C. Rota [161] laid the foundations of the theory of M¨obius function in combinatorial mathematics. His paper is filled with historical comments and examples (not recognizing, however the importance of the fundamental paper by R. Wiegandt), and has an extensive bibliography. A number of applications of the M¨obius functions are contained e.g. in the paper by E. A. Bender and J. R. Goldman [11]. The important papers by D. Smith [184], [187] survey his results in this theory, while the work of H. Scheid [171] has an arithmetical emphasis. In what follows, only the most important notions and results of this theory will be mentioned. P = ∅ will be a partially ordered set (abreviated as poset), with ”≤” a partial ordering. For x, y ∈ P, the set [x, y] = {z ∈ P | x ≤ z ≤ y} will be called an interval determined by x and y. P is called locally finite, if [x, y] is finite for all x, y ∈ P. In what follows, we shall always assume that P is locally finite. 139

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A function f : P × P −→ C such that f (x, y) = 0 whenever x ≤ y will be called an incidence function (or generalized arithmetical function). Let F(P) = { f | f : P × P −→ C incidence function }. The functions

δ(x, y) = e(x, y) =

1, x = y ; 0, x = y 1, x ≤ y ; 0, x ≤ y

clearly are in F(P). Define f + g and f · g naturally in F(P), and define also their convolution by ( f ∗ g)(x, y) = f (x, z)g(z, y), (x, y ∈ P), (1) x≤z≤y

where a sum over an empty set is taken be 0. It can easily proved that (F(P), +, ∗) is a ring with unity element δ (however, this is not necessarily a commutative ring). This ring will be called as the incidence ring of F, or the ring of generalized arithmetical functions defined on P. An incidence function f on P is said to have an inverse, if there exists g ∈ F(P) with f ∗ g = g ∗ f = δ. If there exists an inverse of f , this is unique, let in notation be f −1 . As in the case of classical arithmetical functions, the following result holds: f ∈ F(P) has an inverse iff f (x, x) = 0, ∀x ∈ P.

(2)

The function e has an inverse. This is called the M¨obius function of P, denoted by µ. If f ∈ F(P), then g = f ∗ e ⇐⇒ f = g ∗ µ;

(3)

h = e ∗ f ⇐⇒ f = µ ∗ h.

(4)

If x, y ∈ P, then we say that y covers x if x < y and if there is no element z ∈ P such that x < z < y. In this case 0 = δ(x, y) = (µ ∗ e)(x, y) = µ(x, x) + µ(x, y), 140

¨ THE MOBIUS FUNCTION

hence µ(x, y) = −µ(x, x) = −1.

(5)

If x < y, but y does not cover x, there is nothing we can say generally on the values of µ(x, y). There is one case, however when can determine µ(x, y), that is the case of chains (or totally ordered sets), i.e. when for all x, y ∈ P one has x ≤ y or y ≤ x. If x, y, then the elements of [x, y] can be odered z 0 , z 1 , . . . , z n so that x = z 0 < z 1 < . . . < z n = y. For i = 0, 1, . . . , n − 1, z i+1 covers z i , hence µ(z i , z i+1 ) = −1. Assuming, that y does not cover x, we have n ≥ 2. Then µ(x, x) = 1, µ(x, z 1 ) = −1, µ(x, z 2 ) = −µ(x, x) − µ(x, z 1 ) = 0, and by induction µ(x, z i ) = 0 for all i ≥ 2. In particular, µ(x, y) = 0. The poset P is called a lattice if for all x, y ∈ P, x and y have a least upper bound (which is unique) denoted x ∨ y, and a greatest lower bound, denoted by x ∧ y. For standard and important properties of lattices, see e.g. [16], [201]. Certain examples are important to be noted here (see also [36]). On N, the relation m | n is a partial ordering. Let the resulting poset be denoted by N D . If m | n, then [m, n] = {d : d | n and m | d}, hence N D is locally finit. This is lattice, since x ∨ y = l.c.m.(x, y), x ∧ y = g.c.d.(x, y) (this lattice is even distributive). Let  n  f m , f (m, n) =  0,

if m | n if m |n

,

(6)

Since 1 | n for all n, the mapping f −→ f is one to one, and furthermore preserves addition and convolution, i.e. f + g = f + g, f ∗ g = f ∗ g, 141

CHAPTER 2

where ”*” on the left is a Dirichlet convolution, while the right ”*” is the convolution in F(N D ). If m | n, then d n g = f (m, d)g(d, n) = f ( f ∗ g)(m, n) = m d d|n d|n m|d

=

e| mn

m|d

n n = ( f ∗ g) . f (e)g me m

Let now A be a regular convolution. The relation m ≤ n if m ∈ A(n)

(7)

is a partial ordering on N. The resulting locally finit poset will be denoted by N A . If f is an incidence function of N A , then f is an incidence function of N D . Furthermore, if f, g ∈ F(N A ), then their sum in F(N A ) is the same as their sum in F(N D ), and their convolution in F(N A ) is the same as their convolution in F(N D ). This is because the extra terms which appear in f (m, d)g(d, n) d|n

m|d

are actually equal to zero. Thus F(N A ) can be considered to be a subring of F(N D ). However the element e in F(N A ) is not the same as the element in F(N D ) (unless A = D)! With each arithmetical function f we can associate incidence functions f of N A defined by  n  f m , if m ∈ A(n) , f (m, n) =  0, if m ∈ A(n) The mapping f −→ f is one to one and m ∈ A(n) implies n n ( f + g )(m, n) = f (m, n) + g (m, n) = f +g = m m = ( f + g)(n, m) = ( f + g)(m, n), n = ( f ∗ A g)(m, n). ( f ∗ g )(m, n) = ( f ∗ A g) m We note that (unless A = D), f ∈ F(N A ) and f ∈ F(N D ) are not generally the same. 142

¨ THE MOBIUS FUNCTION

The poset N A is not always a lattice. Since N A is locally finite and since 1 ≤ m and 1 ≤ n, every pair of elements of N A has a greatest lower bound in N A . However, if p is a prime and α = β, then p α and p β have no upper bounds at all in NU . If (P, ≤) is a locally finite partially ordered set, we can define the following generalized convolution (see [101]): ( f ∗ K g)(x, y) = f (x, z)g(z, y)K (x, y; z), (8) x≤z≤y

which is called K -convolution. If K ≡ 1, then we obtain the usual convolution (see [1]). If we have K (x, y; x) = K (x, y; y) = 1 whenever x ≤ y

(9)

then δ serves as the identity with respect to K convolution, f ∗ K δ = δ ∗ K f = f, for all incidence function f . We remark that if K (x, z 2 ; z 1 )K (x, y; z 2 ) = K (x, y; z 1 )K (z, y; z 2 ) whenever x ≤ z 1 ≤ z 2 ≤ y, (10) then the K convolution is associative. It should be noted that the K convolution of incidence functions is a generalization of the K convolution of arithmetical functions (see (35) of 2.2).

2

Factorable incidence functions, applications Let L be a local lattice. An incidence function f of L is called factorable if f (a ∨ b, c ∨ d) = f (a, c) f (b, d),

(11)

for all a, b, c, d ∈ L such that a, b, c, d belong to the same interval in L and c ∧ d ≤ a ≤ c,

c ∧ d ≤ b ≤ d.

An example is in the case of N D . An arithmetical function f is a multiplicative function iff the associated f ∈ F(N D ) is factorable. The following results are true: (1) If L is a locally distributive, local lattice then the inverse of a factorable function f in F(L) is factorable. 143

CHAPTER 2

(2) If L is as above, then the convolution of two factorable functions in F(L) is factorable. (3) If L is a local lattice and µ is factorable, then L is locally distributive. The same if e ∗ e is factorable. (4) Let f be a factorable function in F(L), constant on chains, and let h be a factorable companion of f (i.e. h(x, y) = f (x, y) − 1 whenever y covers x). If x, y, t ∈ L and x ≤ y ∧ t, then f (x, z)µ(z, y) = µ(x, y)µ(y ∧ t, y)h(y ∧ t, y) (12) x≤z≤y

z∧t=x

This important result is due to D. Smith [184]. For an application, let f be a (classical) multiplicative function such that f ( p α ) = f ( p) for all primes p and α ≥ 1. Let h be multiplicative function such that h( p) = f ( p) − 1 for all primes. Then for all m, r one has the identity: r = µ(r )µ(m)h(m), (13) f (d)µ d d|r (n,d)=1

r . (n, r ) We note that (13) could be called as the generalized Brauer-Rademacher identity, 1 r , h(r ) = we get as for f (r ) = ϕ(r ) ϕ(r ) r µ(r )µ(m) d = µ , (14) ϕ(d) d ϕ(m) d|r where m =

(n,d)=1

which is the classical Brauer-Rademacher identity [90]. For another application, let A be regular arithmetic convolution and f be a multiplicative function such that f ( p α ) = f ( p t ) with t = t A ( p α ). Let h be multiplicative such that h( p α ) = f ( p α ) − 1 for all primitive prime powers p α . Then one has the identity r = µ(r )µ(m)h(m); (15) f (d)µ A d d∈A(r ) (n,d) A =1

r . (n, r ) A In his series of papers [184]-[185] D. Smith has found a general context for a common study of Dirichlet, Cauchy, Lucas-Carlitz convolutions (see the preceeding paragraphs). with µ A being the Narkiewicz M¨obius function, and m =

144

¨ THE MOBIUS FUNCTION

3

Inversion theorems and applications

We quote an inversion theorem, essentially due to L. Weisner [215]: 1) Let P be a poset, and f : P −→ C such that f (x) = 0 unless x ≥ t for some t ∈ P. If g(x) = f (z), (x ∈ P) z≤x

then f (x) =

g(z)µ(z, x),

(16)

z≤x

and conversely. 2) Let P, f be as above, but f (x) = 0 unless x ≤ p for some p ∈ P. Then, if g(x) = f (z), z≤x

then f (x) =

µ(x, z)g(z),

(17)

z≤x

and conversely. A parametric M¨obius inversion formula has been recently introduced By B. Chen [39]. Let P be a locally finite poset with incidence set F(P) and convolution (1). Let A be an arbitrary nonempty set and consider the vector space F(P × A) of all complex-valued functions on P × A. The incidence algebra F(P) acts naturally on F(P × A) in the left by D f ξ(x, a) = f ξ(x, a) = f (x, y)ξ(y, a), x≤y∈P

where f ∈ F(P), ξ ∈ F(P × A). Similarly, for η ∈ F(A × P), the action in the right can be defined by η(a, x) f (x, y). D f η(a, y) = η f (a, y) = x≤y,x∈P

Let A and B be nonempty sets. For any a ∈ A and b ∈ B, the following discrete analog of integration by parts is obvious: D f ξ(a, x)η(x, b) = ξ(a, x)D f η(x, b). x∈P

x∈P

145

CHAPTER 2

We assume that all the posets are finite or countable infinite and that all the infinite sums are absolutely convergent, i.e. every infinite series converges to an element which is independent of the combinations of the terms in the sum. The following parametric inversion theorem holds true: For f ∈ F(P), ξ ∈ F(P × A), if f is invertible, and η(x, a) = f (x, y)ξ(y, a) (18) x≤y∈P

is absolutely convergent, then

ξ(x, a) =

f −1 (x, y)η(y, a)

(19)

x≤y∈P

Particularly, if P = N, A = C (set of complex numbers), then we can state the following inversion theorem: Let f be a function on the lattice N × N, and let ξ be a function on N × C. If f (k, k) = 0 for k ∈ N, and the series η(n, a) =

∞

f (n, kn)ξ(kn, a)

(20)

f −1 (n, kn)η(kn, a)

(21)

k=1

is absolutely convergent, then ξ(n, a) =

∞ k=1

Some consequences of this inversion formula has applications in certain inverse problems in Physics. In 1990 Nan-Xian Chen [40] has shown the possibility of application of M¨obius inversion theory to physical problems. These are the inverse problems for obtaining the photon density of states, the inverse black-body radiation problem for remote sensing, and the solution for inverse Ewald summation. In 1993 Shao-jun Liu, Mi Li and Nan-Xian Chen [137] used M¨obius transform for inversion from cohesion to elastic constants, in 1995 M. Li and N.-X. Chen [135] gave a precise method to get the pair potential of diamond-type materials from their cohesive energy, while in 1998 N.-X. Chen and Rong Er-qian [41] found a concise and unified solution (based on M¨obius inversion) of inverse specific heat problems in physics.

4

M¨obius functions on Eulerian posets

We now study certain generalized M¨obius functions on Eulerian posets. A graded poset P is a finite partially ordered set with a unique minimum element 0, a unique maximum element 1, and a rank function ρ : P → N satisfying 146

¨ THE MOBIUS FUNCTION

ρ( 0) = 0, ρ(y) − ρ(x) = 1 whenever y ∈ P covers x ∈ P. The rank ρ(P) of a graded poset P is the rank of its maximum element. Given a graded poset P of rank n + 1 and a subset S ⊂ {1, 2, . . . , n}, define the S-rank-selected subposet of P to be the poset PS = {x ∈ P : ρ(x) ∈ S} ∪ { 0, 1}

(22)

Denote by f S (P) the number of maximal chains of PS . Equivalently, f S (P) is the number of chains x1 < · · · < x|S| in P such that {ρ(x1 ), . . . , ρ(x|S| )} = S. The M¨obius function of a graded poset P is defined recursively for any subinterval of P by   1, if x=y µ([x, y]) = (23) µ([x, z]), otherwise  − x≤z
A graded poset P is Eulerian if µ([x, y]) = (−1)ρ(x,y) ,

(24)

where ρ(x, y) = ρ(y) − ρ(x), for all [x, y] ⊂ P. For a survey of Eulerian posets, we quote R. P. Stanley [189]. In 2000 M. M. Bayer and G. Hetyei [7] have introduced the k-M¨obius function of a graded poset. This is defined recursively by   1, if x = y 1 (25) µk ([x, y]) = µk ([x, z]), otherwise  −1 − k x

S⊂{1,2,...,n}

1 − k

|S|

f Sn+1 (P)

(26)

A graded poset is k-Eulerian if for every interval [x, y] ⊂ P, one has µk ([x, y]) = (−1)ρ(x,y) 147

(27)

CHAPTER 2

From the definitions it follows at once that if P is a k-Eulerian poset of rank n +1, then every interval of P is k-Eulerian;

(28)

and that n

(−1)i−1 f i (P) = k(1 − (−1)n )

(29)

i=1

For the existence of k-Eulerian posets, Bayer and Hetyei obtain the following characterization: For every positive integer n, there exists a k-Eulerian poset of rank n + 1 if and only if k = j/2 for some positive integer j. Moreover, every k-Eulerian poset is 2k-thick (i.e. every nonempty open interval has at least 2k elements).

5

(30)

Miscellaneous results

For a recent paper on M¨obius function of lattices, see A. Blass and B. F. Sagan [18], where the concept of a bounded below set is introduced. This can be used to give a generalization of a theorem of Rota (on ”broken circuits” on a finite lattice). This result can be used to compute and combinatorially explain the M¨obius function in various examples including non-crossing set partitions, schuffle posets and integer partitions in dominance order. This paper also contains many References. In another paper, the homotopy and homology of finite lattices is considered, originated from the M¨obius function of geometric lattices [17]. For a recent paper containing a generalization of M¨obius’ function arising in intuitionistic fuzzy theory (with application in number theory) see K. T. Atanassov [4]. For a study of M¨obius categories, and functors between such categories (containing a general setting for M¨obius inversion which includes the cases of locally finite partially ordered sets and monoids with finite decomposition property), see M. Content, F. Lemay and P. Leroux [54]. However, we do not consider here this very extensive field of study.

2.5

1

M¨obius functions of arithmetical semigroups, free groups, finite groups, algebraic number fields, and trace monoids M¨obius functions of arithmetical semigroups

The theory of arithmetical functions, particularly M¨obius’ function of an arithmetical semigroup was begun by A. Beurling in 1937 [13]. One can find a good account of results in the book by J. Knopfmacker [125]. 148

¨ THE MOBIUS FUNCTION

A free commutative semigroup G with identity element 1G generated by a countable set P is called arithmetical if in addition there exists a real-valued norm mapping | · | on G such that i) |ab| = |a| · |b| for all a, b ∈ G ii) the total number NG (x) of elements n ∈ G of norm |n| ≤ x is finite for each x ∈ R. The elements of P (i.e. the generators of G) are called primes. Clearly, every element n = 1G in G has a unique (up to the order of factors) factorization of the form n = p1α1 . . . prαr

(1)

with pi distinct elements of P. Complex valued functions defined on an arithmetical semigroup are called arithmetical. The M¨obius function µG is defined by   1, if n = 1G µG (n) = (2) (−1)r , if a1 = a2 = · · · = ar = 1 in (1)  0, otherwise Let MG (x) =

µG (n)

(3)

1≤x

be the generalized Mertens function. Assume that the following asymptotic axiom holds true: Axiom. There exist positive constants C and δ, and a constant η with 0 ≤ η < δ such that NG (x) = C x δ + O(x η ) as x → ∞

(4)

It can be shown (see [125], Chapter 6) that if Axiom (4) is true in G, then the prime number theorem holds true in G. In fact, if (4) holds, then the prime number theorem, and the estimate MG (x) = o(x δ )

(5)

are equivalent. It is interesting to note that this equivalence is not true in general arithmetical semigroups, see W. Zhang [222]. Relation (5) can be strengthened to the following: If G satisfies Axiom (4), then for every a > 0 one has MG (x) = O(x δ (log x)−a ) which is due to J. Knopfmacher [126]. 149

(6)

CHAPTER 2

Various estimates on primes, or special arithmetic functions can be found in [125]. See also P. Haukkanen [91] for various generalizations. For some special arithˇ Porubsk´y [152]. For asymptotic results metic functions, (e.g. on k-free integers) see S. on multiplicative functions defined on an arithmetical semigroup, see R. Warlimont [214]. For semigroups with unique factorization, and M¨obius type inversion theorems, see Z. Chen, Y. Shen and N. Chen [42]. In the classical case when P is a multiplicatively independent subset of (1, ∞) and G is the multiplicative semigroup generated by P, Beurling’s classical theorem states that if for N (x) = NG (x), N (x) = D · x + x · ε(x) (x → ∞), where ε(x) = o(1), and if we want (prime number theorem) x P(x) ∼ (x → ∞), log x in order to get this conclusion the condition 1 ε(x) = O (log x)3/2+

(x → ∞)

is sufficient (here a+ denotes any number strictly largen than a), and the condition 1 ε(x) = O (x → ∞) (log x)3/2 is not sufficient. After the work on Beurling (1937), the main contributors in the field were B. Nyman [149], P. Malliavin [140] (who connected estimates on ε(x) and estimates on P(x) − li x), P. T. Bateman and H. G. Diamond [6], H. G. Diamond [62], H. Muller [144] (who looked for more refined conditions on ε(x) implying PNT (prime number theorem). Sz. R´ev´esz [158] went in another direction opened by Beurling: what can we say on P(x) when the assumption on N (x) is of the form N (x) = x Q(log x) + xε(x) (x → ∞), Q(y) being either A · y ρ (ρ > 0) or a more general perturbating factor. J.-P. Kahane used the approach of Bateman and Diamond, in [119] he proved a conjecture by them, namely ∞ dx (ε(x) log x)2 <∞ x 1 implies PNT. For a Fourier formula, see [120], for a theorem of Littlewood for generalized primes see [121]. See also [122] for the role of Sobolev or Wiener algebras in the prime number theory of Beurling. 150

¨ THE MOBIUS FUNCTION

2

Fee abelian groups and M¨obius functions

Let G be a free abelian group on a countable number of generators gi (i = 1, 2, . . . ), and consider a homomorphism N : G → Q+ (positive rationals) such that N (gi ) are all integral. Therefore the images under N of all integral words of G are integers. Next we give a subgroup G of G which is of finite index h = [G : G ], and denote a generic coset of G /G by H . Then the problem which arises is concerned with what can be deduced about the distribution (relative to N ) of generators of G in the cosets of G/G if one has information about the distribution of the integral words of G in these cosets. More precisely, letting w denote integral words, assume that for all cosets H of G/G one has 1 = C H · x + O(x δ ), (7) B H (x) = N w≤x,w∈H

0 ≤ δ < 1, C H ≥ 0, c =

C H > 0 and aim at deriving for each H a result of the

H

form (prime number theorem) x x 1 = dH π H (x) = +o log x log x N g≤x,g∈H

(8)

with c H , d H constants depending on H . For example, in the classical case of primes in arithmetic progressions the c H are all equal and the d H are all equal to 1/ h; i.e. there is ”equi-distribution” of the generators of G over the cosets. In the case where the c H are unequal we may have in particular the other extreme wherein dG = 1 and for the cosets = G , d H = 0, i.e. 1 ”almost all” of the generators of G are in G . In general, for δ < , the situation is a 2 combination of these two. More precisely, consider the character group K of G/G and extend these characters naturally to characters of G; i.e. for any element t ∈ H , χ ∈ K we define χ (t) = χ (H ). Then to each subgroup of K denote by #φ the set of elements t ∈ G such that χ (t) = 1 for all χ ∈ φ. Then #φ is a group and G ⊂ #φ ⊂ G. From the well known properties of the characters of finite abelian groups we then infer that the order of φ equals [G : #φ]. Next, introduce for each χ ∈ K A(χ ) = c H χ (H ) (9) H

When all the c H are equal, for χ = χ0 (principal character), one has A(χ ) = 0. In other cases we may have χ = χ0 for which A(χ ) = 0. The situation is saved, 151

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since for = (G/G ) = the set of characters χ ∈ K with A(χ ) = 0 can be proved (see W. Forman and H. N. Shapiro [71]) that is a subgroup of K

(10)

Setting L = #, the following is true: If (7) holds with 0 < δ < 1/2, then almost all generators of G are in L; furthermore, the generators of G equi-distribute among the cosets of G in L. Thus if G =

n

Lti , L =

m

i=1

(11)

G s j are decompositions into disjoint cosets (t1 =

j=1

1), G=

G ti s j

i=1,n, j=1,m

is such a decomposition for G with respect to G . Theorem (11) then asserts that x 1 · m log x x whereas for the other cosets H , π H (x) = o . Here log x πG s j (x) ∼

m = [L : G ] =

(12)

h h = . [G : L] or d

Thus, in this case   or d , if H ⊂ L dH = h  0, otherwise

(13)

1 If we do not assume δ < , the situation may be more complicated (see [71]). 2 An important step in the above theory is the asymptotic result that for all cosets H we have nG≤x,g∈H

log N g = D(H ) log x + O(1) Ng

(14)

where D(H ) is a ”Dirichlet density”. By using (14) and assuming (7) one can show that (8) holds with d H = D(H ). 152

¨ THE MOBIUS FUNCTION

In the proofs a M¨obius function and a corresponding inversion formula constitute a key method. Let µG (w) be defined by  if w = I (identity of G)  1, (−1)k , if w = g1 . . . gk , where gi are distinct generators (15) µG (w) =  0, otherwise Then it follows that

µG (d) =

d|w

1, w = I 0, otherwise

(16)

where the summation is taken over all integral words d of G which ”divide” w, i.e. w = dd , with d an integral word of G. We then can state the following inversion formula: If P(w) is a complex-valued completely multiplicative function defined over the integral words of G (i.e. P(ww ) = P(w)P(w )) with P(I ) = 1, and f (x) is a complex-valued function defined for x > 0, then x g(x) = (x > 0) ⇔ P(w) f Nw N w≤x f (x) =

P(w)µG (w)g

N w≤x

x (x > 0) Nw

(17)

A character χ belongs to 1 if A(χ ) = 0; to 2 if A(χ ) = 0 and L 0 (χ ) = 0; and χ(w) . to 3 if A(χ ) = L 0 (χ ) = 0. Here L 0 (χ ) = lim x→∞ Nw N w≤x Examples. 1) Let G = Q+ . Fix any prime p, and any positive integer d, and let G be the subgroup of G generated by p d and the other primes. Then [G : G ] = d, and for the coset Hi = pi G (i = 0, 1, . . . , d − 1), n≤x,n∈Hi

1=

p d−i−1 ( p − 1) + O(log x) pd − 1

(18)

Here G/G is cyclic, and the coset H0 = G contains all primes other than p. Thus (G ) = K (G ) and the other sets 2 , 3 are empty. A(χ ) = 0 for all characters, can be seen directly. Also, in this case L = G . 2) G = Q+ , p a fixed prime, d > 1 fixed positive integer. Define G as the set of positive rationals of the form p α a/b where α ≡ 0 (mod d), (a, p) = (b, p) = 1, 153

CHAPTER 2

a ≡ b (mod p). Here G is a subgroup with [G : G ] = d( p − 1) and cosets Hi j = pi j G (i = 0, 1, . . . , d − 1; j = 1, . . . , p − 1). Also n≤x,n∈Hi j

1=

p d−i−1 + O(log x) pd − 1

(19)

In this case the primes are equi-distributed among the cosets H0 j ( j = 1, p − 1), 3 is empty, or d1 = d, and L is the subgroup composed of the elements in the cosets H0 j ( j = 1, p − 1). 3) G = Q+ . For r = π piαi let (r ) = αi . Define G = the set of r such that

(r ) ≡ 0 (mod 2). Hence [G : G ] = 2, the cosets being G and 2G = H . On the assumption of the Riemann hypothesis, x 1 1 and 1 = + O(x 2 +ε ) (ε > 0) (20) 2 n≤x,n∈H n≤x,n∈G = G. In this case, all primes are in H , 2 = ∅, or d1 = or d3 = 1, L = G For equivalent formulations of abstract prime number theorems, see also P. Laborde Montaner and H. N. Shapiro [128].

3

M¨obius functions of finite groups

In his paper on Eulerian functions, P. Hall [86] in 1936 defined the M¨obius function on a subset of a power set, partially ordered by inclusion, and used the inversion formula to compute the number ϕs (G) of ordered s-tuples of elements which generate a finite group G. He illustrated the method by evaluating ϕs (G) when G ∼ = P S L 2 ( p) ( p prime), and in the process computed the M¨obius value µ(G) for these groups. In 1959 W. Gasch¨utz [73] by calculating ϕs (G) for soluble groups, implicitely obtained a formula for µ(G) in terms of the numbers of complements of chief factors. This formula was stated explicitly by Ch. Kratzer and J. Th´evenaz [127]. In what follows, P will be a finite poset and µ P will denote its M¨obius function. If P has a smallest element, say 0, we shall write µ P (x) for µ P (0, x). The µ(G) above referred is in fact µL (1, G), where L = L(G) is the poset of all subgroups of G. When disscussing subposets of L(G), we shall reserve the symbol µ for this meanning, too. Let G be a finite group, and denote by S(G) the poset of all subgroups of G. We will be mainly concerned with subposets P of S(G), and their M¨obius function µ P . The notation µ refers always to µ S(G) and µ(1, H ) will be denoted simply by µ(H ). If L ≤ H ≤ G, then µ(L , H ) can be calculated entirely within S(H ) and is independent of the embedding of H in G; in particular µ S(G) (H ) is an invariant of H . 154

¨ THE MOBIUS FUNCTION

H. H. Crapo [55] proved essentially the following result: Let N G and denote by K (G, N ) the set of all subgroups of G which complement N . Then µ(G) = µ(G/N ) µ(K , G). (21) K ∈K (G,N )

As a corollary, one can state that for a group G with µ(G) = 0, if N ≤ M G with G N G, one has that M/N is complemented in G/N . In particular ( ) = 1 (for N Group theory, see e.g. [116]). Thus µ(G) = 0 whenever G has a Frattini chief factor. If N is an Abelian minimal normal subgroup of G with a complement K , then K is a maximal subgroup of G, so µ(K , G) = −1. We thus have another corollary: Let A be an Abelian minimal normal subgroup of G and let k denote the number of complements to A in G. Then µ(G) = −kµ(G/A).

(22)

This corollary gives an easy proof of a result by P. Hall [85]: Let X be a p-group of order p n . Then µ(X ) = 0 unless X is elementary Abelian, in which case: µ(X ) = (−1)n p (2) . n

(23)

But (22) applied inductively leads also to the Gasch¨utz formula for the M¨obius value of a soluble group: Let 1 = H0 < H1 < . . . < Hn = G be a chief series for the soluble group G, and let ki denote the number of complements to Hi /Hi−1 in G/Hi−1 . Then µ(G) = (−1)n k1 k2 · · · kn .

(24)

¨ The following general theorem is due to T. Hawkes, I. M. Isaacs and M. Ozaydin [105]: Let G be a finite group whose order is divisible by n, and assume that n is divisible by |G| p , the order of a Sylow p-subgroup of G. If H is a subgroup whose order divides n, then [N(H ) : H ] p divides µ(H, X ), where the sum is over all subgroups X of G with |X | dividing n. When H = 1 and n = |G| p , this is exactly a classical theorem by K. S. Brown [22] from 1975. 155

CHAPTER 2

We now turn our attention to applications of M¨obius inversion formulae. For subgroups L and H of a group G, define α L (H, G) = µ(H, X ) ⊆X

where the sum runs over all subgroups X of G containing < H, L >. Note that αG (H, G) = µ(H, G). By applying a classical M¨obius inversion formula for posets, the following result follows. Let L , H ≤ G. Then µ(H, G) = µ(X, G)α L (H, X ) (25) where the sum runs over subgroups X of G containing < H, L >. The following divisibility result is a consequence of (25): Let p be a prime and H a subgroup of G. Set L = O p (G). Then α L (H, G) is divisible by |NG (H ) : H | p .

(26)

Kratzer and Th´evenaz proved essentially the following: Let H be a subgroup of G and let m be the squarefree part of |G : G H |. Then |NG (H ) : H | divides m · µ(H, G).

(27)

¨ This implies (see Hawkes-Isaacs-Ozaydin [105]): Let m be the product of those prime numbers p for which G/O p (G) is elementary Abelian of order p or p 2 . Then |G| divides m · µ(G). Furthermore, for any prime p, |O p (G)| p divides µ(K , G) K

where K runs over all complements to O p (G) in G.

(28)

Frobenius’ classical theorem states that if n is a divisor of the order of a finite group G, then the number of solutions to the equation x n = 1 in G is divisible by n. 156

(29)

¨ THE MOBIUS FUNCTION

This can be proved elementarily by the following result of P. Hall [86]: If X is a group, let ϕs (X ) be the number of s-tuples (x1 , . . . , xs ) of a group elements xi such that X =< x1 , . . . , xs >. Then obviously ϕs (X ). |G| S = x≤G

Set µ∗ (X ) = µ(X, G). Then µ∗ is the M¨obius function associated with the dual poset of S(G) and so µ(G) = µ∗ (1). Applying the M¨obius inversion formula, we get: µ∗ (X ) · |X |s . (30) ϕs (G) = X ≤G

In particular, ϕ1 (G) is zero unless G is cyclic, in which case ϕ1 (G) = ϕ(|G|) the usual ϕ-function of Number theory. If H ≤ G, let [H ] = {H g | g ∈ G}, the conjugacy class of H , and let C(G) denote the set of all conjugacy classes of subgroups of G. We view C(G) as a poset with partial ordering ≤ defined by the rule [H ] ≤ [L] if H ≤ L g for some g ∈ G. In general, the poset C(G) is neither a meet -nor a join- semilattice, but it naturally admits M¨obius function µC(G) , denoted by λG . Also we write λG (G) for λG ([G]). The following important result is true: If G is soluble, then µ(G) = λG (G) · |G |.

(31)

¨ The above result by Hawkes, Isaacs and Ozadyn has been generalized by H. Pahlings [150] as follows. If G is a soluble group and H a subgroup, then µ(H, G) = [NG (H ) : H ∩ G ]λ(H, G)

(32)

where λ(H, G) is the notation for λ([H ], [G]). For example, for the simple groups G = P S L(2, p) ( p > 3 a prime) one has µ(H, G) = [NG (H ) : H ]λ(H, G) for every subgroup H of G. A group A has the N -property with respect to a group B if, by definition |Hom(B, N A (T )/T )| = 1, 157

(33)

CHAPTER 2

for all T with 1 < T ≤ A (this holds for example, when B is a simple group with |B| ≥ |A|). The following result enables us to compute the M¨obius function of a direct product ([105]). If A is nontrivial and has the N − pr oper t y with respect to B, then µ(A × B) = µ(B) · (µ(A) − s),

(34)

where s is the number of epimorphisms from B onto A. For an application, when A and B are non-isomorphic simple groups, then µ(A × B) = µ(A) · µ(B).

(35)

Relation (34) enables us to compute µ(G) when G is a direct product of simple groups: Let A be a non Abelian simple group and G = A × A ×. . .× A, the direct product of n copies of A. Then µ(G) = 0 if µ(A) = 0, or else µ(G) = µ(A)n ·

n−1

(1 − r t),

(36)

r =1

|Aut(A)| . µ(A) For example, when G is a direct power of n copies of A5 , then

with t =

µ(G) = (−60)n · 3 · 5 · . . . · (2n − 1),

(37)

so one can see the interesting fact that there exists a group G and a prime p such that p divides µ(G) = 0, but doesn’t divide |Aut(G)|. For groups whose commutative subgroup is an abelian Hall π -subgroup, the following theorem is true (see M. Bianchi, A. Gillio Berta Mauri and L. Verardi [14]): Let G be (a finite) group, whose commutator subgroup G is an abelian Hall π subgroup (all Sylow subgroups of which are elementary abelian). Let R be the lattice of those subgroups of G which are normal in G, and µR (G ) the corresponding M¨obius function. Then µ(G) = µ(G/G )|G | · µR (G )

(38)

As a corollary of (38) one obtains that if G is a group with G elementary abelian Sylow p-subgroup of order p k ( p prime, k > 1), then G/G is of squarefree order. Further, for all H ≤ G let H < G. Then µ(G) = ±|G | · µ(G ) 158

(39)

¨ THE MOBIUS FUNCTION

The following result gives a condition for µ(G) to be a prime power: Let G be a group such that G is the elementary abelian Sylow p-subgroup. If G/G is of squarefree order, then µ(G) = p k

(40)

for certain k ≥ 1. Generally (see [14]) µ(G) = p k if and only if |G | = p s , s ≤ k; the Sylow p-subgroup is elementary abelian and the other Sylow subgroups are cyclic of prime order.

(41)

In fact, for each prime p and k ≥ 1 there is a metabelian group G such that µ(G) = p n .

(42)

Similarly, if n = p k q h (n, q primes, k, h ≥ 1), then there exists a metabelian group G such that µ(G) = n.

(43)

Therefore for ω(n) ≤ 2, the M¨obius function µ is surjective. It can be proved that this is true for all n ≤ 100 (see [14]), however this may be not true even for simple groups (see [15], where for a such group G one has µ(G) = 0).

4

(44)

M¨obius functions of algebraic number and function-fields

Let K be an algebraic number field of degree k over the rational field. A generalized M¨obius function µ K (a) defined over the integral ideals of K is defined by H. N. Shapiro [175] as follows:  1, if a = 0    m (−1) , if a is squarefree and is the product (45) µ K (a) = of m distinct prime ideals    0, otherwise If one defines F(a) =

µ K (b), then F is multiplicative function and F( p α ) =

b|a

0, so that F(a) =

1, for a = 0 0, otherwise 159

(46)

CHAPTER 2

This enables us to introduce a generalized inversion formula as follows: If f (x) is a given complex-valued function defined for all x > 0, and x , (47) f g(x) = Na N a≤x then f (x) =

µ K (a)g

N a≤x

x . Na

(48)

This inversion formula gives a possibility to estimate various sums involving integral ideals of K , and to deduce Selberg’s identity in case of the algebraic number field K (see [175]): log2 N p + log N p log N q = 2x log x + O(x) (49) N p≤x

N pq≤x

This is equivalent to

log N p ∼ x

(x → ∞)

(50)

N p≤x

i.e. the prime ideal theorem. We note that the proof of (49) uses also an important result from algebraic number theory, namely: 1 = αx + O(x 1−1/k ) (51) N a≤x

where α is a constant depending only on the field K , and α=

2r1 +r2 π r2 Rh √ w |D|

(52)

where r1 = number of real conjugate fields of K , r2 = number of imaginary conjugate fields, w = number of roots of unity in K , h = number of ideal classes (see [218]). The function µ K of M¨obius enables us to introduce a generalized von Mangoldt function, too:  0, if a = 0 a  log N p, if a = p α K (a) = (53) µ K (b) log N =  b b|a 0, otherwise 160

¨ THE MOBIUS FUNCTION

The following identity is true (see [175]): b|a

µ K (b) log2 N

a a = K (a) log N a + K (b) K b b b|a

(54)

Besides Shapiro’s proof, another elementary proof of the prime ideal theorem (in the form (50)) is due to C. Toubi [209], who follows a method due to A. Hildebrand [106]. If P = ∅ is a set of prime ideals p ∈ P, M. Skalba [182] defines   µ(a) if a is an integral ideal of K not divisible by any p ∈ P, µ P (a) =  0, otherwise

(55)

This is applied in the proof of generalized Euler identities. For M¨obius and Euler type functions on algebraic function fields, see e.g. the monograph by M. Rosen [160]. For generalizations of von Mangoldt’s function , see A. Ivi´c [117], [118]; and for a characterization, see P. Haukkanen [102]. For a unitary Mangoldt function, see [30].

5

Trace monoids and M¨obius functions

Finally, we shall deal with M¨obius functions in trace monoids. Given a finite alphabet , a trace monoid M is defined as the quotient ∗ / ≡ I , where I ⊂ × is a symmetric and irreflexive relation, and ≡ I is the smallest congruence extending the set of equations {ab = ba| (a, b) ∈ I }. The elements of M are called traces. These were introduced in 1969 in P. Cartier and D. Foata [37] in order to give a completely combinatorial proof of Mac Mahon’s Master theorem (see G. Lallement [129], ch. XI). They were studied further as a possible model of parallelism by interpretation of generators of the monoid as actions and their commutation as independence of actions (A. Mazurkiewicz [141]). A theory of trace languages was then developed, extending the properties of formal languages. For a survey of results, as well as open problems, see V. Diekert and G. Rozenberg [64]. The pair , I is called independence alphabet and can be represented by an undirected graph, with the set of nodes, and I the set of edges. In what follows, Z M ! will be the ring of formal power series over M with integer coefficients. The support of σ ∈ Z M ! is the set of all t ∈ M with σ (t) = 0. A polynomial is a formal power series having finite support. 161

CHAPTER 2

The M¨obius function of a trace monoid M is defined by   (−1)n , if t = a1 . . . an , where all ai ∈ are different, and (ai , a j ) ∈ I for all i = j, (56) µ M (t) =  0, otherwise with the convention that µ M (1) = 1. If ξ M = t is the characteristic series of M, t∈M

then one has ξ M µ M = µ M ξ M = 1,

(57)

see [37]. The M¨obius function µ M can be represented by several polynomials in Z . Consider the canonical morphism ψ: Z !→ Z M !. Let us say that a polynomial µ ∈ Z ! is a non-commuting lifting of the M¨obius function of M if ψ(µ) = µ M and the restriction ψ : ∗ → M is injective on the support of µ. Such a function is called unambiguous if its inverse ξ satisfies ξ(w) ∈ {0, 1} for all w ∈ ∗ and ψ(ξ ) = t. V. Diekert [129] proved that an unambiguous nont∈M

commuting lifting µ of the M¨obius function of M exists iff the graph , I admits a transitive orientation. Assume that I is a transitive orientation of , I and let be a linear extension of I over . The formal power series given by  (−1)n , if x = x1 x2 . . . xn , {x1 , . . . , xn } is a clique    of , I and x is a lexicographically smallest string (58) µ(x) = in x with respect to ,    0, otherwise is an unambiguous lifting of µ M (see [63]). C. Choffrut and M. Goldwurm [44] call µ as the unambiguous M¨obius function defined by I . In [44] the following result is proved: Let , I be an independence alphabet that admits a transitive orientation I . Then the unambiguous M¨obius function µ defined by I is given by µ = det(In − X ), where X is a matrix and In the identity matrix of order n. In fact X is a minimal acceptor matrix of a lexicographic cross section defined by I . For details, see [44]. For related results on Algebraic topology, see [20], [70], [153], [204], [18] and [17].

162

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Chapter 3

THE MANY FACETS OF EULER’S TOTIENT

3.1

Introduction

Motivated by a generalization of Fermat’s divisibility theorem, in 1760 L. Euler [133] proved that a ϕ(n) ≡ 1(modn) for (a, n) = 1,

(1)

where ϕ(n) denotes the number of positive integers r < n such that (r, n) = 1. Euler studied also other important properties of this function, e.g. the multiplicativity property, which means that ϕ(mn) = ϕ(m)ϕ(n) whenever (m, n) = 1. (2) It was C. F. Gauss (see L. E. Dickson [103]) who introduced the symbol ϕ(n) for Euler’s ϕ-function, making the convention ϕ(1) = 1. In 1879 J. J. Sylvester introduced the word ”totient”, while E. Prouhet (1845) proposed the name ”indicator” (see [103]). Sylvester defined also the ”totatives” of n to be the integers r < n with (r, n) = 1. Euler’s totient is of major interest in number theory, as well as many other fields of mathematics. For example, the apparently simple result (1) allowed mathematicians to build a code which is almost impossible to break, even though the key is made public (see for instance R. Rivest, A. Shamir and L. Adleman [352]). Euler’s totient is connected to many notions and functions of various domains. Gauss proved the identity ϕ(d) = n (3) d|n

179

CHAPTER 3

which by the M¨obius inversion formula (see Chapter 2) gives µ(d) 1 =n ( p prime) 1− ϕ(n) = n d p p|n d|n

(4)

For inequalities and estimates connecting ϕ to the arithmetical functions d (number of divisors), σ (sum of divisors), ω (number of distinct prime divisors), (total number of prime divisors), Jk (Jordan’s function), ψ (Dedekind’s function), (von Mangoldt’s function), ϕ ∗ , d ∗ , σ ∗ (unitary analogs of ϕ, d, σ ), etc., see D. S. Mitrinovi´c, J. S´andor and B. Crstici [291] (which we shall call sometimes in what follows Handbook I). On the other hand, many other aspects of Euler’s totient were not considered in [291], for instance: congruence properties similar to (1) (and generalizations and extensions); other congruence properties of ϕ; identities (such as Kesava Menon or Brauer-Rademacher identities); generalizations and extensions of ϕ; various equations involving ϕ; the cyclotomic polynomial; matrices and determinants (as the Smith determinant); connections to congruences, etc. All these subjects will be treated in what follows. In this Introduction we want also to mention certain unfamiliar occurrences of Euler’s totient.

1

The infinitude of primes

E. Kummer (see [27]) derived the following simple proof of the infinitude of primes: suppose p1 < p2 < · · · < pk form the (finite) set of primes. Then by applying the prime factorization, for n = p1 p2 . . . pk we must have ϕ(n) = 1. On the other hand, one has ϕ(n) = ( p1 − 1) . . . ( pk − 1) > 1, a contradiction. Another proof, based on the divisibility property n|ϕ(a n − 1) (a, n > 1 integers)

(5)

is given in M. Deaconescu and J. S´andor [97] (see also J. S´andor [362]). Properties similar to (5) will be studied in the next paragraph.

2

Exact formulae for primes in terms of ϕ

There exist in the literature various exact formulae expressing the n-th prime pn , or the prime-counting function π(n) (i.e. the number of primes ≤ n). For example by Williams, Min´acˇ , Sierpinski or Gandhi, see R. Ribenboim [349] (pp. 180-186), or [350] (where one can find also the following statement ”... c’est inutile, mais c’est joli...”). In 1992 K. Atanassov [16] has found the formula π(n) =

n

sg(k − 1 − ϕ(k))

k=2

180

(6)

THE MANY FACETS OF EULER’S TOTIENT

0, if x = 0 1, if x = 0 He proved also that

where sg x =

pn =

2n

sg(n − π(k))

(7)

k=0

0, if x ≤ 0 1, if x > 0 M. Vassilev [464] proved that

where sg x =

π(n) =

n ϕ(k) k=2

k−1

(8)

where [x] denotes the integer part of x.

3

Infinite series and products involving ϕ, Pillai’s (Ces`aro’s) arithmetic functions

Euler’s totient occurs in many identities for special functions or constants. In 1857 J. Liouville (see [103], p. 120) proved that ∞ n=1

xn x ϕ(n) = for |x| < 1. n 1−x (1 − x)2

(9)

More generally, E. Ces`aro (see [103], p. 127) showed that ∞ n=1

if F(n) =

∞ xn f (n) = x n f (n), 1 − xn n=1

(10)

f (d). See also G. P´olya and G. Szeg¨o [332] (vol. II, Part 8, paragraph

d|n

7). In [332] one can find also identities like ∞ n=1

xn 1 + x2 ϕ(n) = x (due also to J. Liouville), 1 + xn (1 − x 2 )2 ∞

xn 1+x =x , 1 − xn (1 − x)3

(12)

xn 1 + 2x + 6x 2 + 2x 3 + x 4 = x 1 + xn (1 − x 2 )3

(13)

n=1 ∞ n=1

J2 (n)

(11)

J2 (n)

181

CHAPTER 3

where J2 is the Jordan totient function Jk for k = 2, and |x| < 1. Similarly, ∞ −x (1 − x n )ϕ(n)/n = e 1−x

(14)

n=1

while the infinite product ∞ 1 + x n ϕ(n)/n −1 = e2/(x −x) n 1 − x n=1,odd

(15)

has been proposed by P. J. Forrester and M. L. Glasser [136] (in fact (15) follows from (14) on replacing x by −x). A Mercier [289] generalized (15) as follows: ∞ ∞ 2 F(n)x n /n 1 + x n f (n)/n n=1,odd =e 1 − xn n=1,odd

(16)

where F(n) is given in (10). An interesting infinite product identity, involving also Fibonacci and Lucas numbers Fn and L n is due to D. Redmond [348]: ϕ(n)/n ∞ √ √ Ln = e−(1+ 5)/2 5 (17) √ 5 · Fn n=1 For finite arithmetical products, involving e.g. the gamma-function, too, see [384]. The constant

∞ ∞ 1 1 1+ = 1.78657 . . . (18) = ϕ(n)σ (n) p 2k − p k−1 p prime n=1 k=1 is called also as the ”Silverman constant” (see D. Rusin [357]). For other constants related to the Euler totient function, we quote S. Finch [135]. In the same paper in which he proved (9), Ces`aro considered also the sum of (n, i) for i ≤ n, where (n, i) is the g.c.d. of n and i: n i=1

(n, i) =

d|n

dϕ

n d

(19)

The left side of (19) is called often as the Pillai arithmetic function P(n), who in 1933 [330] studied certain properties (without reference to Ces`aro) of this function. n n For three extensions of P(n), see R. Sivaramakrishnan [420]. The function (i, n) i=1 occurs also in group theory, and for a common generalization of this function and Pillai’s see J. S´andor and A.-V. Kr´amer [382]. A generalization involving regular 182

THE MANY FACETS OF EULER’S TOTIENT

convolutions of Narkiewicz (see Chapter 2) is due to L. T´oth [455] (see [456] for a unitary analogue of P(n), along with asymptotic evaluations). See also [459]. For n other Ces`aro type functions, and particularly for the study of [i, n] (where [i, n] = i=1

lcm(i, n)) see H. W. Gould and T. Shonhiwa [153], [154].

4

Enumeration problems on congruences, directed graphs, magic squares

n 2d − 1 in the first term of the product dϕ in reIf one replaces d with d d lation (19), one obtains the solution of an enumeration problem for a congruence equation. More precisely, R. A. Brualdi and M. Newman [56] have shown that the number of n-tuples (x0 , x1 , . . . , xn−1 ) of nonnegative integers such that

n−1 i=0

is given by

xi = n,

n−1

i xi ≡ 0 (mod n)

(20)

i=0

1 2d − 1 n ϕ n d|n d d

(21)

In fact, this result follows also from a theorem by K. G. Ramanathan [344]. The number (21) gives also the number of formally distinct diagonal products of an n by n circulant matrix (see [56]). In 1994 A. Ehrlich [107] defined for two positive integer a and n the directed graph G(a, n) with vertex set {0, 1, . . . , n − 1} such that there is an arc from x to y if and only if y ≡ ax (mod n). Let C(a, n) denote the number of cycles in G(a, n). Erlich proved that C(2, n) is odd or even according as 2 is or not a quadratic residue mod n. (22) E. Brown [54] has considered the general case, by proving that for C(a, n) one has the formula ϕ(d) C(a, n) = (23) or dd (a) d|n where or dd (a) is the least positive integer r such that a r ≡ 1 (mod d). This implies e.g. that for (a, n) = 1, n odd, one has a 1+ n C(a, n) ≡ (mod 2) (24) 2 a where is the Legendre-Jacobi symbol. n 183

CHAPTER 3

For n even, n = 2k n (n odd) C(a, n) is even, for k = 1; and −1 1− n (mod 2) (25) C(a, n) ≡ 2 A generalization of Euler’s totient is the Schemmel totient function (see [103]), defined by S2 (n) = car d{(x, x + 1) : 0 < x, (x, n) = (x + 1, n) = 1, x ≤ n}. It can be proved that 2 (26) 1− S2 (n) = n p p|n D. N. Lehmer [244] used Schemmel’s totient to the enumeration problem of certain magic squares. Many other generalizations of Euler’s totient will be studied in the next paragraphs of this Chapter.

5

Fourier coefficients of even functions (mod n)

Let n be a fixed positive integer, and let F be a field of characteristic 0 containing the nth roots of unity. In 1952 E. Cohen [76] defined a function f : N → F to be (n, F)-arithmetic, if f (a) = f (b) whenever a ≡ b (mod n). An example of such a function is f (a) = exp(2πi ja/n) which is (n, F)-arithmetic for any integer j. Let An (F) denote the set of all (n, F)-arithmetic functions. If f ∈ An (F), then f is called even (mod n) if f (a) = f ((a, n)), where (a, n) denotes the greatest common divisor of a and n. Taking F = C, the field of complex numbers, let Bn (C) denote the set of even functions (mod n). Let C(a, d) be the Ramanujan sum, defined by C(a, d) = e(ax, d) (27) (x,d)=1, 1≤x≤d

with e(a, b) = e2πia/b . In fact C(a, d) can be defined for all integers a, and positive integers d, leading to a common generalization of Euler’s totient ϕ, and M¨obius’ function µ. Indeed, it is immediate that C(0, d) = ϕ(d), C(1, d) = µ(d) for all d ≥ 1

(28)

E. Cohen [77] proved that if f ∈ Bn (C), then f can be written uniquely as f (a) = α(d)C(a, d), (29) d|n

where the coefficients α(d), d|n are called as the Fourier coefficients of f . These coefficients may be explicitly written (see P. J. McCarthy [382]) as 1 α(d) = f (a)C(a, n) (30) nϕ(d) a (mod n) 184

THE MANY FACETS OF EULER’S TOTIENT

For f, g ∈ Br (C) let us introduce the inner product n n = (ϕ ∗ f g)(n) ϕ(d) f g f, g = d d d|n

(31)

where g(a) = g(a) is the complex conjugate of g(a), and ∗ is the Dirichlet convolution. It is known (see J. Knopfmacher [229]) that Bn (C), ·, · ) is a complex Hilbert space, with norm

1/2 2

f = n |α(d)| ϕ(d) (32) d|n

P. Haukkanen and R. Sivaramakrishnan [189] have introduced a subspace Sq (C) of Bn (C) as follows. Let f ∈ Bn (C) and q be a fixed divisor of n. Associated with f , one can construct an even function f (a, q) (even (mod n)) by defining α(d)C(a, d) (33) f (a, q) = d|q

If d|n, d q, then α(d) = 0 in the representation of f (a, q) as an element of Bn (C) (therefore we have actually a truncated sum). Let Sq (C) be the set of functions of form (33). Then Sq (C) becomes a subspace of Bn (C), of dimension d(q), where d is the number-of-divisor function. By the Riesz representation theorem one can write Br (C) = Sq (C) ⊕ Sq⊥ (C)

(34)

where Sq⊥ (C) is the orthogonal complement of Sq (C) (of dimension d(n) − d(q)). The set 1 C(·, d) : d|n (35) (nϕ(d))1/2 forms an orthonormal basis of the Hilbert space Bn (C) (see [189]). For a measure theoretic approach, as well as linear transformations, see [189]. For multiple trigonometric sums, a similar study has been done in [190].

6

Algebraic independence of arithmetic functions

In 1948 R. Bellman and H. N. Shapiro [36] have introduced a notion of algebraic independence of real valued arithmetic functions. A set of arithmetic functions f i (i = 1, N ), f i : N → R are called algebraic independent over R if there exists no polynomial P(X 1 , . . . , X N ) ≡ 0 with real coefficients, which is irreducible over R N such that P( f 1 (n), . . . , f N (n)) = 0, for all n ∈ N (36) 185

CHAPTER 3

Bellman and Shapiro obtain various general theorems, for example: Let E : N → R be the identity function; i.e. E(n) = n for all n; and let f be a multiplicative function. If E and f are algebraically dependent, then f (n) = n r , n ∈ N (where r is a fixed rational number). (37) Let f k+1 (n) = f k (d), k = 0, N , where f 0 is multiplicative, and suppose d|n

that f 0 ( p) takes on an infinite set of values over some set of primes p ∈ P. Then f 0 , . . . , f N are algebraically independent. (38) As a corollary one gets that the functions E, ϕ, σ are algebraically independent

(39)

Another result says that

as well as:

7

E, ϕ, σ, d, d ∗ are algebraically independent

(40)

E, ϕ, σ, d, d ∗ , µ are algebraically independent

(41)

Algebraic and analytic application of totients

In what follows, in this Introduction we mention an algebraic and an analytic application of totients. Jordan’s totient Jk is defined by Jk (n) = car d{(a1 , . . . , ak ) : 1 ≤ a1 , a2 , . . . , ak ≤ n, (a1 , . . . , ak , n) = 1}

(42)

Clearly J1 ≡ ϕ. This function has applications in the theory of linear groups (see C. Jordan [216]). For some recent applications in group theory, see J. Schulte [400]. R. Guitart [167] has applied this function to explicit computation of internal exponentiation of isomorphic classes of objects in the topos of permutations. See also [168]. The function Jk should not be confused with Jk , given by Jk (n) = car d{(a1 , . . . , ak ) : 1 ≤ a1 ≤ a2 ≤ · · · ≤ ak ≤ n, (a1 , . . . , ak , n) = 1} (42 ) n k µ(d) , one has on the other hand, While Jk (n) = d d|n Jk (n)

=

d|n

µ

n d + k − 1 d 186

k

THE MANY FACETS OF EULER’S TOTIENT

For the more general functions Jkm (n), where in (42) 1 ≤ a1 , . . . , ak ≤ n and (a1 , . . . , ak , m) = 1; and Jkm (n) where in (42’), 1 ≤ a1 ≤ · · · ≤ ak ≤ n, and (a1 , . . . , ak , m) = 1, see T. Shonhiwa [412]. Remark that Jkn (n) = Jk (n) and Jkn (n) = Jk (n). Shonhiwa studies also the functions Jk (n) = car d{(a1 , . . . , ak ) : 1 ≤ a1 < a2 < · · · < ak ≤ n, (a1 , . . . , ak , n) = 1}, and the corresponding function Jk m (n). For example, one has n d Jk (n) = µ d k d|n The sum-of-divisors function σ is strongly related to Euler’s totient (see [291]). Now the following inequality is equivalent to the famous Riemann Hypothesis on the zeta function: The Riemann Hypothesis is true if and only if, for sufficiently large n one has σ (n) < eγ n log log n

(43)

(Here γ is Euler’s constant). This is due to G. Robin [354].

1 Recently A. Verjovsky [470] proved the following result: Let ≤ a ≤ 1. Then 2 the Riemann zeta function has no zeros in the half-plane Re s > a if and only if ∞ n=1

ϕ(n) f

n x

= m 0 ( f, x) + O(x a+ε ),

ε > 0,

(44)

for any smooth function f : R+ → R with compact support, where ∞ 6 m 0 ( f, x) = 2 x 2 t f (t)dt. π 0

8

ϕ-convergence of Schoenberg

In 1959 I. J. Schoenberg [398] defined the ϕ-convergence of a sequence {γn } of real numbers as follows: ϕ − limit γn = λ if and only if lim sn = λ, where n→∞

sn =

1 ϕ(d)γd n d|n

A sequence which is ϕ-convergent, but not convergent is given e.g. by 1, if n = 2 · 3 · 5 . . . pk γn = 0, otherwise 187

(45)

CHAPTER 3

where pk is the kth prime (see the solution of Problem 6090 [361]). Schoenberg proved that if {n k } is a given sequence of increasing natural numbers, then the assumption ϕ − limit γn = λ implies that lim γn k = λ if and only if k→∞

lim inf ϕ(n k )/n k > 0 k→∞

(46)

From the existence of an asymptotic distribution function for ϕ(n)/n, it follows that if the sequence (n k ) has the property that lim ϕ(n k )/n k = 0, then the density of k→∞

(n k ) is 0. Write d − lim γn = λ if d{n : |γn − λ| ≥ ε} = 0 for every ε > 0, where d denotes (asymptotic) density. Now, Schoenberg proves that if ϕ − lim γn = λ, then one has also d − lim γn = λ. (47) 1, n = prime The converse is evidently false (put γn = ). 0, otherwise A criterion for d − lim γn = λ is contained in the following necessary and sufficient condition: 1 lim (eitγ1 + eitγ2 + · · · + eitγn ) = eitλ (48) n→∞ n for every real t (see [398]).

3.2 1

Congruence properties of Euler’s totient and related functions Euler’s divisibility theorem

The most famous congruence property of Euler’s totient is clearly Euler’s divisibility theorem (see also paragraph 3.1): a ϕ(n) ≡ 1 (mod n) for (a, n) = 1

(1)

For the early history of this theorem, one can find in Dickson’s book [103], that in 1776 P. S. Laplace obtained a proof different from Euler’s. In 1831 V. Bouniakowsky gave a proof similar to that by Laplace, while in 1832 Giovanni de Paoli gave another proof. In 1845 L. Poinsot proved Euler’s theorem by geometrical considerations: Let a be prime to n and < n. Join any vertex of a regular polygon of n sides with the ath vertex following it, the new vertex with the ath vertex following it, etc., thus defining a regular star polygon of n sides. With the same a, derive similarly a new n-gon, etc., until the initial polygon is reached. The number k of distinct polygons thus obtained is seen to be a divisor of ϕ(n), the number of polygons corresponding to the various 188

THE MANY FACETS OF EULER’S TOTIENT

a’s. If in the initial polygon we take the a k th vertex following any one, etc., we obtain the initial polygon. Hence a k , so a ϕ(n) has the remainder unity when divided by n. In 1896 H. Weber deduced the first proof of (1) based on Group theoretical considerations. In 1913 G. Frattini noted that if F = F(a, b, . . . ) is a homogeneous symmetric polynomial, of degree g with integral coefficients, in the integers a, b, . . . less than n and prime to n, and if F is prime to n, then kg ≡ 1

(mod n) for any integer k, (k, n) = 1

(2)

In fact, F(a, b, . . . ) ≡ F(ka, kb, . . . ) ≡ k g F(a, b, . . . ) (mod n). So if F = ab . . . , we reobtain Euler’s theorem. There is an interest, even today in the new proofs of Euler’s theorem. See H. Alzer [8] (where, in fact R´edei’s theorem (11) is proved). K. Heinrich and P. Horak [197] obtain a combinatorial proof of (1) when m is a prime power. The proof of the general case follows from this fact by usual methods. For combinatorial and geometrical investigations of congruence theorems, see also K. H¨artig and J. Sur´anyi [180]. R. T. Hansen [176] formulated for general a and n as follows: Let (a, n) = d. (1) n ϕ(n)+1 Then a = 1. (1 ) ≡ a (mod n) iff d, d

2

Carmichael’s function, maximal generalization of Fermat’s theorem

It follows from Euler’s theorem that if (a, n) = 1, there exists the smallest positive exponent r such that a r ≡ 1 (mod n), (3) called the order of a modulo n, and denoted r = or dn (a). It is immediate that a m ≡ 1 (mod n) if and only if m is a multiple of r ; thus in particular, r |ϕ(n). It is natural to ask: given n > 2, does there always exist an integer a, (a, n) = 1 such that or dn (a) = ϕ(n)? When n = p is a prime, such numbers exist, namely the primitive roots mod p. If n = p k , p odd prime, it is also true. However, if n is divisible by 4 p, or pq, where p, q are distinct odd primes, then there is no number a, (a, n) = 1 such that or dn (a) = ϕ(n). Indeed, in 1910 R. D. Carmichael [65] (see also [66]) introduced a function λ, defined as follows: λ(1) = 1, λ(2) = 1, λ(4) = 2, λ(2a ) = 2a−2 (r ≥ 3), λ( pa ) = ϕ( pa ) = pa−1 ( p − 1) (r ≥ 1, p odd prime); λ(2a p1a1 p2a2 . . . psas ) = [λ(2a ), λ( p1a1 ), . . . , λ( psas )] where [. . . ] denotes the least common multiple. 189

(4)

CHAPTER 3

Note that λ(n)|ϕ(n)

(5)

so λ(n) ≤ ϕ(n), but λ(n) = ϕ(n) only for n = 1, 2, 4, pa , 2 pa . Carmichael proved the following divisibility theorem: a λ(n) ≡ 1 (mod n) for (a, n) = 1

(6)

As a corollary of (6) one obtains that, except for the cases n = 1, 2, 4, pa or 2 pa , we have 1 a 2 ϕ(n) ≡ 1 (mod n) (6 ) He also showed that there always exists an element a such that or dn (a) = λ(n). One can find in [103] that E. Lucas (1890) stated, without proof the following result: Let n = p1a1 p2a2 . . . pkak be the prime factorization of n and put H (n) = max{a1 , a2 , . . . , ak }.

(7)

Then for all positive integers a and n a λ(n)+H (n) ≡ a H (n)

(mod n)

(8)

This is a maximal generalization of Fermat’s theorem written in the form a p ≡ a (mod p) for any prime p, any integer a. For a proof of (8), see D. Singmaster [419]. The fact that (8) is maximal generalization, follows also from the following remark (see [419]): Let r > s. Then a r ≡ a s (mod n) for all a and n iff λ(n)|(r − s) and s ≥ H (n). (9) The following inequalities are true: λ(n) + H (n) ≤ ϕ(n) + H (n) ≤ n

(10)

with simultaneous equality iff n is prime, or n = 4. From (8), (9), (10) the following corollaries can be stated: a n ≡ a n−ϕ(n) ≡ a n−λ(n) a ϕ(n)+H (n) ≡ a H (n)

(mod n);

(mod n), for any a and n.

(11) (12)

We note that the first congruence of (11) is attributed to L. R´edei (see T. Szele [451], S. Schwarz [404]), while (12) is mentioned in N. Nielsen [321]. It is interesting to note also that Carmichael’s theorem (6) has applications in the theory of the so-called ”3x + 1 problem” (a famous unsolved problem), see J. C. Lagarias [234] and E. Belaga and M. Mignotte [35]. 190

THE MANY FACETS OF EULER’S TOTIENT

In [404] and [403] S. Schwarz developed a semigroup approach to the Fermat and Euler’s theorems. In 1996 M. Laˇssˇa´ k and S, Porubsk´y [240] generalized the idea to finite commutative rings, and to residually finite Dedekind domains. (For Dedekind domains, see also W. Narkiewicz [314]). F. Smarandache [430] obtained a generalization of Euler’s theorem by applying a terminating divisibility algorithm. For a best possible variant of this result, with an algebraic extension, see the recent paper by S. Porubsk´y [338]. A generalization of another type of Euler’s theorem, involving Lucas sequences, appears in [349] (see pp. 53-63). This result is based on a Lucas sequence-generalization of Euler’s totient function, and was proved essentially by E. Lucas in 1877 (see [103], p. 398).

3

Gauss’ divisibility theorem

In 1770 J. L. Lagrange (see [103]) obtained a common generalization of Fermat’s and Wilson’s divisibility theorem: a p−1 − 1 ≡

p−1 (a − i) (mod p)

(13)

i=1

Indeed, for a prime p, and for (a, p) = 1, a p−1 ≡ 0 (mod p). Comparing the constants on both sides of (13) one obtains Wilson’s theorem ( p − 1)! + 1 ≡ 0

(mod p)

(14)

For the early history of Wilson’s theorem, and generalizations, see [103]. In a little known paper, N. Gruber [158] has proposed to study (13) for composite n (in place of p − 1) in the following manner: a ϕ(n) − 1 ≡ (a − a1 )(a − a2 ) . . . (a − aϕ(n) ) (mod n)

(15)

where a1 , a2 , . . . , aϕ(n) is a complete system of reduced residues (mod n) (i.e. (ai , n) = 1 for i = 1, ϕ(n). It is proved in [158] that (15) holds true for all composite n and all a if and only if n = 4 or n = 2Fk , where Fk is a Fermat prime (i.e. k Fk = 22 + 1, prime). (16) A different kind of generalization of Fermat’s theorem was first stated by K. F. Gauss (see [103]) in 1863: G a (n) = µ(d)a n/d ≡ 0 (mod n) for all a, n. (17) d|n

In fact, Gauss proved (17) only for a = prime, while the case a = p m ( p prime) 1 is due to T. Sch¨onemann (1846), who showed that G a (n) in this case is the number n 191

CHAPTER 3

of all irreducible polynomials over the finite field of p m elements. The first proof for general a is due to J. A. Serret (1855). In 1880 S. Kantor deduced (17) by proving that 1 G a (n) is the number of cyclic groups of order n in any birational transformation of n order a in the plane. In 1899 L. E. Dickson obtained an inductive proof. He showed also that ϕ(d) (18) G a (n) = d|(a n −1), d (a s −1) f or s
In fact, (17) follows at once from (11) by remarking that α α−1 µ(d)a n/d = (a p − a p ), p α n

d|n

and by (17)

α

ap ≡ ap

α−1

(mod p α ).

Thus p α divides the product for every p α n, so b divides the product. In 1891 P. A. McMahon ([273], [103]) showed that the number of circular per1 ϕ(d)a n/d . mutations of a distinct things n at a time, repetitions being allows is n d|n Thus ϕ(d)a n/d ≡ 0 (mod n) for all a, n. (19) d|n

L. Gegenbauer in 1910 ([147], [103]) obtained a common generalization for (17) and (19), by proving that if for an arbitrary integer-valued arithmetic function h one has h(d) ≡ 0 (mod n), then d|n

h(d)a n/d ≡ 0 (mod n)

(20)

d|n

Related to (20), see also L. Carlitz [60]. For a generalization of Gegenbauer’s theorem, see also A. Axer [19], [103]. MacMahon’s result (19) has been recently rediscovered by Chong-Yun Chao [74], who proved also that n − 1 ! ≡ 0 (mod n) (21) (ϕ(d))2 d n/d−1 d d|n Chao proved also a generalization of Euler’s theorem: For all n, a one has r

a where n =

r

i=1

a −1

pi i

(a ϕ(n) − 1) ≡ 0 (mod n),

piai is the prime factorization of n. See also J. Morgado [298].

i=1

192

(22)

THE MANY FACETS OF EULER’S TOTIENT

J. Westlund [481] extended Gauss’ theorem (17) to algebraic number fields. Let A be an ideal of an algebraic number field, A = P1 P2 . . . Pi , where P1 , . . . , Pi are distinct prime factors of A. Let N (A) be the norm of A. If a is an arbitrary algebraic integer, then A| µ(d)a N (A)/N (d) (23) d|A

For finite permutation groups, S. P. Strunkov [442] has recently proved the following theorem. Let G be a finite permutation group and H a subgroup. We define a M¨obius-type function on the subgroup of G by recursion: µ(H, H ) = 1, and µ(H, T ) = − µ(H, K ). Let T < G and c(T ) denote the number of T -orbits. T >K ≥H

Then for any a ∈ Z \ {0}, µ(H, T )a c(T ) ≡ 0

(mod |NG (H )|/|H |)

(24)

T ≥H

4

Minimal, normal, and average order of Carmichael’s function

As we have seen, Carmichael’s λ function is closely related to Euler’s totient. For an applet, which computes the values ϕ(n) and λ(n), see the Internet address: http://www.math-it.de/Mathematik/Zahlentheorie/Zahl/ZahlApplet.html Relations (10) inform us on certain estimates related to ϕ, λ and H . We note that the following elementary properties of H are true (see [419]) a|b ⇒ H (a) ≤ H (b);

(25)

max{H (a), H (b)} ≤ H (a, b) ≤ H (a) + H (b);

(26)

(a, b) = 1 ⇒ H (a, b) = max{H (a), H (b)}

(27)

In what follows we shall investigate the minimal, normal, and average order of Carmichael’s function λ. Estimates for the minimal order are implicitly stated in the analysis of the primality testing arguments of L. M. Adleman, C. Pomerance and R. S. Rumely [1]. These are stated explicitly in P. Erd¨os, C. Pomerance and E. Schmutz [129] as follows: For any increasing sequence (n i ) of positive integers, and any positive constant c0 < 1/ log 2 one has λ(n i ) > (log n i )c0 log log log ni (28) for sufficiently large i. On the other hand, there exists a sequence m 1 < m 2 < · · · < . . . and a constant c1 with λ(m i ) < (log m i )c1 log log log m i for all i. (29) 193

CHAPTER 3

λ(n) was stated without proof in P. Erd¨os The normal order of magnitude of log n [109]. A stronger result appears in [129]: There is a set S of positive integers of aymptotic density 1 such that, for all n ∈ S one has −1+ε λ(n) = n/(log n)log log log n+A+O(log log log n) (30) log p where A = −1 + = .2269688 . . . , and ε > 0 is fixed, but arbitrarily ( p − 1)2 p prime small. In [109] appeared also without proof an estimate for the average order of λ: for all ε > 0, k > 0 and x > x0 (ε, k), 1 x x λ(n) ≤ (log log x)k ≤ log x x n≤x (log x)1−ε The following sharper result appears in [129]: For all x ≥ 16, we have B log log x x 1 exp (1 + o(1)) λ(n) = x n≤x log x log log log x 1 = .34537 . . . ( p − 1)2 ( p + 1) p prime As a corollary of theorem (30) we obtain that ϕ(n) < (log log n)2 , n ∈ S P λ(n)

where B = e−γ

(31)

(32)

1−

(33)

where S has asymptotic density 1, and P(r ) denotes the greatest prime divisor of r . (For many results on the function P, see [291], Chapter 4). Let N (k) be the number of solutions to λ(n) = k. The following results appear in [129] (without proof): N (k) < exp[exp[(log 2 + o(1)) log k/(log log k)],

(34)

for all k ≥ k0 , and N (k) > exp[exp[(c2 − o(1)) log k/(log log k)]] for infinitely many k (where c2 is a constant).

194

(35)

THE MANY FACETS OF EULER’S TOTIENT

Similarly, for car d{n : λ(n) ≤ x} one has the estimates: (c2 − o(1)) log x < car d{n : λ(n) ≤ x} < exp exp log log x (log 2 + o(1)) log x < exp exp log log x

(36)

Let Rλ (x) = car d{m ≤ x : m = λ(n)}. The exact order of Rλ is not known. Clearly Rλ (x) = o(x) (37) since few numbers have a large divisor of the form p − 1 (see P. Erd¨os and S. S. Wagstaff, Jr. [132]). On the other hand, Rλ (x)

x log x

(38)

since this is already attained on the primes. The authors of [129] conjecture that Rλ (x) = x/(log x)c+o(1)

(39)

where c is a constant. Probably c ∈ (0, 1), but the correct value of c remains in doubt.

5

Divisibility properties of iteration of ϕ

Certain properties of the iteration of ϕ were included in [291], Chapter 1. Here we will study the divisibility properties of ϕ ( j) , where ϕ (1) (n) = ϕ(n), ϕ ( j) (n) = ϕ(ϕ ( j−1) (n)) ( j > 1), as well as of certain related functions. W. Sierpinski [414] determined all n such that ϕ(n)|n

(40)

It is not difficult to see that all solutions of (40) are n = 1, and n = 2a · 3b , where a > 0, b ≥ 0. (41) F. Smarandache [431] has considered mϕ(n)|n, for a fixed m, but clearly this implies (40). In 1982 M. Hausman [193] completely described the solutions of ϕ ( j) (n)|n

(42)

This was rediscovered in 1989 by D. Marcu [284]. However, in 1988 V. Siva Rama Prasad and Ph. Fonseca [423] have studied a general class of functions, including (42).

195

CHAPTER 3

For j = 2, the solutions of (42) are n = 1, 2a , 3, 2a · 3b , 2a · 5, 2a · 7, where a, b ≥ 1

(43)

This has been rediscovered by L. T´oth [454], too. A problem proposed by A. H. Stein [438] asks for the determination of all pairs of positive integers m, n such that ϕ(m)|n and ϕ(n)|m

(44)

For m = n this generalizes Sierpinski’s problem (40). It is not difficult to see that the relations of (44) imply that ϕ(ϕ(n))|n, ϕ(ϕ(m))|m so n, m, must be in the sets of (43). Another method is based on the fact that all solutions can be generated by primitive solutions (m, n), i.e. when gcd(m, n) = squarefree. Since ϕ( pn) = pϕ(n) for p|n, p prime, all other solutions can be obtained from primitive ones by the following rule: If p is a prime dividing both m and n, then the pair (m, n) is a solution iff the pair ( pm, pn) is a solution. There are eleven primitive pairs of solutions, namely: (1,1), (1,2), (2,2), (2,3), (2,4), (2,6), (4,6), (4,10), (6,6), (6,14) and (6,18). (45) In 1950 H. N. Shapiro [411] introduced a class F of arithmetic functions f which r are of the form: f (1) = 1 and if 1 < n = piai is the prime factorization of n, then i=1

f (n) =

r

f ( pi )[A( pi )]ai −1

(46)

i=1

where f ( pi ), A( pi ) are integers satisfying 0 < f ( pi ) < pi , 0 < A( pi ) ≤ pi , f ( pi )A( pi ) > 1 for odd primes pi (i = 1, r ), while f (2) = 1, A(2) = 2. Since f (n) < n, whenever n > 2, by denoting by f (k) the kth iterate of f , for n > 2 there is a unique integer k = k f (n) such that f (k) (n) = 2.

(47)

Define k f (1) = k f (2) = 0, and put c f (n) =

if n is odd, k f (n), k f (n) + 1, if n is even

(48)

Shapiro proved that for any f ∈ F, the function c f is completely additive, i.e. c f (mn) = c f (m) + c f (n) for all m, n; iff f ( p) is even, and c f (2A( p)) = c f ( p) + 1 for all odd primes p. (49) 196

THE MANY FACETS OF EULER’S TOTIENT

Let now f ∈ F such that (49) is true, and let {qi } denote the sequence of odd primes in increasing order. Let F1 be the class of all f ∈ F such that f (qi ) = qi − εi and A(qi ) = qi

(50)

where (εi ) are odd integers satisfying 0 < εi < qi , i = 1, 2, 3, . . . Remark. If εi = 1 for all i, then by (46) and (50) one gets f = ϕ, so that ϕ ∈ F1 . Clearly, the set F1 is uncountable. Let f ∈ F1 , and define S f ( j) = {n ∈ N : f ( j) (n)|n}, i.e. the set of n such that (42) holds true. Let P f ( j) denote the set of primitive solutions of (42), i.e. n P f ( j) = n ∈ N : n ∈ S f ( j) and ∈ S f ( j) 2

(51)

(52)

Siva Rama Prasad and Fonseca [423] have proved the following result: For f ∈ F1 , j ≥ 1 one has ∞ 2i P f ( j); (53) S f ( j) = i=0

therefore it is enough to characterize the primitive solutions. For these, the following theorem is valid: Let n ∈ P f ( j). Then: c f (n) < j iff n is odd;

(54)

c f (n) = j iff n = 2m, where m is odd, and c f (m) = j − 1;

(55)

c f (n) > j iff n = 2(εi + 2)l , for some i and l ≥ j.

(56)

(Here c f (n) is given by (48)). Since ϕ ∈ F1 , results (53)-(56) are extensions of the theorems by Sierpinski, Hausman. Another generalization of (42) has been studied by F. Halter-Koch and W. Steindl [175], namely ϕ ( j) (n)|Qn (57) where Q is a fixed positive integer. Let S Q ( j) = {n ∈ N : ϕ ( j) (n)|Qn}, n PQ ( j) = n ∈ N : n ∈ S Q ( j), ∈ S Q ( j) 2 197

(58) (59)

CHAPTER 3

and C(n) = min{ j ≥ 0 : ϕ ( j) (n) ≤ 2}

(60)

S Q ( j) = {2k n : m ∈ PQ ( j), k ≥ 0},

(61)

Since

we need a characterization of primitive solutions given by (59). It can be shown that if C(n) < j, then n ∈ S Q ( j) and n ∈ PQ ( j) only when 2 n. If C(n) ≥ j, Q odd, and n ∈ PQ ( j), then n ≡ 2 (mod 4). For odd n one has n ∈ S2Q ( j) \ S Q ( j) only if 2n ∈ PQ ( j). (62) If Q is odd, then if n ∈ PQ ( j) and C(n) ≥ j, then n ≡ 2 (mod 4). If n ≡ 2 (mod 4), and C(n) = j, then n ∈ PQ ( j). So we may assume in what follows C(n) > j. For odd Q the following complete characterization is true (see [175]): Let Q be odd, and suppose C(n) > j. Then n ∈ PQ ( j) if and only if one of the following cases holds true. a) n = 2 · 3s with s ≥ j + 1; b) n = 2 p, where p is a prime of the form p = 2 · 3s + 1, with s ≥ max{2, j} and 3s+1− j |Q; c) n = 2 j ( p + 1) − 2, where p > 3 is a prime such that p−1 j |(2 − 1)Q, and the numbers 2r p + 2r − 1 (r = 0, 1, . . . , j − 1) 2 are all primes. (63) When Q is even, a similar characterization is an open question. By assuming Schinzel’s Hypothesis H (see e.g. [414]) for linear polynomials, the following can be proved (see [175]): Let p be a prime, and put L = C( p). Let j ≥ L + 2. Then there exist infinitely many even numbers Q such that 2 L Q and for each such Q there is an n ∈ PQ ( j) with p|n, but p Q. (64) For certain remarks on the divisibility properties of the iteration of Euler’s totient, see also W. Steindl [439]. By studying certain asymptotic properties of functions like ϕ ( j) (n)/ϕ ( j+1) (n), P. Erd¨os, A. Granville, C. Pomerance and C. Spiro [124] (see also [291] for details) proved also certain results on prime factors of ϕ ( j) (n). For example: There is a positive absolute constant c > 0, such that the set of natural numbers n, for which there is some j with ϕ ( j) divisible by every prime up to (log n)c , has asymptotic density 1. (65) 198

THE MANY FACETS OF EULER’S TOTIENT

Let (n) = n

∞

ϕ ( j) (n). Then

j=1

log n ω( (n)) ≤ log 2

(66)

(Here ω(m) denotes the number of distinct prime factors of m). In particular, for all n there is some prime p, with p log n log log n such that p (n)

(67)

The following two conjectures are also stated in [124]: 1) For each prime p, let N (x, p) denote the number of n ≤ x with p| (n). Then for every ε > 0, N (x, p) = o(x) uniformly for p > (log x)1+ε ; (68) (69) and N (x, p) ∼ x uniformly for p < (log x)1−ε . 2) For each ε > 0, the upper asymptotic density of the set {n ∈ N : ω(ϕ ( j) (n)) > n ε } tends to 0 as j → ∞ (70) The authors state (without proof) that by sieve methods they can prove (70) for ε > 2/3. Results related to the distribution of the distinct prime divisors of ϕ(n) are summarized in [291]. Recently these have been extended to the iteration function ϕ ( j) (n) as follows (see N. L. Bassily, I. K´atai and M. Wijsmuller [29]): √ For each fixed integer j ≥ 1 let a j = 1/( j + 1)! and b j = 1/ 2 j + 1 · ( j!). Then for each real number z, ω(ϕ ( j) (n)) − a j (log log x) j+1 1 lim car d n ≤ x :
CHAPTER 3

For k = 1 this has been previously proved by I. K´atai [220]. Let now λ( j) be the jth iterate of the Carmichael function and for n ≥ 1 let g(n) be the least positive integer j such that λ( j) (n) = 1. (73) By using some divisibility properties of the λ function, e.g. λ(ab) = [λ(a), λ(b)] for (a, b) = 1

(74)

(implying a|b ⇒ λ(a)|λ(b), which is similar to the corresponding property of Euler’s totient ϕ), I. Niven [322] proved the following fundamental properties of the function g: (the first is similar to (27)) g(ab) = max{g(a), g(b)} for (a, b) = 1;

(75)

g(22n ) = g(22n+1 ) = n + 1, g( p n ) = n − 1 + g( p), for any odd prime p, n ≥ 1.

(76)

Now let C(n) be the integer j such that ϕ ( j) (n) = 2 (n ≥ 3), with C(1) = C(2) = 0, be the Shapiro-Pillai function (see [291], p. 34). Then (see [322]) lim sup(g(n + 1) − g(n)) = lim sup(C(n) − g(n)) = +∞ n→∞

n→∞

(77)

and lim inf(g(n + 1) − g(n)) = −∞,

lim inf(C(n) − g(n)) = −1

n→∞

n→∞

(78)

We note that, in a little known paper, A. Mur´anyi [305] introduced a function in an essentially different manner as Shapiro, Pillai or Niven. For a positive integer n > 1 let m = m(n) be the integer such that ϕ (m) (n) = 2 if n is odd, and ϕ (m) (n) = 1 if n is even. Put m(1) = 0. Then (unlike the function C(n)) m(n) is completely additive function (see [305]), i.e. m(ab) = m(a) + m(b) for all a, b ≥ 1

(79)

Let Mk = {n ∈ N : m(n) = k}, k fixed. If a ∈ Ms and b ∈ Mk , with s ≥ k and b|a (i.e. b divides a), then writing a = b · t one has (for b = fixed) min t = 2s−k ,

max t = 3s−k

(80)

which give the minimum, respective maximum values of a ∈ Ms divisible by a fixed b ∈ Mk . Another result states that the number of elements of Ms which are divisible by a given integer a, is equal to the number of elements of the set Ms−m(a) . (81) 200

THE MANY FACETS OF EULER’S TOTIENT

For the function m the following estimates are true: log n log n ≤ m(n) ≤ log 3 log 2 (82) implies ax log x ≤

m(n) ≤ bx log x

(82)

(82 )

n≤x

with a, b > 0 constants. It is not known if m(n) ∼ cx log x

(82 )

n≤x

for certain constant c (see I. Z. Ruzsa [359]). Finally we note that Mur´anyi used the notation m(n) = grav(n) (i.e. the ”gravaritm” of n), in concordance with the logarithm of n.

6

Congruence properties of ϕ and related functions

We now study other congruence properties of Euler’s totient and related functions. Perhaps, the simplest congruence satisfied by ϕ(n) is ϕ(n) ≡ 0

(mod 2) for n ≥ 3

(83)

From the known form of ϕ (see formula (4) of 3.1) it follows that if for each prime p|n ⇒ p|m, then ϕ(mn) = nϕ(m) (84) Particularly, if n = p (prime), and p|m, then ϕ(mp) = pϕ(m), a formula frequently used in the theory of Euler’s totient. This is due to J. A. Grunert (see [103], p. 117). Another simple, but useful divisibility property of ϕ is m|n ⇒ ϕ(m)|ϕ(n) (so, ϕ(m) ≤ ϕ(n))

(85)

As a consequence of (85), ϕ(m)ϕ(n)|ϕ(mn) ⇔ gcd(m, n) ∈ {1} ∪ {2a · 3b : a > 0, b ≥ 0}

(86)

d , where d = gcd(m, n) (due to ϕ(d) E. Prouhet ([103], p. 118), we must have ϕ(d)|d, and this is Sierpinski’s relation (40), Indeed, by the identity ϕ(mn) = ϕ(m)ϕ(n)

201

CHAPTER 3

having solutions given by (41). A similar property is ϕ(mn) = (m, n)ϕ([m, n]), due to E. Lucas [272]. A counterpart of (85) is m|n, m < n ⇒ n − ϕ(n) > m − ϕ(m)

(87)

(see I. Niven and H. S. Zuckerman [323] (problem 22 of 2.4)). In 1913 U. Scarpis (see [103]) proved that if a is a prime, then n|ϕ(a n − 1) for all n ≥ 1, a ≥ 2.

(88)

In fact, (88) is true for all positive integers a. Indeed, since a n ≡ 1 (mod a n − 1), (a, a n − 1) = 1, but a s ≡ 1 (mod a n − 1) for s < n, so the order of a in Z a∗n −1 (the multiplicative group of reduced classes (mod a n − 1)) is n, which divides the order of the group, i.e. ϕ(a n − 1). More generally, N. G. Guderson [164] proved that if a > b and m = γ (n) = greatest squarefree divisor of n, then n2 |ϕ(a n − bn ), and n|ϕ(a n + bn ) γ (n)

(89)

See also P. Kesava Menon [221] for related results. See also a problem by R. E. Shafer [409] for the second relation of (89). In 1961 A. Rotkiewicz [356], by using (85) and the Zsigmondy-Birkhoff-Vandiver theorem on prime divisors of a n −bn (see e.g. [349], p. 43) deduced that T (n)|ϕ(a n − bn ) if a > b, n ≥ 1

(90)

where T (n) denotes the product of divisors of n. (In fact, T (n) = n d(n)/2 , where d(n) is the number of divisors of n, see Chapter 1). M. Deaconescu and J. S´andor [97] (see also J. S´andor ([363], pp. 232-233)) have considered functions f : N → N such that n| f (a n − 1) for all n, a > 1.

(91)

Relation (91) is true iff ϕ(n)| f (n) for all n. Related to the factors of a n − bn is also the following non-divisibility property (see [364], ([363], pp. 230-231)). Let a > b, (a, b) = 1. Suppose that (a, n) = (b, n) = (a − b, n) = 1. Then n (a n − bn ), and n (a n−ϕ(n) − bn−ϕ(n) )

(92)

n (2n−ϕ(n) − 1) for any n > 1.

(93)

Particularly,

202

THE MANY FACETS OF EULER’S TOTIENT

An improvement of (88) is due to H. Gupta [169]: For all a ≥ 2, p prime and j ≥ 1 one has p j ( j+1)/2 |ϕ(a p − 1) j

(94) j

This implies (88). Indeed, writing n = kp j with (k, p) = 1, since p j |ϕ(b p − 1) for all b ≥ 2, putting b = a k one gets p j |ϕ(a n − 1), and the result follows. Another improvement of (88) is the following divisibility property (see M. Deaconescu and J. S´andor [98]): n|ϕ(φn (a)) for all n ≥ 1, a ≥ 2

(94 )

where φn denotes the nth cyclotomic polynomial. Since φn (a)|(a n − 1), by (85) we get that (88) is a consequence of (94 ) (by the transitivity property of ”|”). Another refinement is n a −1 m|n, m < n ⇒ n|ϕ |ϕ(a n − 1), (94 ) am − 1 see [98]. There are also some congruence properties of the totient function, which are connected to other number-theoretical functions. For example, M. V. Subbarao [443] considered the relations ϕ(n)|(nσ (n) − 2),

(95)

n|(ϕ(n)d(n) + 2)

(96)

Clearly, (95) and (96) are satisfied by any n = p (prime). On the other hand, it is not difficult to see that all composite solutions of (95) are n = 4, 6 and 22. (97) The determination of all composite solutions of (96), however, is an Open problem (see [443]). Though n = 4 is a solution, no other solution is known. If n > 4 composite satisfies (96), Subbarao proves that n must have at least 4 distinct prime factors; n is squarefree; if p is an odd prime divisor of n, then there is no prime divisor of the form px + 1; if ϕ(n)d(n) + 2 = K n, then K and n are of opposite parity, and 4 K . (98) Also, for 1 ≤ K ≤ 1023 the possible number of prime divisors of n is studied. M. Zhang [486] considers the divisibility n|(ϕ(n) + σ (n)),

(99)

and proves that numbers of the form n = p α (α > 1) or n = p α q (α ≥ 1), where p, q are distinct primes, are not solutions of (99). For n < 107 there are 17 composite solutions. (Clearly, all n = p are solutions). The following Open problems are stated: 203

CHAPTER 3

a) Is there a solution n of (99) with ω(n) = 2? b) Is there an odd composite solution? c) Is there a solution n ≡ 2 (mod 6)? d) Are there infinitely many composite solutions? (100) For the values of ϕ at Fibonacci, or Lucas numbers, S. Singh [418] proves the following (note that 4|ϕ(Fn ) for all n ≥ 5, see C. Kimberling [224] (see also V. E. Hogatt and H. Edgar [113])) 2n|ϕ(F2n ), if n = 1, 2, 3, 4, 8, 16;

(101)

80n|ϕ(F5n ), if n > 1 is odd;

(102)

n|ϕ(L n ), if n > 3 is odd;

(103)

10n|ϕ(L 5n ), for all n;

(104)

2n |ϕ(L 2n+1 ), for all n.

(105)

The congruences like ϕ(F2n+1 ) ≡ 0 (mod 2n + 1)), ϕ(L 2n ) ≡ 0 (mod 2n) are not true in general. It is conjectured that 2n |ϕ(L 2n ) for infinitely n

7

(106)

Euler’s totient in residue classes

K. Ford [137] remarks that the following congruences are true: ϕ(n) ≡ 1 (mod 6) iff n = p 2α or 2 p 2α , (107) If n > 6, then 2 where α is a positive integer, and p is a prime, p ≡ 11 (mod 12). Similarly, ϕ(n) (108) ≡ 3 (mod 6) is n = p α or 2 p α , 2 where α is a positive integer, and p ≡ 7 (mod 12) is a prime; or else p = 3 and α > 1. ϕ(n) (109) ≡ 5 (mod 6) iff n = p 2α−1 or 2 p 2α−1 , 2 where α is a positive integer, and p ≡ 11 (mod 12) is a prime. ϕ(n) Relations (107)-(109) together cover all residues of (mod 6). Similar con2 gruences can be determined for any modulus of the form 2q, where q is an odd prime, or more generally to find all integers n such that ϕ(n) ≡k 2

(mod 2q),

204

(110)

THE MANY FACETS OF EULER’S TOTIENT

where k is odd (see [137]). Let Sm,r denote the set of integers n with ϕ(n) ≡ r

(mod m)

(111)

Then e.g. (107) can be rewritten as S12,2 = {3, 4, 6} ∪ {n : n or n/2 = p 2α } where p ≡ 11 (mod 12), etc. Let N (x, m, r ) = car d{n ∈ Sm,r : n ≤ x}, (112) which counts how many members Sm,r has up to x. P. Erd¨os (see [102]) proved the asymptotic result: N (x, m, 0) ∼ x as x → ∞,

(113)

for any integer m. The following asymptotic results are considered in the cases m = 12 or m = 3 (see T. Dence and C. Pomerance [102]) √ x 1 1 + √ ; (114) N (x, 12, 2) ∼ 2 2 2 log x x ; N (x, 12, 4) ∼ c1 log x

(115)

x 3 · ; 8 log x x N (x, 12, 8) ∼ c2 ; log x

(116)

3 x · ; 8 log x x ; N (x, 3, 1) ∼ c1 log x x N (x, 3, 2) ∼ c2 log x

(118)

N (x, 12, 6) ∼

(117)

N (x, 12, 10) ∼

where

√ 2 3 −1/2 c (2c3 + c4 ) and c3 = c1 = 3π 3 c4 =

p≡2(3) prime

1 1− , ( p + 1)2

p≡2(3) prime

(119) (120)

1 , 1+ 2 p −1

√ 2 3 −1/2 c2 = (2c3 − c4 ) c 3π 3

205

(121)

CHAPTER 3

Dence and Pomerance prove also (inspired by an argument by W. Narkiewicz [315]) that if the residue class r (mod m) contains a multiple of 4, then it contains infinitely many numbers ϕ(n). (122) In what follows, by a ”totient” we mean a value taken by Euler’s function ϕ(n). Since 1 is the only odd totient, by result (122) of Dence and Pomerance, it remains to examine residue classes consisting entirely of numbers ≡ 2 (mod 4). K. Ford, S. Konyagin and C. Pomerance [143] have characterized which of these residue classes contain infinitely many totients and which do not. The following negative result is true: For any ε > 0 there exist such m that at least (1 − ε)m residue classes a (mod 4m), 0 < a < 4m, a ≡ 2 (mod 4) are totient-free. (123) As a corollary of (123) one gets that the union of all totient-free residue classes has density 3/4. (124) On the other hand, holds true a positive result: The set of all odd numbers m such that for any s ≥ 1 and for any even a the residue class a (mod 2s m) contains infinitely many totients, has a positive lower density. (125) If a residue class r (mod m) contains infinitely many totients, the authors in [143] remark that using the methods of [102] and [316], it is possible to get asymptotic formulas for N (x, m, r ). A result of H. Delange [101] can be used to give an asymptotic formula for the number of n ≤ x for which m ϕ(n) (where m is fixed). For the distribution in residue classes of values of general multiplicative functions we quote also [315], [316], [101]. It is well known that the set of values of the function ϕ (i.e. the set of totients) has ˇ at [38] prove (asymptotic) density 0 (see [323], section 11.3). H. Berekov´a and T. Sal´ that the sets of positive values of the functions ϕ + d and ϕ − d have density 0. R. B. Killgrove [223] poses the problem if the set of values of ϕ is or not a Dirichlet set?

8

Prime totatives

If we consider the ϕ(30) = 8 integers that are ≤ 30 and relatively prime to 30, we find these to be 1, 7, 11, 13, 17, 19, 23, 29. Thus, apart from 1, all of these integers are primes. It is interesting that 30 is the largest integer with this property, i.e.: The number 30 is the largest integer such that all the integers less than it that are relatively prime to it are prime (apart from 1). (126) Usually, this result is attributed to H. Bonse [46] (who obtained a proof based on 2 his inequality p1 p2 . . . pk ≥ pk+1 , where pk is the kth prime (see [291], Chapter 7)). 206

THE MANY FACETS OF EULER’S TOTIENT

However, this was first proved by S. Schatunowsky (see [103], p. 132). E. Maillet [278] gave a generalization, as a solution to an Open problem posed by G. de Rocquigny [355] (see also [103]). In 1984 H. Iwata [207] rediscovered this result which can be stated as follows: There are only finitely many integers b such that all smaller positive integers n, with (n, b) = 1, are products of at most k prime factors. (127) In 1989 L. Cseh [91] shows that if k = 1 and n, b are restricted to be odd integers, then the largest b is 105. (128) This was an Open question of S. W. Golomb. In fact, Cseh extends the Maillet (Iwata) theorem for Beurling’s generalized primes (see Chapter 2). We note that Bonse [46] proved Maillet’s theorem (127) for k = 1, 2, 3 without using Chebysheff’s theorem, which states that there is always a prime between a and 2a (a ≥ 2). Later, in 1909, R. Remak (see [103], p. 138) proved completely (127) for all k, without using the Chebysheff theorem. Another proof was obtained by E. Landau (see [103], p. 138). For integers a such that (a, n) = (a + 1, n) = 1. L. Carlitz [61] showed the following: Let n be odd. Then ≡ 1 (mod n) and ≡ −1 (mod n) (129) 1≤a
Similarly, for (n, b) = 1,

1≤a

(a + 1) ≡ 1 (mod n)

(130)

1≤a
For k ≥ 1 a fixed integer, let Sk (n) denote the number sets of k consecutive integers each less than n and relatively prime to n (this is a generalization of Euler’s totient). For 1 ≤ r ≤ k let P denote the product of the r th terms of the Sk (n) sets of consecutive integers described above. V. Schemmel [391] proved that p k−1 ≡ {(−1)r −1 (r − 1)!(k − r )!} Sk (n)

(mod n)

(131)

For r = 1, k = 2 this becomes P ≡ 1 (mod n). For squarefree integers which are relatively prime to n, the following result has been proposed by E. B. Cossi [89]. Let R(n) denote the number of ordered pairs of squarefree integers j, k such that j + k = n and ( j, n) = (k, n) = 1. Then R(n) > cϕ(n) for n ≥ n 0 .

(132)

In fact, J. Herzog and P. R. Smith (see the solution from [89]) showed (132) with 1 , n 0 = 6000000. Since R(n) ≥ 2 for 2 < n ≤ 6000000 (see above), (132) c= 200 holds with n 0 = 1 for a suitable constant c > 0. 207

CHAPTER 3

9

The dual of ϕ, noncototients

Relation (11) gives R´edei’s generalization of Euler’s theorem. This relation involves the dual function ϕ(n) = n − ϕ(n),

n≥1

(133)

Certain divisibility properties of ϕ, similar to that of ϕ, are introduced in L. Panaitopol [327]. For example, ϕ(n)|n iff n = p α

( p prime, α ≥ 1).

(134)

The relation

ϕ(m) ϕ(n) = is true iff γ (m) = γ (n), (135) m n where γ (m) denotes the core of n (i.e. greatest squarefree divisor of n; see [291], Chapter 6). Let l(n) = (ϕ(n), ϕ(n + 1)), n ≥ 3. (136) Then for any odd number m, there exist infinitely many integers n such that (137)

m|l(n)

The following Open question is stated: If k is a positive integer, and m is odd, is there an n such that m|(ϕ(n), ϕ(n + 1), . . . , ϕ(n + k))?

(138)

ϕ(n)|(n − 1)

(139)

If n is prime, clearly On the other hand, there are many composite solutions of (139). For example, n = 15, 85, 225 259, 391, etc. In fact, there are 21 such composite numbers n with n < 105 . It is proved that all n satisfying (139) must be odd and squarefree; n−1 must be even, and P(n) > hω(n) (140) h= ϕ(n) where P(n) is the greatest prime factor of n. In 1959 W. Sierpinski asked, whether there are infinitely many integers not of the form ϕ(n) (i.e. noncototients, see [171], p. 91). Recently, J. Browkin and A. Schinzel [53] have shown that the equation ϕ(x) = 2k · 509203 208

(k = 1, 2, . . . )

THE MANY FACETS OF EULER’S TOTIENT

has no solutions. The following Open problem is stated: do the integers not of the form ϕ(n), have a positive lower density? It is not known if there exist infinitely many composite solutions of (139). (141) A number of asymptotic results involving ϕ are included in [328]. For a fixed positive integer m, m ϕ(n) x2 (k − 1)! 2 . +O =x ϕ(n) 2k (log x)k logm+1 x 2≤n≤x k=1

(142)

For any a > 0, n≤x

m 1 (k − 1)! +O = x k a (ϕ(n)) k=1 log x

x logm+1 x

.

(143)

For a fixed positive integer k one has

ϕ(n) · ϕ(n + k) = x

n≤x

2

4 1 c(k) + − 2 3 π

+ O(x 2 log2 x),

(144)

2 1 1 . 1− 2 1+ where c(k) = 3 p prime p p( p 2 − 2) p|k In 1980 P. Erd¨os [110] (see also R. K. Guy [171]) has conjectured that: i) ϕ(n) > ϕ(ϕ(n)) for almost all n; (145) ii) ϕ(n) < ϕ(ϕ(n)) for infinitely many n. Recently, A. Grytczuk, F. Luca and M. W´ojtowicz [160] have settled ii), by proving that there exists an infinite sequence of positive integers (n k ) such that ϕ(n k ) + 2k ≤ ϕ(ϕ(n k ))

(146)

For conjecture i) of (145) it is proved that if n > 2 is such that the odd part of n is squarefull, then n satisfies the inequality of i). (147) Particularly, the lower asymptotic density of numbers n with inequality of i) is ≥ 0.54. (148) It is immediate that lim sup n→∞

ϕ(n) − ϕ(ϕ(n)) =1 n

The exact value of lim inf, however is an Open question (see [160]). n→∞

209

(149)

CHAPTER 3

10

Euler minimum function The Euler minimum function is defined by E(n) = min{k ≥ 1 : n|ϕ(k)}

(150)

It was introduced by P. Moree and H. Roskam [297], and independently by J. S´andor [365] (see also [363], pp. 141-149) as a particular case of the more general function (151) F fA (n) = min{k ∈ A : n| f (k)}, (A ⊂ N) if such a function exists. For A = N, f = ϕ one obtains the function E given by (150). Since by Dirichlet’s theorem on arithmetical progressions there exists a ≥ 1 such that k = an + 1 is prime, by ϕ(k) = an is a multiple of n, so E(n) exists. We note that for A = N, f (k) = k! one obtain the Smarandache function, while for A = P (set of primes), f (k) = k! one gets a new function [365]. There is also a ”dual” of (151), namely G gA (n) = max{k ∈ A : g(k)|n},

(152)

if this is well defined. For A = N, g(k) = k! this is denoted by S∗ (n) and studied in [365]. For f (k) = ϕ(k), A ⊂ N in (151) one obtains a generalization of the Euler minimum function, while from (152) one can introduce the Euler maximal function by (see [366]) E ∗ (n) = max{k ≥ 1 : ϕ(k)|n} (153) Since ϕ(1) = 1|n for all n, this function is well defined. It can be proved that (see [366]) that max{E( pa ), E(q b )} ≤ E( pa · q b ) ≤ [E( pa ), E(q b )] for p = primes

(154)

so one needs to study first E on prime powers. Let q be a prime, and denote by A(q) the set of squarefree numbers composed of only primes p satisfying p ≡ 1 (mod q). For convenience, let the empty set have minimum equal to ∞. Then Moree and Roskam prove ([297]): E(q n ) = min{m, q n+1 }, where m = min{a ∈ A(q) : q n divides ϕ(a)/q n < q} (155) For (u, v) = 1 let p(v, u) denote the smallest prime with p ≡ u (mod v) and pi (v, u) (i ≥ 2) the ith smallest such prime. As a corollary of theorem (155) one has: The largest prime divisor of ϕ(E(q n )) is q; The smallest prime divisor of E(q n ) is ≥ q; 210

THE MANY FACETS OF EULER’S TOTIENT

log q ; If q is odd, then ω(E(q )) < min n + 1, log 2

n

(156)

E(q) = min{q 2 , p(q, 1)}; E(q 2 ) = min{q 3 , p(q 2 , 1), p(q, 1) p2 (q, 1)}. This corollary shows that the behaviour of E is strongly related to the distribution of primes in arithmetic progressions. Another consequence of (155) is that E( pa ) = E(q b ) if p = q (primes); E( pa ) = E( p b ) if a = b

(157)

The prime 2 has the property that E(2n ) = 2n+1 for infinitely many n. Let M denote the set of primes p such that E( p n ) = p n+1 for infinitely many n. Let q be an odd prime, and suppose that there are integers a, d, n 0 such that E(q n ) = min{q n+1 , p(q n , 1)} for n ≥ n 0 , and n ≡ a (mod d). Then q ∈ M. (158) It can be proved that 2, 3, 5, 7, 13, 19, 31 ∈ M, and in [297] it is conjectured that every prime is in M. Another function, connected also to the Euler minimum function is the Euler l.c.m. sequence, defined by ek = lcm(ϕ(1), ϕ(2), . . . , ϕ(k)), Let ak =

k = 1, 2, . . .

(159)

ek

(k ≥ 2). We say that k ≥ 2 is a jumping point if ak > 1. Then ek−1 the following interesting fact is true: The number k ≥ 2 is a jumping point iff k = E( pr ) for some prime p and positive integer r > 1. (160) This implies that for k ≥ 2, ak is a prime, or equals 1. (161) For the growth of the Euler l.c.m. sequence we have (see [297]): exp(k .6687 ) ek exp((1 + ε)k)

(162)

for ε > 0 fixed, but arbitrary. We note that the result on the functions E and e are used in [297] in the study of the discriminator D( j, n) = smallest positive integer k for which the first n jth powers are distinct modulo k. This notion has been introduced by L. K. Arnold, S. J. Benkoski and B. J. McCabe [15]. (For more References, see [297]). The study of the discriminator involves also the Fermat’s quotient of a prime p, related to the congruence a p−1 − 1 ≡ 0 (mod p 2 ). 211

CHAPTER 3

For many results, as well as open problems on Fermat’s quotients, see e.g. [349] (pp. 335-346). Recently, the Fermat’s quotients have been extended to composite moduli m by A. Takashi, K. Dilcher and L. Skula [452] who considered the congruence a ϕ(m) − 1 ≡ 0 (mod m 2 ) (163) for composite m. If m is such an integer, then P(m) = largest prime factor of m, satisfies the congruence a P(m)−1 − 1 ≡ 0

(mod (P(m))2 )

(164)

(i.e., is a Wieferich prime). With a given P(m), all solutions of (163) can be determined (see [452]).

11

Lehmer’s conjecture, generalizations and extensions

Finally, we shall deal with a famous divisibility problem on Euler’s totient, namely Lehmer’s conjecture. We will include also some analogues problems on Euler’s or related totients. In 1932 D. H. Lehmer [250] considered the relation ϕ(n)|(n − 1),

(165)

and asked if there are any composite solutions. This problem is still open (see e.g. [171]), ”many people feeling it is as difficult as the odd perfect number problem” (quotation from C. Pomerance [333]). Let n be a composite solution of (165). Lehmer proved that then n must be squarefree, odd, and ω(n) ≥ 7 (where ω(n) denotes the number of distinct prime factors of n). Further, he showed that if pq|n ( p, q distinct primes), then q ≡ 1 (mod p). (166) In 1944 F. Schuh [399] proved that if 3|n, then k ≡ 1 (mod 3), where k=

n−1 . ϕ(n)

(167) (168)

Clearly, k = 1 iff n = prime. He ”proved” also that ω(n) ≥ 11, (169) but this proof was not correct. In 1970 E. Lieuwens [256] corrected Schuh’s proof, so result (169) was established. Lieuwens proved also that if 3|n, then ω(n) ≥ 212 and n > 5.5 · 10570 212

(170)

THE MANY FACETS OF EULER’S TOTIENT

Similarly, he proved that if 3 n, 5 n (i.e. the least prime factor of n is ≥ 7), then ω(n) ≥ 13. (171) In 1977 M. Kishore [225] proved without any condition that (171) is true, i.e. ω(n) ≥ 13. (172) Even Lehmer ([250]) showed that if k = 3 in (168), then ω(n) ≥ 32, (173) and Kishore proved that for any k > 2 one has ω(n) ≥ 33. (174) In 1980 G. L. Cohen and P. Hagis Jr. [86] improved result (171) to ω(n) ≥ 17, and result (172) to ω(n) ≥ 14. (175) D. W. Wall [472] improved (171) to ω(n) ≥ 26. (176) Cohen and Hagis have obtained without any condition the bound n > 1020 ,

(177)

which apparently is the best current result of this type. In 1985 M. V. Subbarao and V. Siva Rama Prasad [445] have studied the analogue of Lehmer’s problem to the unitary totient function ϕ ∗ , i.e. ϕ ∗ (n)|(n − 1) Since ϕ ∗ (n) =

(178)

( p α − 1), it is immediate that ϕ ∗ (n) = n − 1 if and only if n is

p α n

a prime power: n = p α ( p prime, α ≥ 1). We shall call n a proper solution of (178), if ϕ ∗ (n) = n − 1, i.e. when n−1 >1 (179) k∗ = ∗ (ϕ (n) Let S denote the set of composite solutions of (165) and S ∗ , the set of proper solutions of (178). Since S contains only squarefree numbers n, and ϕ(n) = ϕ ∗ (n), for such n, clearly (180) S S∗ i.e. S is a proper subset of S ∗ . Subbarao and Siva Rama Prasad prove that any solution is odd, and not a powerful number; (181) if 3|n, then ω(n) ≥ 1850. (182) By (180), relation (182) improves on Lieuwen’s bound on ω(n) given by (170). Further, if 3 n, 5 n,then ω(n) ≥ 11; (183) if 3 n, 5 n, then ω(n) ≥ 17; (184) if k ∗ = 3, 4, 5 in (179), then ω(n) ≥ 33. (185) ∗ Similarly: If k = 5 and n is squarefree, then ω(n) ≥ 53. (186) If k ∗ = 6, then ω(n) ≥ 140 or 48, according as n is squarefree, or not. (187) 213

CHAPTER 3

If k ∗ = 7, then ω(n) > 200 or 103 according as n is squarefree, or not. If k ∗ ≥ 8, then ω(n) > 200. Another result says that if 2 < ω(n) ≤ 16, then k ∗ = 2, 3 n, 5|n and 7|n.

(188) (189)

(190) Let Jk be the Jordan totient function (see (42) of 3.1). The analogue of Lehmer’s problem for Jk can be easily settled. Namely, it is not difficult to prove (see [445]) that if k > 1, then Jk (n)|(n k − 1) iff n = prime (191) In 1989 V. Siva Rama Prasad and M. Rangamma [424] have obtained certain results on the form of solutions of Lehmer’s problem (165). Let s be the number of primes of the form p ≡ 2 (mod 3) in the prime factorization of n. If 3 n, then n must have the form 214 · 3 · m + 1 or 214 · 3 · m + 65537, according as s is even, or odd. (192) P. Hagis, Jr. [173] further refined Kishore’s result (174) that for any k > 2, ω(n) ≥ 1991 and n > 108171 . (193) Estimates (182) are further improved to 3|n ⇒ ω(n) ≥ 298848, and n > 101937042

(194)

In an unpublished manuscript, M. Deaconescu and J. S´andor [99] have shown that if a composite n has an odd number of prime factors p ≡ 3 (mod 4), then n cannot be a solution of (165). Recently, A. Grytczuk and M. W´ojtowicz [163] have shown that if n is a composite solution to (168) with least prime divisor p1 = 3, then ω(n) ≥ 3049k/4 − 1509. If p1 > 3, then ω(n) ≥ 143k/4 − 1. C. A. Nicol [318] proved that n ≥ 2 is prime iff ϕ(n)|(n − 1) and (n + 1)|σ (n). We now give estimates for the solutions, or the number of solutions of the considered problems. Let n ∈ S be a solution of Lehmer’s problem (165). Then r

n < r2 ,

(195)

where r = ω(n) (see C. Pomerance [334]). Subbarao and Siva Rama Prasad [445] have refined this (on view of (180)) to: If n ∈ S ∗ is a solution of problem (178), then r −1

n < (r − 1)2

(196)

Let N (x) = car d{n ≤ x : n ∈ S}. By sharpening a result of P. Erd¨os [109], in 1977 Pomerance [334] derived: N (x) = O(x 1/2 (log x)3/4 (log log x)−1/2 ). 214

(197)

THE MANY FACETS OF EULER’S TOTIENT

Shan Zun [410] improved the exponent 3/4 to 1/2. (198) Let P denote the set of all odd primes. Grytczuk and W´ojtowicz [163] have proved that for all m ≥ 2 there exists an infinite subset P(m) of P such that: i) for every distinct primes p1 , . . . , pm ∈ P(m) the square-free number n = p1 . . . pm is not a solution of (168); ii) P(m) is maximal with respect to inclusion. For the corresponding set N ∗ (x) = car d{n ≤ x : n ∈ S ∗ } one has N ∗ (x) = O(x 1/2 (log x)2 (log log x)−2 );

(199)

see [445]. Based on a remark of A. Schinzel (see [171], p. 92) that for n = 2, p or 2 p (where p is an odd prime) one has (ϕ(n) + 1)|n, and if the converse is true, G. L. Cohen and S. L. Segal [88] have studied the relation (ϕ(n) + 1)|n.

(200)

They proved the following theorem: If n satisfies (200), then one of the following assertions is true: a) n = 2, p, or 2 p (where p is an odd prime); b) n = mt, where m = 3, 4, or 6; (m, t) = 1 and t − 1 = 2ϕ(t); c) n = mt, where (m, t) = 1, ϕ(m) = j ≥ 4 and t − 1 = jϕ(t). Thus, in case b) one can write (according to (175)) ω(t) ≥ 14, while in case c) ω(t) ≥ 140. As a corollary, if Lehmer’s problem has no solutions, then n = 2, p, 2 p are the only solutions of (200). If there are other solutions, too, then they have at least 15 distinct prime factors. (201) Let N1 (x) = car d{n ≤ x : n = 2, p, 2 p and satisfies (200)}. Then, (see [88]), N1 (x) = O(x 1/2 (log x)3/4 (log log x)−5/6 )

(202)

Finally, we want to mention certain other analogs of Lehmer’s problem. R. L. Graham (see [171], p. 93) makes the following conjecture: For all k there are infinitely many n such that ϕ(n)|(n − k)

(203)

This is true for k = 0, k = 2α (α ≥ 0) and k = 2α · 3β (α, β > 0). For fixed k, C. Pomerance [335] has shown that the number of elements n ≤ x satisfying (203) is x ; (204) O log x 215

CHAPTER 3

in particular, for fixed k, the numbers n have asymptotic density 0. He also showed that for each k, (203) has at least four solutions. V. Meally (see [171], p. 93) remarks that ϕ(n)|(n + 1)

(205)

has at least two solutions. Are there infinitely many? M. Deaconescu [95] proved the inequality ϕ(n)(ϕ(n) − 1) ≥ (n − 1)S2 (n),

(206)

where S2 (n) is the case k = 2 of Schemmel’s totient ([391]) (see relations (26) of 3.1 and (131)), and with equality iff n = prime. Therefore (n − 1)|ϕ(n)(ϕ(n) − 1) iff n = prime

(207)

Deaconescu conjectures that for n ≥ 2 S2 (n)|(ϕ(n) − 1) iff n = prime,

(208)

and expects that this to be as hard as Lehmer’s conjecture. For an extension of S2 (n) with applications in group theory, see M. Deaconescu and H. K. Du [96]. Let n ≥ 3. Then (ϕ(n))2 |(n 2 + 1). (209) is impossible, since ϕ(n) is even, and −1 is not a quadratic residue mod4. On the other hand, if (ϕ(n))2 |(n 2 − 1), (210) one obtains a difficult (but treatable) problem. By applying primitive divisors of Lucas sequences, related to a Pell equation, as well as a form of the Brun-Titchmarsh theorem (see [291], Chapter 8) due to H. L. Montgomery and R. C. Vaughan [294]. F. Luca and M. Kriˇzek [270] have recently proved that all solutions of (210) are n = 1, 2, 3. (211)

3.3 1

Equations involving Euler’s and related totients Equations of type ϕ(x + k) = ϕ(x)

Perhaps the oldest equation related to Euler’s totient is ϕ(x) = ϕ(x + 1) 216

(1)

THE MANY FACETS OF EULER’S TOTIENT

In 1917 R. Ratat (see [103] p. 140) noted that x = 1, 3, 15, 104 are solutions; a year later R. Goormaghtigh found the solutions x = 164, 194, 255, 495. All other solutions up to x ≤ 10000 are x = 584, 975, 2204, 2625, 2834, 3255, 3705, 5186, 5187 (see V. Klee [227] and L. Moser [302]). (2) In 1972 M. Lal and P. Gillar [236] used an IBM1620, Model1, to determine all solutions for x ≤ 105 . In 1978 M. Yorinaga [485] found 146 solutions for x ≤ 10928925, by using a computer HITAC20. R. Baillie [23] made extensive computations to obtain all solutions for x ≤ 108 . He found 306 solutions of equation (1). (3) In 1999, S. W. Graham, J. J. Holt and C. Pomerance [156] by using C++ programming language to implement simple sieving and scanning procedures to compute the necessary ϕ-values and then look for solutions of (1), have obtained all solutions for x ≤ 1010 . There are 1267 solutions. (4) It is known that ϕ(30n + 1) > ϕ(30n) for all n ≤ 2 · 107 . However, D. J. Newman [317] points out that this kind of numerical evidence may be extremely misleading. In fact, he proves that if a, b, c, d are nonnegative integers with a, c > 0 and ad − bc = 0, then there exists a positive integer n for which ϕ(a · n + b) < ϕ(c · n + d). The equation ϕ(x + 2) = ϕ(x), (5) has the solutions x = 4, 7, 8, 10, 26, 32, 70, 74 for x ≤ 100. There are 80 solutions satisfying x ≤ 10000. L. Moser [302] noted that if p and 2 p − 1 are both odd primes, then x = 2(2 p − 1) is a solution. However, it is not known (and it is an extremely difficult problem) that there are infinitely many such primes. Lal and Gillard [236] obtained all solutions x ≤ 105 , M. Yorinaga [485] those of x ≤ 4 · 106 (in total 7998), while for x ≤ 1010 there are in total a number of 7558421 solutions (see [156]). (6) The equation ϕ(x + 3) = ϕ(x) (7) has the curious property that has only two solutions for x ≤ 10000, namely x = 3, 5 (see A. Schinzel [394]). D. H. Lehmer extended the range to x ≤ 106 (see [171], p. 90). Finally, Graham, Holt and Pomerance considered x ≤ 1011 , (8) without finding any new solutions. In 1956 W. Sierpinski [415] has shown that the equation ϕ(x + k) = ϕ(x)

(9)

has at least one solution for each value of k. In 1958 A. Schinzel [394] showed that there are at least two solutions to (9) for all k ≤ 8 · 1047 . (10) A year later, A. Schinzel and A. Wakulicz [396] have increased the bound to k ≤ 2 · 1058 . (11) 217

CHAPTER 3

In fact, the following two propositions by Schinzel have been applied: Let k be odd. Then suppose that a sequence of primes 3 = p1 < p2 < · · · < pm satisfies the conditions i) ( pi − 2)| p1 p2 . . . pi−1 (2 ≤ i ≤ m), (12) ii) ( pi − 1)|∗ 2 p1 p2 . . . pi−1 (2 ≤ i ≤ m), where a|∗ b means that each prime factor of a also is a prime factor of b. Suppose k is not divisible by p1 p2 . . . pm , and let p j be the smallest prime in the product that does nor divide k. Then pjk (13) x= pj − k is a solution to (9). Let now k be even. Let q1 , q2 , . . . , qm be a sequence of primes such that a) 2qi − 1 is prime (1 ≤ i ≤ m), (14) b) 2qi − 1 = q j (1 ≤ i, j ≤ m). Suppose k < q1 q2 . . . qm . Then there exists a prime q j in the sequence such that q j and 2q j − 1 both do not divide k, and x = (2q j − 1)k

(15)

is a solution to equation (9). Using Mathematica, and the built-in function Prime Q, which test primality using a combination of the Miller-Rabin and Lucas tests, recently J. J. Holt [199] has raised the bound to k ≤ 5 · 1038926 is the result (10) (k = even). (16) There is a conjecture of L. E. Dickson [104] known as the ”prime k-tuple conjecture”. A special case of this conjecture is that there are infinitely many primes p with 2 p − 1 prime. Therefore, this conjecture implies that for k even, there are infinitely many solutions to equation (9) (known as Schinzel’s conjecture). This remark (by Schinzel) appears also in M. Zhang [487]. There are also some solutions of particular type of the considered equation. n D. Klarner [226] remarks that equation (1) has solutions x = 22 − 1 only for n = 0, 1, 2, 3, 4, 5. (17) A. Schinzel [394] shows that for all positive integers n, there exists a positive integer m such that the equation (9) for k = n m has at least two solutions. (18) Another method of finding solutions to equation (9) is due to Graham, Holt and Pomerance [156]: Suppose that k is even and that j and j + k have the same prime factors. Let g = ( j, j + k), and suppose that for a positive integer r , j j +k r + 1 and r +1 g g 218

(19)

THE MANY FACETS OF EULER’S TOTIENT

are both primes that do not divide j. Then j +k x= j r +1 g

(20)

is a solution of equation (9). It is known that, for a given k, there are only finitely many values of j such that j and j + k have the same prime factors (see e.g. R. Tijdeman [453], and the References therein). For example, for each even k ≤ 30 all possible values of j can be completely determined by elementary techniques (based on the study of certain exponential diophantine equations). Related to the asymptotic number of solutions of the considered equations, many conjectures have been stated. Based on the Hardy-Littlewood conjecture on the distribution of prime pairs ( p, 2 p − 1), p ≤ n (see [178]), M. Yorinaga conjectures that P(2, n) ∼

c2 n as n → ∞, · 2 log2 n

(21)

where P(k, n) denotes the number of solutions x ≤ n of the equation (9), and 1 c2 = 1− = 0.6601618 . . . is the twin-prime constant. His ( p − 1)2 p>2 prime conjecture is supported by numerical tables. Without any assumption, Graham, Holt and Pomerance [156] prove the following result. Let P1 (k, n) be the number of solutions x ≤ n of equation (9), which are not in the form (20). In particular, when k is odd, P1 (k, n) = P(k, n). Then for every k, there is some n 0 (k) such that if n ≥ n 0 (k), then P1 (k, n) < n/ exp(log1/3 n). (22) The case k = 1 is due to P. Erd¨os, C. Pomerance and A. S´ark¨ozy [128]. Assume now that the following conjecture of Hardy-Littlewood type (see P. T. Bateman and R. A. Horn [111]) is true: Suppose that a and b are relatively prime, and b < a. Then n p−1 1 1 ∼ 2c2 dt (23) p − 2 log(at) log(bt) 2 r ≤n,an+1,bn+1 primes p|ab(a−b), p>2 Now, as a corollary of theorem (22) one obtains. ∗ ∗ p − 1 ∗ g , where runs over all Let k be even, and let c(k) = j ( j + k) p−2 ∗ runs over all primes p > 2 j such that j and j + k have the same prime factors, such that p| jk( j + k)/g 3 , and g = ( j, j + k). Then 0 < c(k) < ∞; and if conjecture (23) is true, then as n → ∞ P(k, n) ∼ 2c2 c(k) 219

n log2 n

(24)

CHAPTER 3

Let now P0 (k, n) be the number of solutions x ≤ n of equation (9) in the form (20). Then P(k, n) = P0 (k, n) + P1 (k, n), and corollary (24) asserts that if (23) is true, then P(k, n) ∼ P0 (k, n) as n → ∞

(25)

for each k. The following theorem is surprising, since it shows fixed even number P(k, n) ∼ P0 (k, n) or P(k, n) ∼ P0 (k, n) are not that k≤n

k≤n

true: Let c=

k≤n, k even

∗∗ 1≤a
where in c < ∞, and

∗∗

r

k≤n

br 2 (ar + 1)(br + 1)2 γ (a)γ (b)

(26)

one has (b, ar + 1) = 1 and ar + 1, br + 1 are both primes. Then (27)

P0 (k, n) ∼ cn as n → ∞

(28)

k≤n

A generalization of equation (9) to arithmetic progressions is ϕ(x) = ϕ(x + k) = ϕ(x + 2k) = · · · = ϕ(x + qk)

(29)

For k = 1, q = 1, as we have seen from relation (2), x = 5186 is a solution. No similar progression with difference q = 1 is known. Note that a solution with q ≥ 3 immediately implies that ϕ(x) = ϕ(x + 3), so by (8), we must have x > 1011 . (30) P. Erd¨os [111] has conjectured that for k = 1 and arbitrary q, equation (29) is solvable. By numerical verifications many progressions of length 3 (i.e., q = 3) can be found. For example, there is exactly one progression of length 3, when k ∈ {1, 2, 4, 5, 8, 11, 14, 23, 25, 26, 28, 29, 31, 37, 38, 41, 46, 47, 52, 53, 55, 56, 58, 59, 62, 67, 71, 73, 74, 76, 79, 80, 85, 86, 89, 92, 94, 97, 98} (31) There are exactly two progressions of length 3, when k ∈ {16, 17, 22, 32, 34, 43, 44, 61, 82, 83, 88}. (32) There are exactly three progressions of length 3, when k ∈ {64, 68}. (33) The first progression of length 6 has k = 30; x = 583200 (see also the Sloane encyclopedia [429], sequence A050518). An extension of theorem (20) for the equation (29) can be stated as follows (see [156]): 220

THE MANY FACETS OF EULER’S TOTIENT

Suppose that j, j + k, . . . , j + qk all have the same prime factors. Define B = j ( j + k) . . . ( j + qk). For i = 0, . . . , q define B , g = gcd(b0 , b1 , . . . , bq ), bi = j + ik (34) bi B = ai = g ( j + ik)g Suppose that for some positive integer r , a0r + 1, a1r + 1, . . . , aq r + 1

(35)

are all primes that do not divide j. Then x = j (a0r + 1) =

Br +j g

(36)

is a solution to (29). As a corollary one obtains that, if one assumes the conjecture (a particular case of Dickson’s conjecture, or Schinzel’s Hypothesis H [395]) that for distinct positive integers a0 , a1 , . . . , aq , the integers (35) are all primes for infinitely many r , then for any positive integer q there exists a positive integer k such that (29) has infinitely many solutions. An arithmetic progression of length 10 with equal phi-values is the following (see [156]): ϕ(502781177052210 + 210i), 0 ≤ i ≤ 9 (37)

2

ϕ(x + k) = 2ϕ(x + k) = ϕ(x) + ϕ(k) and related equations A similar equation to the above considered ones is ϕ(x + k) = 2ϕ(x)

(38)

A. Makowski [279] has shown that for each positive integer k has at least a solution. Indeed, if (k, 6) = 1, let x = 2k; if k is even, i.e. k = 2a b (a ≥ 1, b odd), put x = 2a b; if k is odd and divisible by 3, i.e. k = 6s + 3, let x = 2s + 1. Therefore, (38) is solvable for any k. (39) J. S´andor [368], as well as L. Cseh and I. Mer´enyi [92] have studied the equation ϕ(x + k) = 3ϕ(x)

(40)

The density of positive integers k for which (40) has at least a solution is at least 1/2; (41) in fact D. B. Tyler [461] stated that the density is at least 0.864 221

(42)

CHAPTER 3

but no doubt, that the equation is solvable for all k, except k = 2. This seems to be still open. Particular solutions for 1 ≤ k ≤ 10 have been analyzed by S´andor [368] and recently by J. S´andor and L. Kov´acs [381]. For k = 2 there are no solutions up to x ≤ 106 , while for k = 1 there are 42, for k = 3 there are 18, for k = 5 there are 83, for k = 7 there are 63 solutions up to x ≤ 106 . For k = 4 the single solution is x = 3; for k = 6 the single solution is x = 3, for k = 8 there are two solutions: x = 5, 6; and for k = 10, x = 4, 9 (where x ≤ 106 ). A. Makowski [280], [281] has studied the equation ϕ(x + k) = ϕ(x) + k

(43)

Since ϕ(x) ≡ 0 (mod 2) for x ≥ 3, it easily follows that for k = odd, equation (43) is solvable only when k + 2 is prime, in which case the only solution is x = 2. (44) For k = 2a or k = 3 · 2a (a ≥ 1) one may take x = 3 · 2a and x = 7 · 3a , so in these cases the equation is solvable. (45) a More generally (see [281]) if p and q = p − p + 1 are prime numbers, for k = ( p − 1) pa one obtains the solution x = q p s . (46) Therefore, the number of solutions of this form depends essentially on the number of known Mersenne primes (see Chapter 1 of this book). Hypothesis H of Schinzel ([395]) states the following: Suppose there are given s irreducible polynomials f 1 (x), f 2 (x), . . . , f s (x) with integer coefficients, positive for large values of x and such that there is no integer > 1 which is a divisor of f 1 (x) f 2 (x) . . . f s (x) for every x. Then there exist infinitely many x such that all of f 1 (x), f 2 (x), . . . , f s (x) are prime numbers. (47) By assuming (47), Makoswski shows that for all even k, equation (43) has infinitely many solutions. (48) For k = 2 the equation (43) is particularly interesting, since all x with (x, x + 2) twin primes, is a solution. But there are also composite solutions, e.g. x = 12, 14, 20, 44. L. Moser [302] has shown that there are no composite odd solutions x < 10000. (49) Perhaps there are no other odd solutions x excepting when (x, x + 2) is a twinpair. It is easy to see that if p > 2 and 2 p + 1 are both primes, then x = 4 p is a solution of the equation. Another open problem related to this question is due to A. Makowski ([414], p. 232): Has the system of equations ϕ(x + 2) = ϕ(x) + 2,

σ (x + 2) = σ (x) + 2

(50)

at least a composite solution? Another equation introduced by A. Makowski [282] is ϕ(x + k) = ϕ(x) + ϕ(k) 222

(51)

THE MANY FACETS OF EULER’S TOTIENT

He showed that at least one solution exists if k is even, or 3 k, or k = i . . . Fsas , where Fi = 22 + 1 is a Fermat number, ai > 1 (i = 0, s), Fs+1 = prime, and (m, 2F0 F1 . . . Fs Fs+1 ) = 1. (52) A particularly interesting case is k = 3, when it is at least not known whether there is a solution (see [282]). (53) In [171] (p. 91) one can read that J. Browkin showed that for k = 3 there is no solution with x < 37182142. Recently J. S´andor and L. Kov´acs [381] have verified this for x ≤ 3 · 108 . P. Jones [214] has considered the case k = 3, too. She showed that any solution, if exists, must be of the form x = 2 p α or x = 2 p α − 3, where p > 3 is a prime. (54) Further, if x is a solution, then (i) x or x + 3 has at least 33 distinct prime factors, or (ii) x = 2 p α (α odd), p ≡ 2 (mod 3), x > 1011 and x + 3 has at least 9 distinct prime factors. (55) For the general equation (51), Jones has proved that for k = 3m odd it has a solution, in the following cases: a) p α k, p β = q − 2, α > β and q k; b) p k, p = 3q − 4, and q k; (56) c) p k, p = 9q − 16, and q k; d) p k, p = 3α q − 2a r, 3α − 1 = 2a−1 (r + 1), q k, and r k. Finally, if 2m + 1 = 3α n, where (3, n) = 1, α ≥ 0; and if we suppose there exists a positive integer j such that j − ϕ( j) = n and 3α j − 2m+1 = p, then if k = 3 pv, with (3v, 2 pj) = 1, the equation (51) has at least a solution. (57) A more general context of the above equation is the theory of so called ϕ-partitions introduced in 1991 by P. Jones [215]. A partition n = a1 + a2 + · · · + ai of n is said to be a ϕ-partition if

m F0a0 F1a1

ϕ(n) = ϕ(a1 ) + ϕ(a2 ) + · · · + ϕ(ai )

(58)

In the above paper there are studied characterizations of positive integers which have at least one ϕ-partition and those which have exactly one ϕ-partition, the number of ϕ-partitions of prime numbers, as well as techniques for constructing ϕ-partitions and reduced ϕ-partitions for various types of integers. A ϕ-partition is said to be reduced if each summand has exactly one ϕ-partition. In 1996 C. Powell [340] proved that Jones’ recursive algorithm gives a unique reduced ϕ-partition. Powell characterizes also the integers having exactly one reduced ϕ-partition. (59) A number x is called a Phibonacci number if satisfies the equation ϕ(x) = ϕ(x − 1) + ϕ(x − 2), 223

x ≥3

(60)

CHAPTER 3

A. Bager [22] raised the problem if there exists any composite Phibonacci number? The answer is yes, 1037 being the smallest one. P. J. Weinberger found 70 such numbers < 200, 000, 000. All known Phibonacci numbers are odd. An even Phibonacci number, if it exists, must exceed 101600 . C. A. Nicol [320] proved that there exist infinitely many numbers n such that ϕ(n) ≤ ϕ(n − k) for all 1 ≤ k ≤ n − 1, and infinitely many m such that ϕ(m) ≥ ϕ(k) + ϕ(m − k) for 1 ≤ k ≤ m − 1. The equation of k variables ϕ(x1 ) + ϕ(x2 ) + · · · + ϕ(xk ) = ϕ(x1 x2 . . . xk )

(61)

has been solved independently by J. S´andor [367] and F. Luca [261]. There are at most a finite number of solutions, which can be found step-by-step. For example, when 2 ≤ x1 ≤ x2 ≤ · · · ≤ xk , the inequality ϕ(x1 ) + · · · + ϕ(xk ) ≤ ϕ(x1 x2 . . . xk )

(62)

holds except for k = x1 = 2 and x2 = odd; and becomes an equality only in the following three instances: ϕ(2·2) = ϕ(2)+ϕ(2), ϕ(3·4) = ϕ(3)+ϕ(4), ϕ(2·2·3) = ϕ(2)+ϕ(2)+ϕ(3). (63) Essentially, besides k = 1 and x1 , and the above, all other solutions can be obtained by ”completing with 1”, i.e. whenever (x1 , . . . , xl ) is not in the above class, since u = ϕ(n 1 . . . n l ) − (ϕ(n 1 ) + · · · + ϕ(n l )) > 0, set k = l + u and xl+1 = · · · = xl+u = 1. (64) The equation xϕ(x) = yϕ(y) (65) of two arguments has the only solutions x = y. For this well-known property see e.g. [323] (Problem 19 of section 2.4). Related to the function f (n) = nϕ(n) (n = 1, 2, . . . ) in W. Janous [212] it is proved that if F(x) = car d{n > 0 : f (n) ≤ x 2 }, then 1 F(x) 1 = (66) lim 1+ √ − x→∞ x p p( p − 1) p prime By the same method, K.-W. Lau [241] showed that if f a,b (n) = n α (ϕ(n))β (α, β ≥ 0, αβ = 0), then putting Fa,b = car d{n > 0 : f a,b (n) ≤ x a+b }, one has Fa,b (x) = lim {1 + ( p − 1)−b/(a+b) · p −a/(a+b) − p −1 } (67) x→∞ x p 224

THE MANY FACETS OF EULER’S TOTIENT

P. Erd¨os and C. W. Anderson [123] propose that for a given positive integer a there are only a finite number of integers b, with (b, a) = 1, such that the equation bϕ(x) = ax

(68)

x has solutions. R. B. Eggleton [106] states that ϕ(x) = for infinitely many x, but 3 x ϕ(x) = for all x. 4

3

Equation ϕ(x) = k, Carmichael’s conjecture Another old equation with a considerable interest is ϕ(x) = k

(69)

In 1900 A. Pichler (see [103], pp. 134-135) noted that for k > 1 there is no solution, while for k = 2n has the solutions x = 2a bc . . . (a = 0, 1, . . . , n + 1) if u v b = 22 + 1, c = 22 + 1 are distinct Fermat primes, and 2u + 2v + · · · = n or n − a + 1, according as a = 0 or a > 0. For k = 2 · q n (q > 3, prime) the equation has no solutions if p = 2q n + 1 is not prime; if p is prime, then has two solutions: x = p, 2 p. For q = 3, p = prime, it has the additional solutions x = 3n+1 , 2 · 3n+1 . For k = 2q n the equation has no solutions if no one of the numbers ps = 2n−s q + 1 (s = 0, 1, . . . , n − 1) is prime and q is not a prime of the form 2h + 1, h = 2l ≤ n; but if q is such a prime, or if at least one ps is prime, then the equation has solutions of forms bq 2 , with ϕ(b) = 2n−h ; resp. aps , with ϕ(a) = 2s . For k = 2qr (q, r primes), the equation has no solutions if p = 2qr + 1 = prime and r = 2q + 1. If p is a prime, but r = 2q + 1, the two solutions are x = p, 2 p. If p = prime, but r = 2q + 1, the two solutions are x = r 2 , 2r 2 . If p = prime and r = 2q + 1, all four solutions occur. In 1907 R. D. Carmichael proved that (see [103], p. 137) for k = 2m, with m > 1 odd, x must have the forms x = p α or x = 2 p α , where p is a prime of the form p ≡ 3 (mod 4). In 1908 he gave a method of solving equations like (69), based on the testing of the equation for each factor x of a definite function of k. He solved also the equation ϕ( p α ) = ϕ(q β ) ([68]). In the same year, A. Ranum (see [103], p. 137) discovered a method for finding solutions of (69) when there are known all solutions of the equation ϕ(x) = m with m < k. This can be summarized as follows. When x is a prime power solution, i.e. x = n p , one can have solutions only if k = p n−1 ( p − 1). Let now x = ab, (a, b) = 1, 1 < a < b. Then ϕ(a)ϕ(b) = k. Hence, either ϕ(a) and ϕ(b) are both even or ϕ(a) = 1, 225

CHAPTER 3

i.e. a = 2. In the latter case, b is odd, yielding two solutions ϕ(b) = ϕ(2b) = k. In the former case ϕ(b) and ϕ(a) are both even, so k is a product of two even integers. This gives the following procedure for obtaining new solutions. Write k = (2u)(2v) and solve the equations ϕ(a) = 2u, ϕ(b) = 2v. We obtain new solutions by multiplying together relatively prime solutions for each of the two equations. For a discussion, with numerical examples, see N. S. Mendelsohn [288]. In 1950 H. Gupta [170] showed that for k = m! (m > 1) the equation is solvable. Namely, x = (m!)2 /ϕ(m!) is an integer, and a solution to (69). See also P. Erd¨os [112]. Similarly, for k = m! · 2m , the equation has the solution x = 4m (m!)2 /ϕ(m! · 2m ), proved by C. S. Venkataraman [468]. In 1977 A. Aigner [2] showed that if a > 1 is a squarefree integer, then there exist infinitely many solutions x, when k is of the form k = ab2 (where b ≥ 1), and also infinitely many solutions x = y 2 (y = 1, 2, . . . ), when k = ac2 (c ≥ 1). There are integers k such that (69) has exactly 2 or 3, etc. solutions. Let Vϕ be the valence function of ϕ, i.e. let Vϕ (k) denote the number of solutions x of equation (69). For example, Vϕ (1) = 2, Vϕ (2) = 3, Vϕ (4) = 4, Vϕ (10) = 2, Vϕ (14) = 0, Vϕ (20) = 5, Vϕ (22) = 2 etc. H.-J. Kanold and W. Sierpinski (see P. Erd¨os [113]) have proved that Vϕ (n) = 2 for infinitely many n, such are the integers n = 2 · 36k+1 (k = 1, 2, . . . ). Similarly, Vϕ (n) = 3 for infinitely many n, due to A. Schinzel [392] (we may take n = 12 · 712k+1 , k = 0, 1, 2, . . . ). S. S. Pillai [331] was the first to prove that lim sup Vϕ (n) = +∞ n→∞

and that for almost all n, Vϕ (n) = 0. This result was given by A. Schinzel [393] (see also [414], p. 233). P. Erd¨os [113] proved that if Vϕ (n) = a, then there exist infinitely many such integers n. In 1935 P. Erd¨os [114] showed that there exists δ > 0 such that Vϕ (m) > m c for infinitely many m, and in 1979 K. R. √Wooldridge [484] proved that the above statement holds in fact for every δ < 3 − 2 2 (by using estimates from Selberg’s upper bound sieve). 226

THE MANY FACETS OF EULER’S TOTIENT

C. Pomerance [336], by using certain improvements on average in the BrunTitchmarsh theorem due to Hooley, together with Bombieri’s sieve, obtained the refinement δ < 0.55655 This implies

Vϕ (m) > m 5/9 for infinitely many m.

A. Balog in 1984 obtained δ < 0.65 (see [349], p. 321). This has been further refined by J. B. Fridlander [144]. As a consequence of the above results one has Vϕ (n) = c0 x + (x 5/9 ) n≤x

where c0 = ζ (2)ζ (3)/ζ (6). P. T. Bateman [30] has shown Vϕ (n) = c0 x + O(x exp{−c1 (log x log log x)1/2 }), n≤x

√

where c1 < 1/ 2 is arbitrary. C. Pomerance in [336] proves Vϕ (n) ≤ n exp(−(1 + o(1)) log n log log log n/ log log n) and conjectures, that this is best possible. This if true, would imply log log log x Vϕ (n) = c0 x + x exp −(1 + ε) log x log log x n≤x for every ε > 0. H. Heilbronn and H. Davenport (see [113]) proved 1 (Vϕ (n))2 → ∞ as x → ∞ x n≤x and conjectured that

(Vϕ (n))2 x 1+c for some c > 0. n≤x

P. Erd¨os [113] conjectures that (92) is true for every c < 1. The value c = 1/9 appears in [336]. If Vϕ∗ (k) denotes the number of squarefree solutions to (69), then Vϕ∗ (n) ≤ n/e(1+o(1)) log n log log log n/ log log n , see [336]. 227

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CHAPTER 3

Let Vϕ# (k) be the number of 1 ≤ m ≤ k such that Vϕ (m) > 0. P. Erd¨os (see [349], p. 321) showed in 1935 and in 1945, respectively that Vϕ# (n) <

n , (log n)1−ε

(71)

for all n > n(ε) (ε > 0 arbitrary, but fixed) Vϕ# (n) >

n(log log n)k log n

(72)

for every k > 0 and sufficiently large n. These results were further improved by P. Erd¨os and R. R. Hall [126]. H. Maier and C. Pomerance [277] deduced Vϕ# (n) =

n exp{C + o(1)(log log n)2 } log n

(73)

where C = 0.81781465. K. Ford [138] further improved this to n exp C(log3 n − log4 n)2 + D log3 n− Vϕ# (n) = log n 1 (73 ) − D + − 2C log4 n + O(1) 2 where logk n denotes the kth iterate of the logarithm of n, and D = 2.17696874 . . . K. B. Stolarsky and S. Greenbaum [441] denoted by L(k) the l.c.m. of the solutions of equations (69). If G(k) = gcd of all numbers a k = 1 (1 ≤ a ≤ L(k), G(k) can be arbitrarily large. (a, L(k)) = 1), then they proved that the ratio L(k) In 1907 R. D. Carmichael (see [103], p. 137) proposed as an exercise to prove that for every n ≥ 1 it is possible to find an m such that ϕ(n) = ϕ(m), i.e. Vϕ (n) = 1 for all n.

(74)

For a few years it was thought that Carmichael had proved this. (74) is the famous Carmichael conjecture. (75) Already in 1922 Carmichael [67] showed that if Vϕ (n) = 1, then n > 1037 . In 1947 V. L. Klee [228] showed n > 10400 , while P. Masai and A. Valette [285] in 1982 raised the bound to n > 1010000 . (76) In 1994, still based essentially on Klee’s method, A. Schlafly and S. Wagon [397] 7 improved this to n > 1010 . (77) K. Ford [138], [139] showed that if there is a counterexample to Carmichael’s conjecture, then a positive proportion of totients are counterexamples. He improved 10 (77) to 1010 . (78) 228

THE MANY FACETS OF EULER’S TOTIENT

Ford ([138], [139], [140]) has proved also a famous conjecture of Sierpinski, namely that every integer n ≥ 2 is a value of the function Vϕ (i.e. all integers ≥ 2 occur as multiplicities). (79) The corresponding conjecture (due also to Sierpinski) for the function σ (i.e. that for every n ≥ 0 there is a number m for which σ (x) = m has exactly n solutions) has been settled by K. Ford and S. Konyagin [142]. There are examples of even numbers n such that there is no odd number m such that ϕ(m) = ϕ(n). L. Foster (see [171], p. 94) has given n = 29 · 2572 as the least such number. (80) Some conditions for the validity of Carmichael’s conjecture are given in W. A. Ramadan-Jradi [343]. M. V. Subbarao and L.-W. Yip [447] have considered the equation Sm (x) = k

(81)

where Sm (n) denotes the Schemmel totient function. Let VSm (k) denote the number of solutions of equation (81). They proved the following result: Let m ≥ 3 be of the form p α − 2, where p is an odd prime, and α ≥ 1. The H-Hypothesis by Schinzel implies that for any given integer n > 1, there exist infinitely many integers k such that VSm (k) = n. (82) They conjecture that this is true for all m ≥ 1 fixed. (83) ∗ Let ϕ be the unitary analogue of the totient function. M. Ismail and M. V. Subbarao [206] consider the equation ϕ ∗ (x) = k

(84)

Since for k odd, (84) has a solution iff k = 2m − 1, the unitary analogue of the Carmichael’s conjecture is clearly false. However, for k = even, there are some evidences that the conjecture could be true (see [206]). As we have seen, equation (69) is not solvable for odd k > 1. But this may be true for even values of k, too. In 1956 A. Schinzel [392] showed that for every a ≥ 1 and k = 2 · 7a , (69) is not solvable. (85) More generally, for every positive integer s there exists a positive integer k divisible by s such that the equation ϕ(x) = k has no solutions. (86) D. Lind [258] proved that if m is a prime such that 2m + 1 is composite, then ϕ(x) = 2m has no solutions. (87) Since, by Dirichlet’s theorem on arithmetic progressions, there are infinitely many primes of the form 3n + 1, and 2(3n + 1) + 1 = 3(2n + 1) is composite, there are an infinite number of primes satisfying theorem (87). In 1963 J. L. Selfridge [406], and also P. T. Bateman [31] gave the solution of a problem proposed by O. Ore: 229

CHAPTER 3

For every a ≥ 1 there exists an odd integer ka such that for k = ka · 2a , equation (69) is not solvable. (88) (i.e. ka · 2a is not a value of Euler’s function, or it is a ”nontotient”). In 1976 N. S. Mendelsohn [288] proved that there exist infinitely many primes p such that for every a ≥ 1, p · 2a is a nontotient. (89) More general theorems have been obtained in 1989 by K. Spyropoulos [436]. For example, let p1 , . . . , pm be distinct odd primes, and let a(1), . . . , a(m) be arbitrary positive integers. Suppose that pi ≡ 1 (mod L), i = 1, 2, . . . , m, where L = lcm{1 + 2, 1 + 22 , . . . , 1 + 2n }. Then 2n p1a(1) . . . pma(m) is a nontotient. (90) M Zhang [488] proves that a nontotient can have an arbitrary divisor, and gives two classes of odd numbers such that for the odd number k of the first class 2α k is a nontotient for a given positive integer α, while for the odd number k of the second class, 2α k is a nontotient for arbitrary positive integer k. (91) A (n) m Finally, if Am (n) = car d{x ≤ n : ϕ (k) (x) = m for some k}, then lim is n→∞ n s equal to 1 if m = 2 , and to 0 otherwise. See E. Jacobson [209].

4

Equations involving ϕ and other arithmetic functions

There are many equations on Euler’s or related totient, connected to other arithmetic functions. In 1894 A. P. Minin (see [103], p. 313) proved that all solutions of ϕ(x) = d(x)

(92)

are x = 1, 3, 8, 10, 18, 24 and 30. In fact ϕ(n) ≥ d(n) for n odd, with equality only for n = 1, 3;

(93)

ϕ(n) > d(n) for all n > 30,

(94)

and se e.g. [363], pp. 110-111. By answering a problem by Jankowska, P. Erd¨os [115] proved that the system of equations of two arguments ϕ(x) = ϕ(y),

d(x) = d(y),

σ (x) = σ (y)

(95)

has infinitely many solutions. More generally, for every k there are k (squarefree) integers x1 , . . . , xk satisfying ϕ(xi ) = ϕ(x j ),

d(xi ) = d(x j ),

σ (xi ) = σ (x j ), 230

1≤i < j
(96)

THE MANY FACETS OF EULER’S TOTIENT

Further, he proves (see [116]) that for every k there are squarefree integers x1 , x2 , . . . , xk satisfying (xi , x j ) = 1,

ϕ(xi ) = ϕ(x j ),

d(xi ) = d(x j ),

1≤i < j ≤k

(97)

The same result holds, when we replace ϕ(n) by σ (n). For further properties of related nature, see P. Erd¨os, K. Gy¨ory and Z. Papp [125]. The equation ϕ(x)d(x) = ϕ(x) + x − 1 (98) has the general solutions x = 1, or x = prime; and in fact ϕ(n)d(n) ≥ ϕ(n) + n − 1 for all n ≥ 1

(99)

(see [363], p. 112), improving ϕ(n)d(n) ≥ n, due to R. Sivaramakrishnan. All solutions of ϕ(x)(ω(x) + 1) = x

(100)

are given by x ∈ {1} ∪ {2a · 3b : a > 0, b ≥ 0} (see [363], p. 112), and in fact ϕ(n)(ω(x) + 1) ≥ n for all n ≥ 1

(101)

σ (x) = ϕ(x) + d(x)(x − ϕ(x))

(102)

The equation has the solutions x = 1, or x = prime. In fact, σ (n) ≤ ϕ(n) + d(n)(n − ϕ(n)) for all n ≥ 1,

(103)

improving nd(n) ≥ ϕ(n) + σ (n), due to C. A. Nicol (see [363], p. 114). Let ψ be the Dedekind arithmetical function, given by (1 + 1/ p). ψ(n) = n p|n, p prime

The equation

ψ(x) = ϕ(x) + 2ω(x)

(104) ω(n)

has x = 1 or x = prime, as general solutions. In fact, ψ(n) ≥ ϕ(n)+2 , improving (by ψ(n) ≥ σ (n)) relation σ (n) ≥ ϕ(n)+2ω(n) , due to C. A. Nicol (see [363], p. 196). All solutions of ψ(x) = 3ω(x) ϕ(x) (105) are x = 1 and 2. Indeed, for n even one has ψ(n) ≤ 3ω(n) ϕ(n) with equality only for n = 1 or n = prime, while for n odd one can write ψ(n) ≤ 2ω(n) ϕ(n) (see [363], p. 195) so (105) is impossible for x odd, x = 1. 231

CHAPTER 3

All solutions of

σ (x) + ϕ(x) = x · 2ω(x)

(106)

are x = 1 or x = prime, see C. A. Nicol [319]. Let k > 1 be a fixed integer. The equation σ (x) + ϕ(x) = kx

(107)

has been studied by C. A. Nicol, too. He proved that all solutions must satisfy ω(x) >

log(k − 1) log 2

(108)

and conjectured that for k > 2, all solutions are even. If k is odd, then x is even, or the square of an odd composite number. For k > 2, x cannot be squarefree. (109) For k = 3, a particular solution of equation (107) is given by x = 2α · 3 · q, if one assumes that q = 7 · 2α−2 − 1 is a prime (α > 2). (110) In 1971 S. A. Sergusov [408] proved that √ ϕ(x) = x + 1 − 2 x + 1 (111) iff x = p · q, where p < q are twin primes (i.e. p, q are primes with q = p + 2). A similar result is true for √ σ (x) = x + 1 + 2 x + 1 (112) W. G. Leavitt and A. A. Mullin [242] proved that the solutions of (x − 1)2 − σ (x)ϕ(x) = 4

(113)

are the products of two twin primes. In fact, if x = pq, with p − q = m ( p, q primes), then x is a solution of the more general equation (x − 1)2 − σ (x)ϕ(x) = m 2 (114) For particular m, however there are also solutions of other type. For example, when m = p k − 1, with p and 2 p − 1 being primes, x = p k (2 p − 1) is a solution of (114). The equation doesn’t have solutions of type x = pq r for r ≥ 2, and for any m ( p < q primes). (115) The complete study of equation (114) remains however, open. Another interesting equation is ([371]) σ (x) = ϕ(x)d(x)

(116)

In 1988 J. S´andor [369] (see p. 11) proved that σ (n) ≤ ϕ(n)d(n) for all n odd 232

(117)

THE MANY FACETS OF EULER’S TOTIENT

with equality only for n = 1, 3 (see also [363], p. 211, and [370], p. 65). Therefore the odd solutions are x = 1, 3. Equation (116) is included also in D. Wells [480], p. 127, where it is mentioned that the least three solutions are x = 1, 3, 14. The only even solutions of type x = 14k, where k is odd and 7 k are x = 14 and 42 (see [371]). (118) S´andor proves also that there are no even solutions x with 3 x and ω(x) ≥ 3; and also with 6|x and ω(x) ≥ 4. (119) All even solutions up to x ≤ 105 are x = 14 and 42. A further refinement of (117) can be found in [372]. Let ϕ ∗ , d ∗ , σ ∗ be the unitary analogues of the functions ϕ, d, σ . Note that, d(n) ≥ d ∗ (n) and σ (n) ≥ σ ∗ (n), but ϕ(n) ≤ ϕ ∗ (n). Since

and

σ (n) σ ∗ (n) ≤ ∗ , with equality only for squarefree n, d(n) d (n)

(120)

σ ∗ (n) ≤ d ∗ (n)ϕ(n) for all n ≥ 3, odd,

(121)

with equality only for n = 3, we get σ (n) σ ∗ (n) ≤ ∗ ≤ ϕ(n) for all n ≥ 3, odd d(n) d (n)

(122)

with equality only for n = 1 and 3. Relation (120) is proved also in [373]. Results of type (123) d ∗ (n) · n ≤ ϕ ∗ (n)(d ∗ (n))2 ≤ n 2 as well as additive variants, are included in [385]. The inequality √ σ (n) ( n − 1)2 √ ≥ + n, d(n) d(n)

(124)

with equality only if n is 1, p, or p 2 , where p is a prime, is due to E. S. Langford [239]. r r If n = piai is the prime factorization of n > 1, let B(n) = ai pi (see [291], i=1

pp. 143-147), B(1) = 0. Then the only solutions of the equations

i=1

ϕ(x)σ (x) + 1 = x B(x)

(125)

ϕ(x) + σ (n) = x + B(x)

(126)

and are x = p (prime), see K. T. Atanassov [17]. 233

CHAPTER 3

Let pn be the nth prime. L. Moser [303] proposed the equation 1 + ϕ(1) + ϕ(2) + · · · + ϕ(x) = px

(127)

and showed that x = 1, 2, 3, 4, 5, 6 are the only solutions. In fact 1 + ϕ(1) + ϕ(2) + · · · + ϕ(n) > pn for n > 100

(128)

A famous, unsolved equation is (see e.g. [171], p. 94) ϕ(x) = σ (y)

(129)

Since for p, q primes ϕ( p) = p − 1 and σ (q) = q + 1, clearly p − 1 = q + 1 iff p = q + 2, so any pair of twin primes is a solution. If M p = 2 p − 1 is a Mersenne prime, then by ϕ(2 p+1 ) = 2 p = σ (M p ), x = 2 p+1 , y = M p is a solution pair of (129). On the other hand, it is not known if there exist infinitely many twin primes, or infinitely many Mersenne primes.

5

The composition of ϕ and other arithmetic functions

In what follows we shall study equations, and related inequalities, for the composition of ϕ and other functions or sequences. In a paper by L. Alaoglu and P. Erd¨os [3] from 1944 one can read that P. Poulet gave many solutions to the equation ϕ(σ (x)) = x

(130)

For x ≤ 2500 all solutions are x = 1, 2, 8, 12, 128, 240, 720. Two further solutions are x = 215 and 231 . They note also that it can be shown that for every c > 0, σ (ϕ(n)) ≥ cn

(131)

except for a set of density zero. Based on numerical investigations and particular cases by K. Kuhn, A. Makowski 1 and A. Schinzel [283] conjecture that (131) holds true for all n, with c = : 2 n σ (ϕ(n)) ≥ for all n ≥ 1 (132) 2 Kuhn showed that this is true for all n with ω(n) ≤ 6. Further, she found the i solutions x = 22 +1 − 2 (0 ≤ i ≤ 5) of the equation σ (ϕ(x)) = 234

x 2

(133)

THE MANY FACETS OF EULER’S TOTIENT

After Makowski and Schinzel’s paper, inequality (132) was first investigated in 1988 and 1989 by J. S´andor [374], [375], [376]. Similar problems, with stronger conjectures (σ replaced with ψ) were included also in [369] and [377]. In [376] the following lemma is proved: Let A denote the set of all numbers m with the property σ (ϕ(m)) ≥ m, m odd (134) Then for all even numbers n, having greatest odd divisor in A one has (132). Since for odd m one has ϕ(2m) = ϕ(m), this easily implies that (132) is true iff (134) is true. This has been rediscovered in 1997 by G. L. Cohen [85]. Let a ∧ b denote the property that there exists at least a prime divisor of a which doesn’t divide b. Let Js (s ≥ 1 integer) be the Jordan totient. The following is proved in [376]: Let S denote the set of all m > 1 such that σk (Js (m)) ≥ m ks · 2−(k−1)ω(m) d k ). (where k, s ≥ 1 are positive integers, and σk (a) =

(135)

d|a

Let p denote a prime. Then, if m ∈ S and p|m, then mp ∈ S, too. If n ∈ S, p m, and ( p s − 1) ∧ Js (m), then mp ∈ S, too. As a corollary one obtains a set B ⊂ S of odd numbers, for which inequality (135) is true. (Note that for k = s = 1, this inequality reduces to (134)). If n ≥ 2 is an even number with greatest odd divisor m ∈ B, then σk (Js (n)) ≥

n ks (2s − 1)k · 2−(k−1)ω(n) 2ks − k + 1

(136)

We conjecture that (135) holds true for all odd m ≥ 1, and (136) holds true for all even n ≥ 2. Inequality (136) reduces to (132) for k = s = 1. In Guy [171] one finds that Selfridge, Hoffman and Schroeppel found 24 solutions to equation (130). Cohen [85] notes that Terry Raines found 10 further solutions, and that he has found 8 more. There are all solutions up to 109 . (137) In [429], sequence A001229 one finds that F. Helenius has found 365 solutions. Any solution of (130) gives a solution to σ (ϕ(x)) = x

(138)

Indeed, let x = σ (y), where y is a solution to (130). Then ϕ(x) = y, so σ (ϕ(x)) = σ (y) = x. For further results on the Makowski-Schinzel conjecture we note that, by using Brun’s sieve, C. Pomerance [337] proved that inf

σ (ϕ(n)) >0 n 235

(139)

CHAPTER 3

In 1992 M. Filaseta, S. W. Graham and C. Nicol [134] verified (132) for n = p1 p2 . . . pk for k ≥ k0 , where pi is the ith prime. U. Balakrishnan [24] proved the same for all squarefull integers n. More generally, if n is k-full (k ≥ 2), then 1 σ (ϕ(n)) ≥ n ζ (k)

(140)

Various classes of integers n for which (132) is true, are described also in [85]: 1◦ Any positive integer n of the form 2a m, where i) the distinct prime factors of m are either Fermat primes or primes p ≡ 1 (mod 3), with at most eight of the latter; ii) m is a product of primes of the form 2b r + 1 with b ≥ 1 and r prime (similar example is given in [375]); 2◦ Any positive integer n which is a product of primes less than 1780. (141) In [161] A. Grytczuk, F. Luca and M. W´ojtowicz prove that the lower density of the set of integers satisfying the inequality is greater than 0.74. (142) In [162] they give various sufficient conditions for the validity of inequality (132). For example, if the prime factorization of n > 1 is n = p1a1 p2a2 . . . prar , where 2 ≤ p1 ≤ · · · ≤ pr , and if p1 ≥ ω(n), then the inequality is true. (143) More generally, if r −1 1 1 2 ≤1− r + , (144) p 2 2 pr i=1 i then the inequality is true. Other results state that if n ≥ 2, and if 1 1 ≤ , a p 2 pa n

(145)

then (132) is true. For n = m! (m ≥ 1), relation (132) is true, with equality only for m = 2 or 3. Recently, K. Ford [141] has shown that (131) is true with c = 1/39.4 for all n. Further results can be found in F. Luca and C. Pomerance [271]. Cohen [85] proves also that (134) holds true for m equal to a product of Fermat primes. For such a product, there is equality iff m = F0 F1 . . . Fk for 0 ≤ k ≤ 4. (146) S. W. Golomb [150] remarks that if p > 3 and 2 p − 1 are primes, and m ∈ {2, 3, 8, 9, 15}, then x = (2q − 1)m is a solution to ϕ(σ (x)) = ϕ(x)

(147)

There are also other solutions, like x = 1, 3, 15, 45. Are there infinitely many? He gives also the particular solutions x = 1, 87, 362, 1257, 1798, 5002 and 9374 to 236

THE MANY FACETS OF EULER’S TOTIENT

the equation σ (ϕ(x)) = σ (x)

(148)

If p and (3 p − 1)/2 are both primes, then x = 3 p−1 are solutions to σ (ϕ(x)) = ϕ(σ (x))

(149)

In fact, it can be proved that for all primes p ≥ 5, σ (ϕ( p)) > p > ϕ(σ ( p))

(150)

ϕ(σ (2 p−1 )) > 2 p−1 > σ (ϕ(2 p−1 ))

(151)

and for all primes p,

For these, and related results, see H. Iwata [208]. For two arithmetic functions f, g : N → N, put f,g (n) = | f (g(n)) − g( f (n))|,

n = 1, 2, . . .

(152)

J. S´andor [378] (see also [363], pp. 175-178) has proved that lim sup σ,ϕ (n) = +∞

(153)

lim inf σ,ϕ (n) = 0

(154)

n→∞

and conjectures that n→∞

He proves also that lim sup d,ϕ (n) = +∞, n→∞

lim inf d,ϕ (n) = 0 n→∞

(155)

and remarks that x = 1 and x = 2 p−1 ( p ≥ 3 prime) are solutions to d(ϕ(x)) = ϕ(d(x))

(156)

and asks for the general solutions. Let S(n) be the Smarandache function, defined by S(n) = min{k ≥ 1 : n|k!}. Without any assumptions, one has lim sup S,ϕ (n) = +∞, n→∞

lim inf S,ϕ (n) ≤ 1 n→∞

(157)

Now, x = p prime is a solution of the equation S(ϕ(x)) = ϕ(S(x)) 237

(158)

CHAPTER 3

What are the most general solutions of this equation? The equation σ (x) =x (159) ϕ x x (where [a] denotes the greatest positive integer ≤ a) has the single solution x = 1, since A. Oppenheim [326] showed that σ (n) ϕ n ≤ n for all n, (160) n with equality only for n = 1. A similar inequality is valid when σ is replaced with ψ (see [369], p. 7), so x = 1 is again the solution of the similar equation to (159), when σ is replaced with ψ. Also n ϕ n ≤ (ϕ(n))2 , (161) d(n) with equality only for n = 1, see [379]. The equation ϕ(|x m − y m |) = 2n

(162)

where m, n are given positive integers and m ≥ 2 has been studied by F. Luca [262]. He first proves that it suffices to find those (x, y) for which x > y ≥ 1; (x, y) = 1, and m = 4. (163) All solutions satisfying (163) are given by k−1

k−1

k

(x, y) = (22 + 1, 22 − 1), where k ≥ 1 and 22 + 1 prime; k or (x, y) = (22 , 1) for k = 0, 1, 2, 3

(164)

The determination of all positive integers x, y, m, n of the equation ϕ(x m − y m ) = x n + y n

(165)

is the main objective of another paper by F. Luca [263]. These solutions are given by (x, y, m, n) = (2k + 1, 2k − 1, 2, 1)

k ≥ 1 integer

(166)

The equation ϕ(|x m + y m |) = |x n + y n |

(167)

where x, y are integers, and m, n positive integers, has been studied in [264]. It suffices to consider x ≥ 0 and x ≥ |y|. A solution (x, y, m, n) of (167) is called trivial, if (k, 1 − k, 1, 1) for some positive integer k, or (168) (x, y, m, n) = (1, 0, m, n) for some positive integers m, n 238

THE MANY FACETS OF EULER’S TOTIENT

All nontrivial solutions (i.e. satisfying x ≥ 0, x ≥ |y| and (168)) are given by  (2, 0, n + 1, n) for some positive integer n, or    (2, 2, n + 1, n) for some positive integer n, or (x, y, m, n) = (169) (3, 3, n + 1, n) for some positive integer n, or    (3, 1, 2, 1) It is not known (see [265]) if the equation ϕ(5m − 1) = 5n − 1

(170)

in positive integers m, n, has any solutions at all. The equation n ϕ = 2a (171) k n n has been considered in [266]. Since = , it suffices to suppose n ≥ 2k. k n−k Let A = {k ≥ 1 : ϕ(k) is a power of 2} (172) Then all solutions of equation (171) for n ≥ 2k are the following: 1) k = 1 and n ∈ A; 2) k = 2 and n is either a Fermat prime, or n ∈ {22 , 2 · 2 3 4 5 (173) 3, 22 , 22 , 22 , 22 }; 3) k = 3 and n ∈ {6, 10, 17, 18, 257, 65537}. Remark. It is well known that n ∈ A iff n either is a power of 2, or n = 2α p1 . . . pt , for some α ≥ 0, t ≥ 1, where p1 < · · · < pt are Fermat primes (i.e. n = 1, 2 or for n ≥ 3, the regular polygon with n sides can be constructed using only the ruler and compass), due to A. Cunningham from 1915 (see [103]. p. 140). Equations on the composition of Euler’s totient with Fibonacci or Lucas sequences appear in [267] and [268]. For example, in [267] it is proved that ϕ(Fn ) ≥ Fϕ(n) for all n ≥ 1,

(174)

with equality only for n = 1, 2, 3. A similar inequality is valid for d (with equality for n = 1, 2, 4), while for σ the sign of inequality is reversed (with equality for n = 1, 3). In [268] one can find: ϕ(L m ) = L n

(175)

iff (m, n) = (0, 1), (1, 1), (2, 0), (3, 0) (where L n+2 = L n+1 + L n , L 0 = 2, L 1 = 1 is the classical Lucas sequence); ϕ(Fm ) = L n 239

(176)

CHAPTER 3

iff (m, n) = (1, 1), (2, 1), (3, 1), (4, 0), (5, 3), (6, 3) (where F2 = Fn+1 + Fn , F0 = 0, F1 = 1 is the classical Fibonacci sequence). Let r and s be two non-zero integers with r 2 + 4s > 0. A binary recurrence sequence (u n )n≥0 is a sequence such that u 0 and u 1 are integers and u n+2 = r u n+1 + su n ,

n≥0

Let α, β denote the two roots of the equation x 2 − r x − s = 0. It is well known that u n = aα n + bβ n (n ≥ 0), where a, b are two constants. If (r, s) = 1, u 0 = 0, u 1 = 1, then (u n ) is called a Lucas sequence of the first kind; while if (r, s) = 1, u 0 = 2, u 1 = r , then (u n ) is called a Lucas sequence of the second kind. In [269] the following is proved: Let (u n ) be a Lucas sequence of the first kind such that its characteristic equation has real roots. Then ϕ(|u n |) ≥ |u ϕ(n) | for all n ≥ 1 (177) with equality iff: i) n = 1; ii) n = 2 and |r | = 1, 2; iii) n = 3, |r | = 1, and s = 1. A binary recurrent sequence is called nondegenerate if ab = 0. In [268] there are studied equations of type ϕ(|au n |) = |bvn |

(178)

where (u n ), (vn ) are certain binary recurrent sequences, and (u n ) is nondegenerate.

6

Perfect totient numbers and related results

As we have seen in 5. of 3.2, the iteration of Euler’s totient has some interesting congruence properties. In 1975 T. Venkataraman [469] defined a perfect totient number n by T (n) ≡ ϕ(n) + ϕ (2) (n) + ϕ (3) (n) + · · · + ϕ (r ) (n) = n

(179)

where r = r (n) is the smallest integer such that ϕ (r ) (n) = 1. We note that the function r (n) was first considered by S. S. Pillai, for its properties see e.g. [291], p. 35. Venkataraman proves that if n = 3 · (4 · 3m−1 + 1) with 4 · 3m−1 + 1 = prime, then n is a perfect totient number. (180) The study of PTN (for perfect totient numbers) however was initiated by L. P. Cacho [59], who in fact proved that 3 p, for an odd prime p, is a PTN if and only if p = 4n + 1, where n is a PTN. Then Venkataraman’s result (180), follows as a corollary. A. L. Mohan and D. Suryanarayana [292] proved that 3 p ( p = odd prime) is not a PTN, if p ≡ 3 (mod 4). They found also sufficient conditions on an odd prime p for 32 · p and 33 · p to be PTNs. For example, if b ≥ 0 is an integer, and 240

THE MANY FACETS OF EULER’S TOTIENT

if q = 25 · 3b + 1, p = 2 · 32 · q + 1 are both prime, then 32 p is a PTN; and if q = 24 · 3b + 1 and p = 22 · q + 1 are both prime, then 33 · p is PTN. (180’) In a recent paper, D. E. Iannucci, D. Moujie and G. L. Cohen [205] have obtained similar results. They gave also a table on all PTN less than 5 · 109 , which are not powers of 3 (3k is PTN, as immediately follows). There are 30 such PTN. We quote the following results from [205]: If b ≥ 0 and q = 23 · 3b + 1 and p = 2q + 1 are both prime, then 32 · p is PTN. If r = 22 · 3b + 1, q = 24 · r + 1, p = 22 · q + 1 are all prime, then 33 p is PTN. There are no PTNs of the form 3k p (k ≥ 4), where p = 2c · 3d · q + 1 and q = 2a · 3b + 1 are primes with a, c ≥ 1 and b, d ≥ 0. (180”) Equation (179) has been considered also by D. L. Silverman [416]. P. Erd¨os and M. V. Subbarao [130] note that T (n) = (1 + o(1))ϕ(n), for almost all n,

(181)

T (n) < n, for almost all n.

(182)

so that They also note that

T (n) >

3n for infinitely many n, 2

(183)

and that

T (2n) =1 (184) 2n n→∞ Let F(x, c) = car d{n ≤ x : T (n) > cn}, where c > 0. Then for every 3 1 < c < we have for every t > 0 and ε > 0, that 2 x x (185) (log log x)t < F(x, 1 + c) < log x (log x)1−ε lim sup

for all x ≥ x0 (c, t, ε). Further, we have F(x, 1) = (c + o(1))

x log log log log x

(186)

The details of proofs for (183)-(186) are not worked out in [130], but the authors state that they follow by the methods of this paper and that of [117]. 3 Erd¨os and Subbarao conjecture that for 1 < c1 < c2 < , 2 lim

x→∞

F(x, 1 + c1 ) =∞ F(x, 1 + c2 ) 241

(187)

CHAPTER 3

x 3 =o F x, 2 log x Some other unanswered questions are: r (n) have a distribution function? Does log n r (n) approach a limit for almost all n? Does log n ∗ Let r ∗ = r ∗ (n) be the smallest integer such that (ϕ ∗ )(r ) = 1, where ϕ ∗ unitary analogue of Euler’s totient. It is not known if and that

r ∗ (n) < c log n has infinitely many solutions for some c > 0.

(188)

(189) (190) is the (191)

M. Lal [235] computed for n ≤ 105 the maximum and minimum values of n for ∗ a given r ∗ such that (ϕ ∗ )(r ) = 1. He obtained the bounds log n < r ∗ (n) < 2.5 log n for n ≤ 105

3.4 1

(192)

The totatives (or totitives) of a number Historical notes, congruences

As we have mentioned in the introduction, in 1879, J. J. Sylvester called the positive integers r < n which are coprime with n the totatives (or ”totitives”, see e.g. [103], p. 124, or [194]). We note that theorem (126) of 3.2 gives all totatives of a number, which are prime numbers (except 1), a result due to S. Schatunowsky. In 1888 H. W. Lloyd Tanner (see [103], p. 131) studied the group G of the totatives of a number, finding all of its subgroups and the simple groups whose direct product is G. nϕ(n) , (1) It is easy to see that the sum of totatives of n is 2 due to A. L. Crelle from 1845. Let t (n) denote the set of totatives of n, i.e. t (n) = {k : 1 ≤ k < n, (k, n) = 1}. In 1850 A. Thacker introduced the function φ j (n) = t j , j ≥ 0, (2) t∈t (n)

(i.e. the sum of jth powers of totatives of n), and noted that for j = 0 it reduces to Euler’s totient. Let n = p1a1 . . . prar be the prime factorization of n, and put s j (m) = 1 j + 2 j + · · · + m j . Thacker proved the following formula: j n j j n + − ... (3) φ j (n) = s j (n) − p1 s j p1 p2 p1 p1 p2 p1 , p2 p−1 242

THE MANY FACETS OF EULER’S TOTIENT

where the summation indices range over the combinations of p1 , p2 , . . . one, two,... at a time. Various authors, as J. Binet, J. Liuoville, E. Lucas, E. Ces`aro, L. Gegenbauer, etc. obtained other formulae, by using Bernoulli numbers or binomial coefficients. W. Brennecke in 1852 proved that 1 2 ω(n) γ (n) (4) φ2 (n) = ϕ(n) n + (−1) 3 2 1 φ3 (n) = nϕ(n)[n 2 + (−1)ω(n) γ (n)] (5) 4 where γ (n) = p1 p2 . . . pr is the ”core” of n, while ω(n) denotes the number of distinct prime factors of n. For new proof, see [98]. J. Binet in 1851 proved that φk (n) ≡ 0 (mod n) if k is odd;

(6)

and Mennesson showed in 1878 that 1 φk (n) ≡ ϕ(n k+1 ) (mod k), if k is odd 2

(7)

N. Nielsen proved in 1915 that φ2k (n) ≡ 0 (mod n),

φ2k+1 (n) ≡ 0

(mod n 2 )

(8)

p1 − 3 , with p1 = p(n) = smallest prime divisor of n. J. Liouville where 1 ≤ k ≤ 2 in 1857 proved the identity n φk (d) = sk (n) = 1k + 2k + · · · + n k , (9) d d|n which for k = 0 reduces to Gauss’ formula

ϕ(d) = n.

d|n

In 1932 H. Davenport [94] rediscovered Thacker’s function and proved an asymptotic formula on it. For generalizations, in the regular convolutions setting, see L. T´oth and P. Haukkanen [459]. In 1985 P. S. Bruckman [57] proposed for the Dirichlet series of φk (n) given by ∞ φk (n) the identity f k (s) = ns n=1 k+1 k+1 1 Bk+1− j ζ (s − j) f k (s) = 1 + j (k + 1)ζ (s − k) j=1 243

CHAPTER 3

where Bm are Bernoulli numbers. For a generalization, see H. W. Gould and T. Shon hiwa [153], where functions of type f (t) and f (t) are also 1≤t≤n,(t,a)=m

studied. (Here (t, a) = gcd(t, a), [t, a] = lcm(t, a)). In 1889 C. Leudesdorf [253] introduced the function ψ j (n) =

1 , tj t∈T (n)

1≤t≤n,[t,a]=m

j ≥0

(10)

which is another extension of Euler’s totient. He proved that for j = odd one has ψ j (n) =

1 2 1 An − jnψ j+1 (n) 2 2

(11)

for certain integer A. As a corollary he deduced, that if n = p s q, where q is not divisible by the prime p > 3, then ψ j (n) ≡ 0 (mod p 2s ). (12) unless ( j, p) = 1 and ( p − 1)|( j + 1). For example, ψ j ( p) ≡ 0 (mod p 2 ). Similarly, if p = 3, ψ j (n) ≡ 0 (mod 32s ) if j is an odd multiple of 3; (13) and ψ j (n) ≡ 0 (mod 22s−1 ) if p = 2, except when q = 1

(14)

For j = 1, by putting S = ψ1 (n) one obtains (see also [179]) S ≡ 0 (mod n 2 ) 1 S ≡ 0 (mod n 2 ) 3 1 S ≡ 0 (mod n 2 ) 2 1 S ≡ 0 (mod n 2 ) 4

if 2 n, 3 n; if 2 n, 3|n; if 2|n, 3 n, n not a power of 2

(15)

if n = 2a

We note that the congruence S ≡ 0 (mod p 2 ) (n = p = prime) was first proved by J. Wolstenholme [483] (see also [367]). Extensions of Leudesdorf’s theorem were obtained e.g. by M. Rama Rao [345], I. Sh. Slavutskii [427]. See also [428]. For example, Slavutskii [427] proves the following:  n (1 − p t−1 )Bt (mod n 2 ) for 2| j     p|n  ψ j (n) ≡ (15 )  t  t−2 2    2 n (1 − p )Bt−1 (mod n ) otherwise p|n 244

THE MANY FACETS OF EULER’S TOTIENT

where t = (ϕ( pl ) − 1)n (l ≥ 1 integer), and (n, 6) = 1. Here Bt denotes the tth Bernoulli number (the Bernoulli numbers (Bn ) are generated by x/(e x − 1) = ∞ Bn x n /n!, |x| < 2π , see Chapter V). n=0

By applying the von Staudt-Clausen theorem for Bernoulli numbers (see Chapter V), (15’) implies the following corollary: 1) If 2| j and ( p, n) = 1 for all prime numbers p such that ( p − 1)| j, then ψ j (n) ≡ 0 (mod n) 2) Let j be odd. a) If p − 1 does not divide j + 1 for any prime p with p|n, then ψ j (n) ≡ 0 (mod n 2 ) b) If p| j for all primes p such that ( p − 1)|( j + 1) and p|n, then ψ j (n) ≡ 0

(mod n 2 ).

For a positive integer x, and t ∈ t (n), let (x − t) f n (x) =

(16)

t∈t (n)

By the classical Lagrange theorem one has f n (x) ≡ x ϕ(n) − 1 (mod n), if n is prime

(17)

However, for all n, the congruence (17) is not valid. In 1902 M. Bauer proved that if pa n ( p prime), then f n (x) ≡ (x p−1 − 1)ϕ(n)/( p−1)

(mod pa )

(18)

Particularly, f pa (x) ≡ (x p−1 − 1) p

a−1

(mod pa )

(19)

Another result by M. Bauer states that for m > 2 even and 2a n one has f n (x) ≡ (x 2 − 1)ϕ(n)/2

(mod 2a )

(20)

(mod 2a )

(21)

Particularly, a−2

f 2a (x) ≡ (x 2 − 1)2

For proofs of Bauer’s theorems see Hardy-Wright [179]. 245

CHAPTER 3

Recently A. Junod [217] has extended (18) as follows: If p is a prime and m, n ≥ 0 are integers, and f (x) ∈ Z p [x]; or d p (m) ≥ 1, then

f (x − km) ≡ f (x)n

(mod

0≤k
mn Z p [x]) 2

This implies that if p|n, then

ϕ(n)/( p−1) f (x − t) ≡ f (x − t) t∈t (n)

(mod

t∈t ( p)

n Z p [x]) 2

(18 )

giving for f (x) = x, result (18).

2

The distribution of totatives

The study of distribution of totatives was initiated by D. H. Lehmer in 1955 [245]. Lehmer introduced the function ∗ φ(n; k, l) = 1, 0 ≤ l < k, (22) nl
where the summation is over t ∈ t (n). This is another generalization of Euler’s totient, since φ(n; 1, 0) = ϕ(n). Define the following sets: Ak = {n : k 2 |n or there exists a prime p|n with p ≡ 1 (mod k)}, ϕ(n) for 0 ≤ l < k}, Bk = {n : φ(n; k, l) = k Ck = {n : k|ϕ(n)}

(23)

It is clear that Ak ⊂ Ck , Bk ⊂ Ck , and Lehmer proved that Ak ⊂ Bk ⊂ Ck

(24)

It is not difficult to prove that C p ⊂ A p for a prime p, so A p = Bp = C p

(25)

In 1957 P. J. McCarthy [70] proved that Ak = Bk if k is not squarefree,

(26)

and asked if the result could be extended to all composite k. This was done by P. Erd¨os [118] who proved that the set Bk \ Ak is infinite for all composite k, B2 p = C2 p for an odd prime p, and Bk = Ck for k = p, 2 p. (27) 246

THE MANY FACETS OF EULER’S TOTIENT

Some parts of the proof given by Erd¨os have to be corrected; this has been done recently by R. R. Hall and P. Shiu [174]. For a fixed modulus k, Hall and Shiu write x < y (mod k) to mean that the least non-negative residue congruent to x is less than that congruent to y. Then they prove the following theorem: Let a, b, c be positive integers which are distinct (mod k) and satisfying (ab, k) = 1,

c ≡ 0 (mod k),

a + b ≡ c

(mod k)

(28)

Then there exists x such that ax < cx < bx (mod k). As a corollary, they obtain the proof of a conjecture by Erd¨os: Let p, q be distinct odd primes such that pq ∈ Ak and pq ≡ −1 (mod k). Then pq ∈ Bk . (29) The Lehmer, McCarthy, and Erd¨os theorems, along with numerical examples, and particular cases, are presented also in the book by L. Holzer [200] (with other notations). Let t (n) = {t1 < t2 < · · · < tϕ(n) }. In 1940 Erd¨os [119] conjectured that for some positive c one has (30) (ti+1 − ti )2 < cn 2 /ϕ(n) where the sum is taken for 1 ≤ i ≤ ϕ(n) − 1. In 1962 C. Hooley [204] showed that for 1 ≤ α < 2, (ti+1 − ti )α n(n/ϕ(n))α−1 (31) and

(ti+1 − ti )2 n(log log n)2

(32)

M. Hausman and H. N. Shapiro [195] obtained the bound log p n2 max 1, c ϕ(n) p−1 p|n

(32 )

in the right side of (30). R. C. Vaughan [465] established the conjecture ”on the average”, and finally in 1986, H. L. Montgomery and R. C. Vaughan [295] completely proved (30). E. Jacobsthal [210] in 1960 defined a function g(n) as the least positive integer so that among g(n) consecutive positive integers there is at least a totient of n. For results on g(n), see [291], p. 34. Another function by Jacobsthal is J (n) = max{ti+1 − ti : i = 1, . . . , ϕ(n)−1}. There are not known nontrivial bounds on J (n). Erd¨os asked if, for infinitely many x, there are two integers a and b, where a ln x. (33) 247

CHAPTER 3

3

Adding totatives

In 1976 B. R. Santos [388] considered the problem of finding the largest integer with the property that, when added to each of its totatives, yields a prime number. He proved that a largest such integer does exist, and conjectured that 12 was actually this largest integer. In 1978 M. Hausman and H. N. Shapiro [194] proved that the Santos conjecture is correct, and considered similar and more general questions. First they prove that only for n = 1, 2, 4, 6, 10, 12, 18 does there exist a value of k for which kn plus each totative of n is prime. For n = 1, 2, 4, 6, 10, 12, the smallest such k is k = 1, and for n = 18, the smallest such k is k = 892. (34) A similar question (raised in [388]) is of adding prime totatives to n either to yield only primes, or only composites. Hausman and Shapiro answer this questions in the following way: There exist infinitely many integers n to which each of its prime totatives can be added to yield a composite. (35) There is a largest integer n with the property that for some positive integer k each of the prime totatives of n added to kn always is a prime. (36) For adding composite totatives to n either to yield only primes or only composites, one has: There is a largest integer n with the property that for some positive integer k each of its composite totatives added to kn always is a prime. (37) There exists a largest integer n to which each of its composite totatives can be added to obtain a composite. However, in the general case, there exist infinitely many integers n for which there exists a positive integer k such that kn added to each of the composite totatives of n is composite. (38) The proofs of (34) and (35) are elementary, and based on a simple consequence of theorem (126) from 3.2 of Schatunowsky. Namely, let p ∗ (n) be the smallest prime totative of n. Then, √ p ∗ (n) < n for n > 30 (39) Theorems (36) and (37) are based on more advanced results of Number theory, as the Siegel-Walfisz theorem; as well as an improvement of (39) for large values of n: p ∗ (n) < c log n (c > 0, constant)

(40)

In fact, the arithmetical function p ∗ was introduced by B. R. Srinivasan [437], who proved that p ∗ (n) lim sup = 1, (41) n→∞ log n 248

THE MANY FACETS OF EULER’S TOTIENT

and

p ∗ (n) ∼ Ax,

(42)

n≤x ∞ pr +1 − pr , and pr is the r th prime. p p . . . pr r =1 1 2 In 1969 Y. Sankaran [387] improved (42) to

where A = p1 +

p ∗ (n) = Ax + O(log x)

(43)

n≤x

4

Adding units (mod n)

When n ≥ 2, for the set of totatives of n, t ∈ t (n), the classes t in the ring Zn of residue classes modulo n, form a multiplicative group Z∗n , called also as the group of units of the ring Zn . If 1 ≤ k ≤ n − 1, one can ask how many solutions (t, t ) ∈ Z∗n × Z∗n there are, such that

t +t =k

(44)

Let s(k) be the number of solutions to (44). Then M. Deaconescu [95] proves that s(k) is given by ϕ(n) s(k) = ψ(d, n) (45) ϕ(n/d) where d = (k, n), and ψ(d, n) is the number of those automorphisms of the additive group of Zn , having exactly d fixed points (i.e. with the property f (k) = k for such r piai is the prime an automorphism f of (Zn , +)). In [96] it is proved that if n = factorization of n, and d =

r

i=1

pibi

is a divisor of n (0 ≤ bi ≤ ai ), then

i=1

ψ(d, n) =

piai −bi −1 ( pi − 1)

pi |n/d pi |d

a −1

pj j

( p j − 2)

(46)

p j |n/d p j d

A corollary of (45) and (46) is the following proposition: 1) If n ≥ 2 is odd, then every element of Zn is a sum of two units (i.e. (44) has always solutions). (47) 2) If n ≥ 2 even, then k ∈ Zn is a sum of two units if and only if k is even. 249

CHAPTER 3

5

Distribution of inverses (mod n)

For each t ∈ t (n) there exists a unique totative t, which is the multiplicative inverse of t (i.e. t · t ≡ 1 (mod n)). Then Z. Zheng in 1993 [493] proved that when n is odd, √ 1 car d{t ∈ t (n) : t + t ≡ 1 (mod 2)} = ϕ(n) + O( n · d(n) log2 n) 2

(48)

The proof makes use of estimates for character and Kloosterman-sums. When n is a prime power, or a product of two distinct primes, similar results have been deduced by W. Zhang [489]. In the same year, he settled also the general case (i.e. n = odd), too, see [490]. A generalization of (48) is given in [494]: car d{t ∈ t (n) : t +t ≡ 1 (mod 2), t ≤ m} =

√ m ϕ(n) · + O(d(n) n ·log2 n) (49) 2 n

where n is odd, and 1 ≤ m < n is a fixed integer. For m = n − 1, one reobtains (48). Another generalization of (48) has been obtained by I. Z. Ruzsa and A. Schinzel [360]: For every choice of ε = 0, 1 and d = −1, +1 and odd n we have t car d t ∈ t (n) : t + t ≡ ε (mod 2), =δ = n =

√ ϕ(n) cn,δ + O(2ω(n) n · log2 n) 4

(50)

t denotes a where cn,δ = 1 + δ if n is a perfect square; and =1, otherwise. Here n Legendre symbol, and ω(n) is the number of distinct prime factors of n. Let n > 2 be arbitrary and 0 < δ ≤ 1 be fixed. Then √ car d{t ∈ t (n) : |t − t| < δn} = δ(2 − δ)ϕ(n) + O(d 2 (n) n · log3 n) (51) This is due to W. Zhang [491]. In [492] W. Zhang shows that for n > 2,

(t − t)2k =

t∈t (n)

ϕ(n)n 2k + O(4k d 2 (n)n (4k+1)/2 log2 n) (2k + 1)(k + 1)

where k ≥ 1 is a fixed positive integer. 250

(52)

THE MANY FACETS OF EULER’S TOTIENT

Let n > 2 be an odd integer such that there exists a primitive root mod n. Let A denote the set of all primitive roots modulo n which are < n. Then t ∈ A ⇒ t ∈ t (n), and ∗

(t − t)2k =

t∈A

ϕ(ϕ(n))n 2k + O(4k d 2 (n)n 2k+3/4 log2 n) (2k + 1)(2k + 2)

(53)

where the sum is taken for t + t ≡ 1 (mod 2) (i.e. t and t are of opposite parity). Formula (53) is due to H. Wang, Z. Hu and L. Gao [474].

3.5 1

Cyclotomic polynomials Introduction, irreducibility results

The nth cyclotomic polynomial n (x) is given by (x − ζt ) n (x) =

(1)

t∈t (n)

where ζt are the primitive roots of unity of order n, i.e. ζt = e2πit/n for 1 ≤ t < n, (t, n) = 1. Another notation used in the literature, is Fn (x) for n (x), which has an old and long history. Since every nth root of unity is a primitive dth root of unity for some uniquely determined d|n, one has xn − 1 = d (x), (2) d|n

which by the M¨obius inversion formula (see Chapter 2) implies log n (x) = {log(x n/d − 1)}µ(d), d|n

giving n (x) =

(−1 + x n/d )µ(d)

(3)

d|n

If the prime factorization of n > 1 is n = p1a1 p2a2 . . . prar , then (3) implies n (x) =

(x n − 1)(x n/ p1 p2 − 1)(x n/ p1 p3 − 1) . . . (x n/ p1 − 1)(x n/ p2 − 1) . . . (x n/ p1 p2 p3 − 1) . . .

(4)

Formula (4) was proved by A. Cauchy in 1829 (see [103], p. 184), and it implies immediately that n (x) is of integer coefficients (with leading coefficient 1) with degree ϕ(n). 251

CHAPTER 3

From (3) (or (4)) follows that np (x) =

n (x p ) , if p n ( p prime), n (x)

np (x) = n (x p ), if p|n

(5) (6)

For an odd prime p, (4) gives p (x) = 2 p (x) = 4 p (x) =

xp − 1 = x p−1 + x p−2 + · · · + x + 1, x −1

(7)

(x 2 p − 1)(x − 1) = x p−1 − x p−2 + · · · − x + 1, (x p − 1)(x 2 − 1)

(x 4 p − 1)(x 2 − 1) = x 2 p−2 − x 2 p−4 + · · · − x 2 + 1, (x 2 p − 1)(x 4 − 1)

(8)

etc., so one can write the following special cyclotomic polynomials: 1 (x) = x − 1, 2 (x) = x + 1, 3 (x) = x 2 + x + 1, 4 (x) = x 2 + 1, 5 (x) = x 4 + x 3 + x 2 + x + 1, 6 (x) = x 2 − x + 1, 7 (x) = x 6 + x 5 + x 4 + x 3 + x 2 + x + 1, 8 (x) = x4 + 1, 9 (x) = x 6 +x 3 +1, 10 (x) = x 4 −x 3 +x 2 −x +1, 11 (x) = x 10 +x 9 +x 8 +x 7 +x 6 + x 5 +x 4 +x 3 +x 2 +x +1, 12 (x) = x 4 −x 2 +1, 15 (x) = x 8 −x 7 +x 5 −x 4 +x 3 −x +1, 20 (x) = x 8 − x 6 + x 4 − x 2 + 1, 21 (x) = x 12 − x11 + x 9 − x 8 + x 6 − x 4 + x 3 − x + 1. For n > 1, n (0) = 1 (i.e. constant term =1); p, if n = pa ( p prime) (9) n (1) = 1, otherwise (i.e. ω(n) ≥ 2) An important result is the irreducibility of the cyclotomic polynomial n (x) over the rational field. For n = prime, this was established by C. F. Gauss [146] in 1801. The first proof of the general case was given by L. Kronecker [232] in 1854. He proved actually a stronger result, namely that n is relatively prime to the discriminant of a polynomial with integer coefficients f (x), then n (x) remains irreducible over the field generated by any root of f (x). (10) There are many other irreducibility proofs for n (x). In 1857, 1858, resp. 1859 R. Dedekind [100], F. Arndt [14], resp. V. A. Lebesgue [243]; for a survey of proofs up to 1907 see M. Ruthinger [358]. Further proofs are due to K. Grandjot (1924) [155], H. Sp¨ath (1927) [435], E. Landau (1929) [238], I. Schur (1929) [401], F. Levi (1931) [254], T. Skolem (1949) [426], etc. L. Weisner [475] considered quadratic fields over which the cyclotomic polynomials are reducible, and more generally polynomials f (g(x)) reducible in fields in which f (x) is reducible [476]. The factorization of n (x) in quadratic fields and their composites was studied by D. Pumpl¨un [341], 252

THE MANY FACETS OF EULER’S TOTIENT

who generalized previous work by H. Petersson [329]. Simple proofs of factorization (mod p) of n (x) may be found in R. Ballieu [25] or W. J. Guerrier [166]. In 1960 I. Seres [407] considered questions concerning the irreducibility of polynomials of the form n (P(x)), where P(x) is a polynomial. In [478] one can read that C. Nicol proved that p (x) + q (x) is irreducible for p, q primes; while generally for m, n > 1, if m (x) + n (x) factors, then for m, n ≤ 150, the factors contain a cyclotomic polynomial, e.g. 7 (x) + 22 (x) = 4 (x)(x 8 − x 7 + 2x 4 + 2)

(11)

Is this generally true (or what conditions should be assumed)?

2

Divisibility properties

There are many divisibility properties of the cyclotomic polynomials or related objects. The following theorem is due essentially to L. Kronecker [233] (see also T. Nagell [306], p. 164-167): If p n is a prime, then n (x) ≡ 0 (mod p)

(12)

is solvable iff p ≡ 1 (mod n). If p ≡ 1 (mod n), the solutions of congruence (12) are the numbers which belong to the exponent n(modulo p). If x is a solution, then the number n (x) is divisible by exactly the same power of p as x n − 1. (13) The particular case n = p k was discussed by A. S. Bang in 1886 (see [103], p. 385). Bang proved also that if a > 1, n ≥ 3, then n (a) has a prime factor ≡ 1 (mod n) except for 6 (2). (14) In 1905 L. E. Dickson (see [103], p. 388) showed that for a > 1, n (a) has a prime factor not dividing a m − 1 (m < n) except in the cases n = 2, a = 2k − 1, and n = 6, a = 2. (15) The following result is due essentially to Dickson (see also [306] p. 166): Suppose p is a prime factor of n, and put n = pa n 1 , where p n 1 . Then the congruence (12) is solvable iff p ≡ 1 (mod n 1 ). In this case, the solutions are the numbers which belong to the exponent n 1 (modulo p). If x is a solution, then n (x) is divisible by p (16) and not by p 2 , provided that n > 2. More general results for the homogeneous form of n (x) were obtained by R. D. Carmichael in 1909. As a corollary of theorem (13) we note that: For n > 1 there are infinitely many primes ≡ 1 (mod n). (17) Indeed, if there would be only a finite numbers of such primes, let P be their product. Put x = n P y (y > 0, integer). Then by (13), every prime factor of n (n P y) must 253

CHAPTER 3

be ≡ 1 (mod n). But this is impossible, since n (n P y) ≡ n (0) ≡ 1 (mod n P) (see [306], p. 168). Similar proofs of theorem (17) (i.e. a particular case of Dirichlet’s Theorem on primes in arithmetic progressions), by use of cyclotomic polynomials were obtained by A. S. Bang and J. J. Sylvester (see [103], p. 418). In 1896 A. Hurwitz (see [103], p. 378) proved the following primality criterion: If there exists an integer a such that n−1 (a) ≡ 0 (mod n) then a = prime. For example, when n = 2k + 1, since 2k (x) = x 2 theorem on the primality of Fermat numbers. We note that for n odd one has 2n (x) = n (−x), and that for all n, 1 ϕ(n) = n (x) x n x

(18) k−1

+ 1, we get a (19) (20)

which can be proved also by (3). (For an irrationality result based on (20), see [363], p. 286). In 1880 A. E. Pellet (see [103], p. 245) considered the equation f n (y) = 0,

(y = x +

1 , n (x) = 0) x

(21)

i.e. the equation of degree ϕ(n) derived from n (x) = 0 by the substitution y = 1 x + . He proved that if p n is a prime, and x f n (y) ≡ 0

(mod p)

(22)

has an integral root a, then n (x) is divisible modulo p by x 2 − 2ax + 1. Either the latter has two real roots and n (x) and f n (y) have all their roots real and p − 1 is divisible by n, or it is irreducible and n (x) is a product of quadratic factors (modulo p) and the roots of f n (y) are all real and p + 1 is divisible by n. If n divides neither p + 1 nor p − 1, f n (y) is a product of factors of equal degree (modulo p). (23) 1 J. J. Sylvester (1880) introduced similarly gn (y) by setting y = x + in x n (x)/(x ϕ(n)/2 ) (see [103], p. 384). He stated that every divisor of gn (y) is of the form ±1 (mod n), with the exception that, if n = p k ( p ∓ 1)/m, then p is a divisor (but not p 2 ). Conversely, every product of powers of primes of the form ±1 (mod n) is a divisor of gn (y). (24) 254

THE MANY FACETS OF EULER’S TOTIENT

In 1888 L. Kronecker and M. Bauer (see [103], p. 385) remarked that x+y ϕ(n) (x − y) n = h n (x, y 2 ) x−y

(25)

is a polynomial with integer coefficients, involving only even powers of y. For the prime factors of h n (x, s), the following is true (s is given): If p is prime, and p n, p s, then s (mod n) (26) p≡ p In 1912 G. Fonten´e (see [103], p. 390) considered the homogeneous form a n (a, b) derived from n (x) by setting x = . If a and b are relatively primes, b then every prime divisor of n (a, b) is of the form ≡ 1 (mod n), unless it is divisible by the greatest prime factor p of n. It has this factor p if p −1 is divisible by n/ p α (where p α n), and a, b satisfy n/ pα (a, b) ≡ 0 (mod p), the latter having for each (b, p) = 1 a number of roots a equal to the degree of the congruence. In particular, if n = pa ( p prime), every prime factor of n is of the form ≡ 1 (mod n), with the exception of a divisor p occurring if a ≡ b (mod p), and then to the first power if n = 2. (27) When a, b are special complex numbers, results on the prime factors of n (a, b) have been obtained by T. N. Shorey and C. L. Stewart [413]. Let a, b ∈ C such that a (a +b)2 and a ·b are nonzero, relatively prime integers, with not a root of unity. Let b P(m) denote the greatest prime divisor of m, and let by convention P(0) = P(±1) = 1. Put Pn = P( n (a, b)) for n ≥ 3. Then for any k with 0 < k < 1/ log 2, and any integer n > 3, with at most k log log n distinct prime factors, we have Pn > C(ϕ(n) log n)/2ω(n)

(28)

where C > 0 is effectively computable in terms of a, b and k. For almost all integers n, Pn > n(log n)2 /k(n) log log n,

(29)

where k : N → R is any function with the property k(n) → ∞ as n → ∞. Finally we state certain simple divisibility properties related to n . As we have seen by relation (94 ) (see 5 of 3.2) one has n|ϕ( n (a)) for any a > 1. A similar property is due to A. Bartholom´e [28]: k| n (a k ) for infinitely many n for any (a, n) = (2, 1). 255

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CHAPTER 3

R. Wang [473] proves that if p is a prime, and n > 2, n| p − 1; if p1 , . . . , pϕ(n) are primes satisfying n ( pi ) ≡ 0 (mod p),

i = 1, ϕ(n),

(31)

then ϕ(m) ≡ α or 1 (mod p) according as ω(n) = 1 and n is an αth power of a prime, or ω(n) > 1, with m = p1 . . . pϕ(n) . E. Vantieghem [463] proves the congruence (x − y t ) ≡ n (x) (mod n (y)) for n ≥ 2 (32) t∈t (n)

By using (32) with x = 1, y = 2 he proves the following primality criterion: Let p > 2. Then p is prime iff p−1 (2k − 1) ≡ p

(mod (2 p − 1))

(33)

k=1

H. W. Leopoldt [252] rediscovered Bang’s result (14), and used it to solve a problem by Kostrikhin: find all pairs (a, n) with a, n > 1 such that for all primes (34) p|(a n − 1) one can find an m < n such that p|(a m − 1). For n = 2 all pairs (2s − 1, 2) with s ≥ 2 satisfy the property, while for n ≥ 3 the only pair is (2,6). (35) Cyclotomic polynomials are also used for factorization methods of integers, see e.g. E. Bach and J. Shallit [20], or R. P. Brent [52].

3

The coefficients of cyclotomic polynomials We will now treat the coefficients of cyclotomic polynomials. Let n (x) =

ϕ(n)

a(m, n)x m

(36)

m=0

If n is prime, then a(m, n) = 1 for all m. When n = pq, with p, q distinct primes, then A. S. Bang [26] and A. Migotti [290] have proved that a(m, n) ∈ {−1, 0, +1} for all m

(37)

Migotti showed also that a(7, 105) = −2. Since for all n ≤ 104, (37) holds true (see P. Erd¨os [120]), 105 (x) is the first cyclotomic polynomial which has other coefficients than 0 and ±1. The number m = 7 is minimal, with |a(m, n)| > 1 for some n, see M. Endo [108]. In fact, every integer appears as a coefficient of a certain cyclotomic polynomial, see J. Suzuki [450]. 256

THE MANY FACETS OF EULER’S TOTIENT

Now, when n = pq, besides (37) there are known some special results. M. Beiter [34] has proved that a(m, n) =

(−1)δ , if n = αq + βp + δ in exactly one way, 0, otherwise

(38)

where α, β are nonnegative integers and δ = 0, 1. L. Carlitz [62] proved that if p < q, the number of positive coefficients is equal to θ = ( p − u)(uq + 1)/ p (39) where u is defined by qu ≡ −1 (mod p), 0 < u < p. Therefore, the total number of nonzero coefficients is 2θ − 1 = 2( p − u)(uq + 1)/ p − 1

(40)

T. Y. Lam and K. H. Leung [237] have recently rewritten the results in another form. Write ( p − 1)(q − 1) = r p + sq (41) and let 0 ≤ k ≤ ( p − 1)(q − 1). Then a(m, n) = 1 iff k = i p + jq for some i ∈ [0, r ] and j ∈ [0, s]; a(m, n) = −1 iff k + pq = i p + jq for i ∈ [r + 1, q − 1] and j ∈ [s + 1, p − 1]; otherwise a(m, n) = 0. (42) The number of terms having ak = 1 is (r + 1)(s + 1), and those with ak = −1 is ( p − s − 1)(q − r − 1). Furthermore, for p < q, the middle coefficient is (−1)r . (43) Let A(n) = max{|a(m, n)| : m = 0, 1, . . . , ϕ(n)}. The following result is essentially due to Bang [26]: If n has three distinct odd prime factors p < q < r , then An ≤ p − 1 (44) If n has fewer than three odd prime factors, then An = 1. (45) D. M. Bloom [45] showed that if An = p − 1 in (44), with p ≥ 5, then neither q nor r is ≡ ±1 (mod p). (46) Bloom proved also that if n has four odd prime factors p < q < r < s, then An ≤ p( p − 1)( pq − 1)

(47)

In 1931 I. Schur showed that (see [251]) lim sup An = +∞ n→∞

257

(48)

CHAPTER 3

by proving that if m is any odd number and if n = p1 p2 . . . pm , p1 < p2 < · · · < pm , p1 + p2 > pm ( pi primes), then a( pm , n) = m − 1

(49)

E. Lehmer [251] showed that (An ) is unbounded, even when n is a product of only three primes, by proving that if p, q = kp + 2 and r = (mpq − 1)/2 are all primes, and if h = ( p − 3)(qr + 1)/2, then a(h, pqr ) =

p−1 2

(50)

W. Bosma [47] has recently used Magma system for computational algebra to list for |a| ≤ 50 the least n for which a appears as a coefficient of n (x). In fact, values n ≤ 26565 suffice for all a with |a| ≤ 50 with the exception of a = −50, which occurs for the first time in 40755 (x) as coefficient of x 1082 . ϕ(n) n has ϕ(n) roots x j , j = 1, ϕ(n). Let αk = αk (n) = x kj be the Ramanujan j=1

sum (denotes also as C(k, n)). By using Newton’s formulae concerning symmetric polynomials, we can write α1 b0 + b1 = 0 α2 b0 + α1 b1 + 2b2 = 0 ... αi b0 + αi−1 b1 + · · · + ibi = 0 ... αϕ(n) b0 + αϕ(n)−1 b1 + · · · + ϕ(n)bϕ(n) = 0

(51)

where n (x) =

ϕ(n)

a(m, n)x m = b0 x ϕ(n) + b1 x ϕ(n)−1 + · · · + bϕ(n) = 0

(52)

m=0

We may view (51) and (52) as a system of equations in the ϕ(n) + 1 unknowns b0 , b1 , . . . , bϕ(n) (x being an arbitrary primitive root of unity), and since b0 = 1, the determinant of thus homogeneous system must be zero. By developing this determinant under the ϕ(n) + 1 th row (i.e. the one which contains x ϕ(n) , x ϕ(n)−1 , . . . , 1) one gets that n (x) = x ϕ(n) +

(−1) (−1)i (−1)ϕ(n) A1 x ϕ(n)−1 +· · ·+ Ai x ϕ(n)−i +· · ·+ Aϕ(n) (53) 1! i! n! 258

THE MANY FACETS OF EULER’S TOTIENT

where Ai (obtained by Laplace’s rule) is given by α1 1 0 0 0 ... 0 α2 α1 2 0 0 ... 0 α3 α α 3 0 . . . 0 2 1 Ai = ... ... ... ... ... αi−1 αi−2 αi−3 . . . . . . . . . i − 1 α αi−1 αi−2 . . . . . . . . . α1 i

(i = 1, ϕ(n))

(54)

Since by H¨older’s theorem n n /ϕ αk (n) = ϕ(n)µ (n, k) (n, k)

(55)

by (54), all coefficients of n (x) can be determined (at least, in theory), see M. Deaconescu and J. S´andor [98] or D. H. Lehmer [246]. For example, in (52) one has

b1 = −µ(n),

 u(u − 1), if n is odd    u(u − 1) , if n = 2m, m odd b2 = 2  u   − , if n = 2k m, m odd, k > 1 2

(56)

where u = µ(m). A complicated formula can be written for b3 . When (n, 6) = 1, 1 b3 = − (u 3 − 3u 2 + 2u) 6

(57)

for a particular table of values up to b10 , see [246]. By using Bernoulli and Stirling numbers, Lehmer determines exactly also the coefficients of n (x + 1). For example, the constant term is e(n) ; the coefficient 1 of x is ϕ(n)e(n) , where (n) is the von Mangoldt function. (These were known 2 essentially to Lebesgue and H¨older). The coefficient of x k is e(n) Rk {ϕ(n), J2 (n), J4 (n), . . . , J2h (n)}

(58)

where 2h ≤ k < 2h + 2, Rk is a polynomial with rational coefficients, and Jm (n) is Jordan’s totient function. Bounds for the coefficients a(k, n) of x k in n (x) given by (36) were obtained by P. Erd¨os and R. C. Vaughan [131]: |a(k, n)| < exp{ck 1/2 + c1 k 3/8 } 259

(59)

CHAPTER 3

and

1/2 k 1 , lim sup |a(k, n)| > exp c2 625 log k n→∞

k > k0 ,

(60)

1/2 2 1− 2 , and c1 , c2 are other constants. where c = 2 p +p p prime When n (x) is given by (52), H. M¨oller [293] proved that

|bi (q1 q2 . . . qs )| ≤

2q1 q1 − 1

2s−4 s−2 s−k−1 −1 · (qk − 1)2 ,

(s ≥ 4)

(61)

k=1

where q1 < · · · < qs are arbitrary odd primes. Schur’s theorem (48) was improved for the first time in 1946 by P. Erd¨os [120]: An > exp{c1 (log n)4/3 } for infinitely many n

(c1 > 0)

Later [121] he refined this to c log n for infinitely many n An > exp exp log log n P. T. Bateman [32] proved that log n An < exp exp(log 2 + o(1)) log log n

(c > 0)

(62)

(63)

(64)

and R. C. Vaughan [466] showed that the constant log 2 is best possible. Similarly c = log 2 in (63) is best possible. (65) A(n) depends essentially on the squarefree kernel of n. Bateman, Pomerance and Vaughan [33] have showed that if s ≥ 3, and q1 < · · · < qs are odd primes, then s−2 S(q1 . . . qs ) s−k−1 −1 < A(q1 . . . qs ) < qk2 q1 . . . qs k=1

where S(n) =

ϕ(n)

(66)

|a(m, n)|.

m=0

In 1990 H. Maier [274] stated the following result: Let ε(n) be a function defined for all positive integers n such that ε(n) → 0 as n → ∞. Then A(n) ≥ n ε(n) for almost all n 260

(67)

THE MANY FACETS OF EULER’S TOTIENT

In another paper [275] it is shown that for any N > 0 there exist numbers c(N ) > 0, x0 (N ) ≥ 1 such that car d{n ≤ x : A(n) ≥ n N } ≥ c(N )x

(68)

In [276] it is considered the following counterpart of (67): Let ψ(n) be a function defined for all positive integers n such that ψ(n) → ∞ for n → ∞. Then A(n) ≤ n ψ(n) for almost all n. Some recent work has been concerned with B(m) = max{|a(m, n)| : n ≥ 1} = lim sup |a(m, n)|. n→∞

In 1985 H. L. Montgomery and R. C. Vaughan [296] proved that log B(m) m 1/2 (log 2m)−1/4 (69) log B(m) m 1/2 (log m)−1/4 √ which show that log B(m) has order of magnitude m(log m)−1/4 for large m. G. Bachman [21], improving (69) showed that

√ log log m m 1+O (70) log B(m) = C (log m)1/4 log m where C is an explicitly given constant. The proof includes the Hardy-Littlewood circle method and estimates on exponential sums with multiplicative coefficients. Finally we note that L. Carlitz [63] studied questions related to the sum of squares of n (x).

4

Miscellaneous results This last section includes miscellaneous results on cyclotomic polynomials. Certain inequalities for n (x) are: (x − 1)ϕ(n) < n (x) < (x + 1)ϕ(n) for x > 1, n > 1

(71)

(see [252]) and

x −1 x

2r −2

x2 − 1 x2

2r −2 x

ϕ(n)

< n (x) <

where r = ω(n) (see R. Richter [351]). 261

x x −1

2r −1

x ϕ(n) , x > 1, n > 1, (72)

CHAPTER 3

If n ≡ 1 (mod 2), then

x x +1

2r −1 x

ϕ(n)

< n (−x) <

x +1 x

2r −2

x2 + 1 x2

2r −2

x ϕ(n) ;

(73)

if n ≡ 2 (mod 4), n > 2, then

x −1 x

2r −3

x2 − 1 x2

2r −3 x

ϕ(n)

< n (−x) <

x x −1

2r −2

x ϕ(n)

(74)

If n is an odd multiple of a prime ≡ 1 (mod 4), then for sufficiently large n (see [247]): n (−2) > n(2n + 1) (75) Consider the function f : [1, ∞) → R, f (t) = n (t), where n is the nth cyclotomic polynomial. K. Motose [304] proved that f is a strictly increasing function for all n. (76) A theorem of T. M. Apostol [11] and F.-E. Diederichsen [105] gives the value of the resultant of cyclotomic polynomials m and n for m > n > 1 as follows:

m p ϕ(n) , if n|m and is a power of a prime p, ρ( m (x), n (x)) = (77) n 1, otherwise For a new proof, see S. Louboutin [260]. More generally, Apostol [12] showed that for all m, n ≥ 1 and arbitrary nonzero complex numbers a, b n ρ( m (ax), n (bx)) = bϕ(m)ϕ(n) [ m/δ (a d /bd )]µ( d )ϕ(m)/ϕ(m/δ) (78) d|n

where δ = (m, d). When m and n are distinct primes p and q, then (78) simplifies to  pq  a − b pq a − b · , for a = b ρ( q (ax), p (bx)) = (79) a p − b p a q − bq  a ( p−q)(q−1) , for a = b M. Kaminski [218] proved that if P(x) = x is a monic irreducible polynomial with integer coefficients such that ρ(P(x), n (x)) = ±1 for infinitely many n, then P(x) is a cyclotomic polynomial. A connection between cyclotomic polynomials and the Riemann hypothesis has been studied by F. Amoroso [9], [10]. Let FN (z) = n (z) and put n≤N

1 h(FN ) = 2π

π

−π

log+ |FN (eiθ )|dθ

262

(80)

THE MANY FACETS OF EULER’S TOTIENT

(a function introduced by K. Mahler in 1964 and M. Mignotte in 1979 in the Theory of Diophantine approximations). Amoroso shows that the inequality h(FN ) N λ+ε

(ε > 0)

(81)

is equivalent to the assertion that the Riemann zeta function does not vanish for Re z ≥ λ + ε. (82) 1 (Particularly, the Riemann hypothesis states (82) for λ = ). 2 Another result in the same vein is that if λ < 1 and if log |FN (α)| −N λ for every root of unity α of order ≤ N , then for any ε > 0, the Riemann zeta function does not vanish for Re z ≥ λ + ε. (83) V. N. Sorokin [434] proved the linear independence over Q of the set {log d (a, b) : d|n} ∪ {1}, where n is a positive integer; a, b are two nonzero rational 1/( p−1) n+1 n integers satisfying |b| > cn |a| , where cn = 12(2nεn ) , with εn = p (p p|n

prime). (84) R. Creutzburg and M. Tasche [90] proved that a Gaussian integer e is a primitive nth root of unity modulo a Gaussian integer m iff m is a divisor of n (e)/q, where q is either 1, p or π , where p is the largest prime divisor of n, and π is a Gaussian prime divisor of p. (85) Let α be a unit of degree d in an algebraic number field and assume that α is not a root of unity. Let U (α) be the number of values of n for which n (α) is a unit. Then J. H. Silverman [417] proved that U (α) ≤ cd 1+0.7/ log log d ,

(86)

where c is an effectively computable constant.

3.6 1

Matrices and determinants connected with ϕ Smith’s determinant

In 1876 H. J. S. Smith [432] discovered his famous determinant theorem det[(i, j)]n×n = ϕ(1)ϕ(2) . . . ϕ(n)

(1)

where (i, j) is the greatest common divisor (GCD) of i and j. In fact (see [432] and [103], p. 128) Smith proved the more general result det[ f (i, j)]n×n = g(1)g(2) . . . g(n), where g(d), m = 1, 2, . . . f (m) = d|m

263

(2)

CHAPTER 3

(and where we have denoted by f (i, j) the value taken by f at (i, j), i.e. f ((i, j))). When f (m) = m, by Gauss’ identity (1) follows. It should be noted that relation (2), or particular cases of it, has been rediscovered many times in the literature. Let f (n) = n k (k ∈ R). Then m k = g(d) gives by M¨obius inversion g(n) = d|m m k µ(d) 1 k 1 − = Jk (m), i.e. the extension of µ(d) = mk = m d dk pk p|n d|n d|n Jordan’s totient for k ∈ R. One obtains (3) det[(i, j)k ]n×n = Jk (1)Jk (2) . . . Jk (n), rediscovered by another method by B. Gyires in 1957 [172]. In 1965 I. Gy. Maurer and M. Ve´egh [286] gave two new proofs of (3); one is based on an inductive argument; the other proof is in fact the classical one. For a survey of generalizations of Smith’s determinant up to 1961, see Gy. Ol´ah [325]. Let f (n) = d(n), the number of divisors of n. Then det[d(i, j)]n×n = 1,

(4)

which is included in P´olya-Szeg¨o (1924 the first edition) Part VIII, Problem 1.31. Problem 1.33 is in fact (2), with a proof different from Smith’s. The proof is based on a remark that [ f (i, j)]n×n can be written in the form [ f (i, j)]n×n = B · C T

(5)

where B and C are lower triangular matrices given by bi j = g( j) if j|i; and 0, otherwise; ci j = 1 if j|i; and 0, otherwise. L. Carlitz [64] observed that [ f (i, j)]n×n = (diag(g(1), g(2), . . . , g(n))C T

(6)

where C is the triangular matrix from (5). Relations (3), (4) appear also in [323], where one can find also some other particular cases. Put e.g. f (n) = σ (n), the sum of divisors of n. Then, since g(d) = d, one has det[σ (i, j)]n×n = n! (7) Let f (n) = µ(n), the M¨obius function. Then it is immediate that g( p) = −2, g( p 2 ) = 1 and g( pr ) = 0 for r ≥ 3 ( p prime), and g is multiplicative, so one has det[µ(i, j)]n×n = k(n),

(8)

where k(n) = 1 for n = 1; −2 for n = 2; 4 for n = 3; 4 for n = 4; −8 for n = 5; −32 for n = 6; 64 for n = 7; and 0 for n ≥ 8. (9) 264

THE MANY FACETS OF EULER’S TOTIENT

Let f (n) = γ (n) =

n, the core of n. Then g( p) = p − 1, g( pr ) = 0 for

p|n

r > 1, so:

det[γ (i, j)]n×n = l(n),

(9)

where l(n) = 1, for n = 1; 1 for n = 2; 3 for n = 3; and 0 for n ≥ 4. [i, j] 1 1 = Let f (n) = . Since , where [i, j] denotes the l.c.m. of i, j; by n (i, j) ij 1 , 1− using properties of determinants, and the fact that g(n) = f (n) f ( p) p|n when f is multiplicative, one can deduce: det [i, j] = ϕ(1)ϕ(2) . . . ϕ(n)γ (1) . . . γ (n)(−1)ω(1)+···+ω(n) (10) n×n

where γ (k) is the core of k, and ω(k) is the number of distinct prime factors of k. We note that (10) is due also to Smith [432], who used the notation γ (k)(−1)ω(k) = π(k) (i.e. it is a multiplicative function satisfying π(1) = 1, π( pr ) = − p for r ≥ 1). Example (1), (4), (7), as well as a proof of (2) can be found also in E. D. Schwab [402]. The determinant of (10) has been presented also in Problem 10232 of American Math. Monthly 96(1992), no. 6, as a subject for evaluation. Problem 6089 by R. M. Redheffer [347] however shows that a small change in the determinant from (4) can lead to a serious difference. Namely, det[d(i, j)]2≤i, j≤n = µ2 (n), (11) i≤n

i.e. if one deletes the first line and first row in (4), then one obtains that the value of the obtained determinant is equal to the number of squarefree integers from 1 to n. In 1972 T. M. Apostol [13] by using the idea that [ f (i, j)]n×n can be written as a product of two triangular matrices, generalized (2) as follows: Let f, g and h satisfy g(d)h( j/d) (12) f (i, j) = d|(i, j)

Then det[ f (i, j)]n×n = g(1)g(2) . . . g(n)(h(1))n

(13)

By taking g(n) = n (n = 1, . . . ) and h = µ, one obtains det[C(i, j)]n×n = n!

(14)

where C(i, j) is Ramanujan’s sum (denoted by C(n, d) in (27) of 3.1). A more general Ramanujan sum is Ck (n, d) = e(nx, d k ), (15) (x,d k )k =1

265

CHAPTER 3

where the sum is over all x in an arbitrary reduced (d, k)-residue system, and (a, b)k is the largest kth power divisor of a and b. Then (see McCarthy [69]) det(Ck (i, j)]n×n = 0

(16)

for all n, k ≥ 2. For more general results, where Apostol’s evaluation is extended to the case of even arithmetical functions, see P. J. McCarthy [71]. G. Daniloff [93] found an analogue of Smith’s determinant as follows: Let Dk (n) = m if n = m k for some positive integer m, and =0 otherwise. Let f (i, j) = Dk (i/d)Dk ( j/d)g(d) (17) d|(i, j)

Then det[ f (i, j)]n×n = g(1)g(2) . . . g(n)

(18)

The unitary analogue of Smith’s determinant was introduced by H. Jager [211]. Let (i, j)∗ denote the greatest common unitary m divisor of i and j. Let d m denote = 1; see Chapter 2). If that d is a unitary divisor of m (i.e. d|m, d, d f (i, j) = g(d), (19) d (i, j)∗

then (20) det[ f (i, j)]n×n = g(1)g(2) . . . g(n) Let ϕ ∗ be the unitary totient function. Since ϕ ∗ (d) = m, Jager deduced from d m

(20) the analogue of (1): det[(i, j)∗ ] = ϕ ∗ (1)ϕ ∗ (2) . . . ϕ ∗ (n)

(21)

K. Nageswara Rao [308] gave an A-analogue of Smith’s determinant, where A denotes Narkiewicz’s regular convolution. See also C. R. Wall [471]. The classical and the unitary cases are particular cases of this A-analogue determinant. Multidimensional Smith’s determinants have been considered by L. Gegenbauer [148], D. H. Lehmer [248], N. P. Sokolov [433], P. Haukkanen [181], R. Vaidyanthaswamy [462].

2

Poset-theoretic generalizations

Smith deduced his result (2) in a slightly more general setting, namely if x1 , x2 , . . . , xn are distinct positive integers with the property that d|xi ⇒ d = x j for some j = 1, 2, . . . , n, 266

(22)

THE MANY FACETS OF EULER’S TOTIENT

then det[ f (xi , x j )]n×n = g(x1 )g(x2 ) . . . g(xn )

(23)

Such a set S = {x1 , x2 , . . . , xn } (i.e. satisfying (22)) is called factor-closed set. For example, (24) det[(xi , x j )]n×n = ϕ(x1 )ϕ(x2 ) . . . ϕ(xn ) for a factor closed set S. The idea has been extended to the lattice-theoretical context by H. S. Wilf [482], B. Lindstr¨om [259], P. Haukkanen [182], B. V. Rajarama Bhat [342], P. Haukkanen [183], P. Haukkanen, J. Wang and J. Sillanp¨aa¨ [192]. For example, a common generalization of (13), (20) and (23) is obtained in the context of meet semilattices. Let (P, ≤) be a poset (see Chapter 2). Then P is called a meet semilattice if for any x, y ∈ P there exists a unique z ∈ P such that: 1) z ≤ x and z ≤ y; 2) if w ≤ x and w ≤ y, for some w ∈ P, then w ≤ z. In such a case z is called the meet of x and y, and denoted by x ∧ y. Let S ⊂ P. The set S is called lower-closed if for every x, y ∈ P with x ∈ S, y ≤ x, we have y ∈ S. We note that in the case of (N, |), the lower closed sets coincide with the factorclosed sets of (22). The following result is true: Let P be a finite meet semilattice, and let {x1 , x2 , . . . , xn } be a lower-closed subset of P. Let F, G : P × P → C be incidence functions of P, and let A be the n × n matrix defined by ai j = F(x, xi )G(x, x j ). (25) x≤xi ∧x j

Then det A =

n

F(xk , xk )G(xk , xk )

(26)

k=1

Let e D (d, m) = 1 is d|m; =0 otherwise in the poset (N , |). Letting F(d, m) = e D (d, m), G(d, m) = e D (d, m)g(d). Then f (xi , x j ) = n F(d, xi )G(d, x j ) = F(xk , xi )G(xk , x j ), and the above theorem yields d|(xi ,x j )

k=1

(23). Let eu (d, i) = 1 if d i; =0 otherwise in the poset (N , ). Then as above, we reobtain from (26) relation Jager’s theorem (20). If F and G are incidence functions of (N , |), let F(d, i) = e D (d, i)g(d); G(d, j) = e D (d, j)h( j/d). Then we reobtain Apostol’s theorem (13). Let C ∗ (i, j) = dµ∗ ( j/d) be the unitary analogue of Ramanujan’s d|i,d j

sum, where µ∗ is the unitary M¨obius function (see Chapter 2). Denoting F(d, i) = e D (d, i)d and G(d, j) = eu (d, j)µ∗ ( j/d). By writing C ∗ (i, j) = 267

CHAPTER 3 n

F(k, i)G(k, j), it follows that

k=1

det[C ∗ (i, j)]n×n = n!

(27)

For a last application, let e N (k, i) = 1 if k ≤ i; =0 otherwise on the poset (N, ≤). Then n min{i, j} = eN (k, i)eN (k, j) = eN (k, i)eN (k, j), k≤min{i, j}

k=1

so: det[min{i, j}]n×n = 1

(28)

The subset S ⊂ P of a finite meet semillatice will be called meet-closed (see [192]) if for every x, y ∈ S we have x ∧ y ∈ S. Let f : P → C. Then the matrix, defined by (S) f = (si j ) with si j = f (xi ∧ x j ) (1 ≤ i, j ≤ n) is called the meet-matrix of S with respect to f . The following theorem is due to Rajarama Bhat [342]: Let S = {x1 , . . . , xn } ⊂ P be a meet-closed subset, and let g(xi ) be defined by g(x j ), (29) g(xi ) = f (xi ) − x j ∈S,x j <xi

(x j < xi means x j ≤ xi and x j = xi ), where f is given above. Then det(S) f = g(x1 ) . . . g(xn )

(30)

The following corollary of (30) contains earlier results by Wilf [482] and Lindstr¨om [259]: Let S be lower-closed subset of P. Then det(S) f = g(x1 ) . . . g(xn ), where g(xi ) =

(31)

f (x j )µ(x j , xi ),

x j ≤xi

µ being the M¨obius function of P. This is a poset-theoretic generalization of Smith’s theorem. The incidence matrix of two subsets S = {x1 , . . . , xn } and T = {y1 , . . . , ym } is defined by E(S, T ) as an n × m matrix whose i, j entry is 1 if y j ≤ xi ; and 0, otherwise. Let S = {x1 , . . . , xn , xn+1 , . . . , xn+r } be the unique (up to isomorphism) minimal meet semilattice containing S. If g : S → C is defined as in (29), then (S) f = E diag(g(x1 ), g(x2 ), . . . , g(xn+r ))E T 268

(32)

THE MANY FACETS OF EULER’S TOTIENT

where E = E(S, S) and E T is the transpose of E. This result is from [342], and is a generalization of Carlitz’s theorem (6). Theorems (30) and (32) extend results of S. Beslin and S. Ligh [41], [42], where the classical (N, |) poset is considered. In [192] the determinant of (32) is calculated: ∗ det(E (k1 ,...,kn ) )2 g(xk1 ) . . . g(xkn ) (33) det(S) f = ∗ where in one has 1 ≤ k1 < k2 < · · · < kn ≤ n + r , and E (k1 ,...,kn ) is the submatrix of E = E(S, S) consisting of the k1 th, . . . ,kn th columns of E. If f and g are real valued functions and g(x) ≥ 0, then det(S) f ≥ g(x1 ) . . . g(xn )

(34)

with equality iff S is a meet-closed subset. Let P be a finite meet semillatice, S = {x1 , . . . , xn } ⊂ S, and f : P → C. The following generalized totient function has been introduced in [342]. Let ϕ S, f be defined inductively by ϕ S, f (xi ) (35) ϕ S, f (x j ) = f (x j ) − xi <x j

(or f (x j ) =

ϕ S, f (xi )).

xi ≤x j

Remark that when S is a factor-closed set of positive integers ordered by | in N, and f (x) = x for all x, then ϕ f,S = ϕ, Euler’s totient function. When S is meet-closed and i < j whenever xi < x j , then f (w)µ(w, z), (35) ϕ S, f (x j ) = z≤x j ,xxt t< j

w≤z

while when S is lower-closed, then ϕ S, f (x j ) =

f (xi )µ(xi , x j ),

(36)

xi ≤x j

which is a consequence of (35). For a proof of (35), see [183]. By using the evaluation (32), P. Haukkanen [183] proved that if S is meet-closed, then n ϕ S, f (xk ) (37) det(S) f = k=1

269

CHAPTER 3

3

Factor-closed, gcd-closed, lcm-closed sets, and related determinants

We now return to the classical poset (N, |), where further analogous notions and results are obtainable. Let S{x1 , . . . , xn } be a set of distinct positive integers. Let the G ⊂ D matrix whose i, j-entry is (xi , x j ) be denoted simply by (S), and in the general case of f (xi , x j ), by (S) f ( f being an arithmetic function). Similarly, let the LCM matrix whose i, j-entry is [xi , x j ] be denoted by [S], and in the case of f [xi , x j ], by [S] f . As we have seen in section 2, S is said to be factor-closed if d ∈ S whenever xi ∈ S and d|xi . H. J. S. Smith showed that if S is factor-closed, then (2) holds. S. Beslin and S. Ligh [41] have called S gcd-closed if (xi , x j ) ∈ S, whenever xi , x j ∈ S. Note that if S is factor-closed, then it is gcd-closed. too. They extended (2) to the case of gcd-closed sets (in fact, (30) of section 2 for meet-matrices, generalizes this result): If S is gcd-closed set, then det(S) f =

n ( f ∗ µ)(d) i=1

(38)

d

where d runs through the divisors of xi such that d xt for xt < xi . We note that Beslin and Ligh obtained (38) for the case f (n) = n, this general form is due to Bourque and Ligh [48]. By extending (10) of Smith to gcd-closed matrices, they proved also that (see [49]) det[S] =

n i=1

xi2

ϕ(d)γ (d)(−1)ω(d) /d 2

(39)

d

where the sum over d is as in (38). For determinants of matrices on gcd-closed sets, and associated to other arithmetic functions, see S. Hong [202]. See also [203] by Hong on LCM matrices. When S is factor-closed, and f is a multiplicative function, in another paper [50] they showed that det[S] f =

n

f (xi )2 g(xi ),

(40)

i=1

1 ∗ µ, and f (n) = 0 for all n. f For f (n) = n k this gives a result (similar to (10)) due also to H. J. S. Smith. In 1996 P. Haukkanen and J. Sillanp¨aa¨ [187] extended (40) to gcd-closed sets S, but for quasi-multiplicative functions, defined as follows: f is said to be quasimultiplicative, if f is not identically zero, and if there exists a non-zero constant q such that f (a) f (b) = q f (ab) (41) where g =

270

THE MANY FACETS OF EULER’S TOTIENT

whenever (a, b) = 1. When q = 1 this gives the usual notion of multiplicative functions. Now let S be a gcd-closed set, and suppose that f is quasi-multiplicative, and f (n) = 0 for all n. Then det[S] f =

n

f (xi )2

g(d),

(42)

d

i=1

where g is given as in (40), and the sum over d is as in (38). Haukkanen and Sillanp¨aa¨ introduced also the concept of lcm-closed sets S, and obtained results for det(S) f and det[S] f when f is completely multiplicative. S is said to be lcm-closed if [xi , x j ] ∈ S for every xi , x j ∈ S. If S is lcm-closed, then xi | max S for all xi ∈ S. Let us denote max S = xn = m. Now the following theorems are valid: Let S = {x1 , . . . , xn } be an lcm-closed set of distinct positive integers, and let f be completely multiplicative with f (n) = 0 for all n. Then det(S) f = f (m)

−n

n

f (xi )2

( f ∗ µ)(d)

(43)

d

i=1

m m such that d for xi < xt . xi xt 1 Let S and f be as above, and let g = ∗ µ. Then f

where d runs through the divisors of

det[S] f = f (m)n

n i=1

g(d)

(44)

d

where in the sum d is as in (43). Regarding unitary Smith determinants, we have stated results like (19) and (27). In what follows, (a, b)∗ will denote the greatest common unitary divisor (gcud) of a and b, while (a, b)∗ will denote the semi-unitary greatest common divisor (sugcd), which is the greatest unitary divisor of b which is a divisor of a. The least common unitary multiple (lcum) of a and b will be defined by [a, b]∗ = ab/(a, b)∗ . The set S = {x1 , . . . , xn } of distinct positive integers will be called ud-closed if d ∈ S whenever d xi for xi ∈ S. The set S is said to be gcud-closed if (xi , x j )∗ ∈ S for every xi , x j ∈ S; and sugcd-closed if (xi , x j )∗ ∈ S for every xi , x j ∈ S. Haukkanen and Sillanp¨aa¨ [187] prove the following theorems: If S is gcud-closed, and f is an arbitrary arithmetic function, then det(S) f [ f ((xi , x j )∗ )]n×n =

n ( f ⊕ µ∗ )(d) i=1

271

(45)

CHAPTER 3

where ⊕ is the unitary convolution, µ∗ the unitary M¨obius function, and the sum is over all d xi , d ∦ xt for xt < xi . As a corollary, (45) implies that if S is ud-closed, then det[ f ((xi , x j )∗ )]n×n =

n

( f ⊕ µ∗ )(xi )

(46)

i=1

containing the particular case ∗

det[(xi , x j ) ]n×n =

n

ϕ ∗ (xi )

(47)

i=1

For the next theorem we must assume that f is completely multiplicative. Let S be gcud-closed, and let f be completely multiplicative with f (n) = n for all n. Then n 1 (48) f (xi )2 det[ f ([xi , x j ]∗ )]n×n = ⊕ µ∗ (d), f i=1 where the sum is over d as in (45). Result (45) holds true also if S is sugcd-closed set. The corresponding result to (48) for sugcd-closed sets, however, is an open problem. For the singularity or nonsingularity of matrices related to (S) and [S], see [49], [188], [201], [192]. The reciprocal GCD and LCM matrices have been introduced by S. Beslin [40]. These are the matrices 1 1 , resp. (49) (xi , x j ) n×n [xi , x j ] n×n For example, when S is factor closed, then n 1 = x −2 ϕ(xi ) det [xi , x j ] n×n i=1 i

(50)

For more general results, with [xi , x j ]r in (50), see E. Altinis¸ik and D. Tas¸ci [6]. For determinants of type f (i, j) (51) det ij i|n, j|n see P. Codec´a and M. Nair [75]. In [231] A. Krieg studies the number N (x) of matrices M in the modular group S L 2 (Z) whose maximum norm ≤ x. It is shown that N (x) is basically the partial sum of Euler’s totient function, therefore N (x) ∼

96 2 x π2 272

(x → ∞)

(52)

THE MANY FACETS OF EULER’S TOTIENT

4

Inequalities

Relation (34) gives an inequality for det(S) f , when S is a subset of a meet semilattice. In [48] Bourque and Ligh proved that if S = {x1 , . . . , xn } is a set of distinct positive integers, and if ( f ∗ µ)(d) > 0 for all d ∈ {d : d|x, x ∈ S}, ( f : N → R), then n det[ f (xi , x j )] ≥ ( f ∗ µ)(xk ) (53) k=1

with equality only if S is factor-closed. This result was first proved by Z. Li [255] in the case f (x) = x for all x. P. Haukkanen [183] obtained an extension to the case of meet-closed sets (see section 2): Let T = {y1 , . . . , ym } be a meet-closed set containing S = {x1 , . . . , xn } and let f (w)µ(w, z) > 0 for all z ∈ {z : z ≤ y, y ∈ T }. Then w≤z

det(S) f ≥

n

ϕT, f (xk ),

(54)

k=1

with equality iff S is meet-closed, and ∀ x ∈ S: ∀ z ≤ x : z ∈ T \ S (see (35) for the definition of ϕT, f ). Letting T = S in (54) one obtains: If S = {x1 , . . . , xn } and f (w)µ(w, z) > 0 for all z ∈ {z : z ≤ y, y ∈ S}, then w≤z

det(S) f ≥

n

ϕ S, f (xk )

(55)

k=1

This gives in the case of the poset (N, |) an improvement of (53): det[ f (xi , x j )] ≥

n ( f ∗ µ)(d)

(56)

k=1

where the sum is taken over all d|xk , d yt for yt < xk ; and all conditions are as in theorem (53). Z. Li [255] proved also the upper bound n! (57) 2 In [48] it is proved that for GCD matrices associated with arithmetical functions satisfying the condition ( f ∗ µ)(d) > 0 for all d ∈ {d : d|x, x ∈ S}, we have that (S) f is positive definite, and thus det[(xi , x j )] ≤ x1 x2 . . . xn −

det[ f (xi , x j )] ≤ f (x1 ) . . . f (xn ) 273

(58)

CHAPTER 3

In [183] it is shown that this holds also for the more general case of meet matrices of poset-theoretic setting: If S ⊂ P and f (w)µ(w, z) > 0 for all z ∈ {z : z ≤ w≤z

x, x ∈ S}, then (S) f is positive definite, so

det(S) f ≤ f (x1 ) . . . f (xn )

(59)

By [50] similar bounds are also true for det[ f [xi , x j ]]: Let f : N → R be multiplicative and assume that (g ∗ µ)(d) > 0 for all d ∈ {d : d|x, x ∈ S}, with 1 g = . Then [ f [xi , x j ]] is positive definite, and f n n 1 2 f (xk ) f (xk ), (60) ∗ µ (xk ) ≤ det[ f [xi , x j ]] ≤ f k=1 k=1 with equality on the left side iff S is factor closed. In [6] it is proved that n n Jr (xk ) 1 1 ≤ ≤ det 2r [xi , x j ]r xr xk k=1 k=1 k

(61)

where r > 0, and Jr is Jordan’s totient function. The reciprocal GCD and LCM matrices were in [40]. These are de introduced 1 1 , resp. [S −1 ] = . See also [49] and [230]. noted by (S −1 ) = (xi , x j ) [xi , x j ] Let · 2 denote the Euclidean norm of a matrix. D. Bozkurt and S. Solak [51] have shown that ! 5π 2

[S −1 ] 2 ≤ −4 (62) 6 Recently, E. Altinisik, N. Tuglu and P. Haukkanen [7] have generalized and improved (62) to ζ (r p)3/ p

[S −r ] p ≤ , (r p > 1), (63) ζ (2r p)1/ p where ζ is the Riemann zeta function, and · p is the l p norm ( p ≥ 1) of matrices

1/ p n 1 p −r . For r = 1, |ai j | ). Here [S ] = (if A = [ai j ], then A p = [xi , x j ]r i, j=1 √ 15 p = 2, the right side of (63) gives π , improving (62). 6 The spectral norm of a matrix A = [ai j ] is defined by √

A ∗ = max{ λ : λ is an eigenvalue of A∗ A}. 274

THE MANY FACETS OF EULER’S TOTIENT

Let 2r > 1. Then (see [7])

[S −r ] ∗ ≤

ζ (2r )3/2 ζ (4r )1/2

(64)

For inequalities on GCUD-reciprocal LCUM determinants, see the recent paper by A. Nalli [311].

3.7 1

Generalizations and extensions of Euler’s totient Jordan, Jordan-Nagell, von Sterneck, Cohen-totients

There exist many generalizations, extensions, or analogues functions to Euler’s totient. In Chapters 1 and 2, as well as in the former paragraphs of this Chapter 3, there have been included many notions or results related to totients. Perhaps the most important generalization of Euler’s totient is the Jordan totient function Jk with Jk (n) defined as the number of ordered sets of k elements from a complete residue system (mod n) such that the g.c.d. of each set is prime to n. In fact Jk = µ ∗ E k

(see 2.2.1), so 1 1− k Jk (n) = n k p p|n

(1)

For Jordan’s totient, see 2.2.1, 2.2.7 of Chapter 2, and 3.1.1, 3.1.7, 3.2.11, 3.3.5, 3.5.3, 3.6.1 of Chapter 3. An extension of Jordan’s totient, namely the Jordan-Nagell totient occurred at 2.3.3 of Chapter 2. Two generalized totients strongly related to Jordan’s are the von Sterneck and the Cohen totients. In 1894 R. D. von Sterneck (see [103], p. 151) defined Hk (n) = ϕ(d1 )ϕ(d2 ) . . . ϕ(dk ) (2) n=[d1 ,d2 ,...,dk ]

where the summation is over all sets of k positive integers d1 , . . . , dk with their lcm equal to n. He proved that (3) Hk (n) = Jk (n) In 1952 E. Cohen [78] introduced the notion of a k-reduced residue system (mod n). Let a, b be integers, not both zero. (a, b)k denotes the g.c.d. of a and b which is also a kth power. If (a, b)k = 1, a is said to be relatively k-prime to b 275

CHAPTER 3

and vice-versa. The set of all integers x from a complete residue system (mod n) such that (x, n k )k = 1 is called a k-reduced residue system (mod n). The number of elements of a k-reduced residue system (mod n) is denoted by ϕk (n). Clearly, ϕ1 ≡ ϕ. More generally, Cohen proves: ϕk (n) = Jk (n)

(4)

For many other identities involving Jk , see [103], Chapter 4; and R. Sivaramakrishnan [421]. For new proofs of (3) and (4), see [79], [80]. P. Haukkanen [184] further generalized ϕk (n) and Jk (n) by introducing the function ϕk,u (n) = number of u-tuples x1 , x2 , . . . , xu (mod n k ) such that (gcd(x1 , . . . , xu ), n k )k = 1. One has ϕ1,u (n) = Ju (n); ϕk,1 (n) = ϕk (n). In fact (see [184]) ϕk,u (n) = d ku µ(n/d), (5) d|n

where µ is the classical M¨obius function. For a generalization of ϕk,u to the setting of Narkiewicz’s regular convolution, see [185]. Another generalization is the JordanNagell totient of 2.3.3. As we have mentioned in section 3.1 (see relation (42’)), in 1999 T. Shonhiwa has considered the generalized totient Jkm (n) = car d{1 ≤ a1 , . . . , an ≤ n, (a1 , . . . , ak , m) = 1}, which for m = n gives Jk (n). Then Jkm (n) =

µ(d)

d|m

n k d

(1 )

which for m = n yields relation (1).

2

Schemmel, Schemmel-Nagell, Lucas-totients

In 1869 V. Schemmel (see [103], p. 147) introduced the function Sk (n), as the number of sets of k consecutive numbers each less than n, and relatively prime to n. It is known that k (6) Sk (n) = n 1− p p|n A generalization of Schemmel’s totient is the Schemmel-Nagell totient θs (M, N ) of 2.3.3 from Chapter 2. For Schemmel’s totient, see also 3.1.4, 3.2.8, 3.3.3 of Chapter 3. 276

THE MANY FACETS OF EULER’S TOTIENT

In 1891 E. Lucas (see [103], p. 147) obtained a generalization of Schemmel’s totient. Let e1 , e2 , . . . , ek be any integers, and denote by ψk (n) the number of integers h, chosen from 0, 1, 2, . . . , n − 1 such that h − e1 , h − e2 , . . . , h − ek are relatively prime to n. Then ψk is multiplicative, and λ (7) 1− ψk (n) = n p p|n where λ is the number of distinct residues of e1 , . . . , ek (mod p). For k < n, e1 = 0, e2 = −1, . . . , ek = −(k − 1) we reobtain Schemmel’s totient. Schemmel’s and Lucas’ totient functions have been further generalized by K. Nageswara Rao [309], in the setting of Cohen’s relatively k-primality (see (4)). Let Sk(s) (n) be the number of sets of k consecutive integers each less than n s and which are s-prime to n s . Then k (s) s 1− s (8) Sk (n) = n p p|n Remark that Sk(1) ≡ Sk . A similar generalization to (7) holds true.

3

Ramanujan’s sum

An important generalization of Euler’s totient is the Ramanujan sum C(n, r ) defined by C(n, r ) = e(nx, r ) (9) (x,r )=1,1≤x≤r

where n ∈ Z, r ∈ N and e(a, b) = e2πia/b . In fact, Ramanujan’s sum is a common generalization of Euler’s totient ϕ and M¨obius’ function µ, since C(0, r ) = ϕ(r ),

C(1, r ) = µ(r )

(10)

Ramanujan’s sum, with certain generalizations or analogues sums occured in 2.3.6 of Chapter 2, as well as in 3.1.5, 3.5.3, 3.6.1, 3.6.2 of Chapter 3. Many other generalizations of Ramanujan’s sum, due to E. Cohen, K. Nageswara Rao, M. Sugunamma, M. V. Subbarao - V. C. Harris, A. C. Vasu, L. Berardi - M. Cerasoli - F. Eugeni, etc. are included (with References) in [184]. R. Sivaramakrishnan, J. Hanumanthachari, or J. Hanumanthachari - V. V. Subrahmanya Sastri obtained Ramanujan sums for square reduced or k-free integers. V. Sita Ramaiah, P. J. McCarthy, P. Haukkanen have generalized Ramanujan’s sum in the setting of regular A-convolutions. We note that the unitary analogue of C(n, r ) originates to E. Cohen. See also [69]. For unitary analogues of generalized Ramanujan sums, see also K. R. Johnson [213]. For Ramanujan sums on semigroups, see A. Grytczuk [159]. 277

CHAPTER 3

For generalizations of Ramanujan’s sum to matrices, see V. C. Nanda [312]. He studied also general arithmetic functions of integer matrices [313]. See also G. Bhowmik [43] and G. Bhowmik and O. Ramar´e [44] on average orders of certain arithmetic functions of integer matrices (among others, the Euler totient of matrices). There is a vast literature on the Ramanujan expansions of arithmetic functions, i.e. representations in the form ar C(n, r ). For a survey of results, see e.g. W. r

Schwarz [405]. See also [159].

4

Klee’s totient

In 1948 V. L. Klee defined the function Tk (n) as the number of integers 1 ≤ h ≤ n such that (h, n) is kth powerfree. Then n Tk (n) = µk (d) (11) d d|n where µk is Klee’s M¨obius function, given by µk (n) = µ(n 1/k ) if n is a kth power; 0, otherwise. See 2.3.5 of Chapter 2, where Suryanarayana’s generalization is included, too. It can be proved that (see e.g. [310], along with other identities) 1 Tk (n) = n 1− k (12) p k p |n U. V. Satyanarayana and K. Pattabhiramasastry [390] proved that the average order of Tk (n) is n2 + O(n log n) (13) Tk (m) = 2ζ (2k) m≤n A. C. Vasu [467] defined an extension Tk,l of Klee’s totient as the number of ordered sets of l integers less than n which have with n the greatest common divisor kth power free. See also Subbarao-Harris’ ϕk,q of 2.3.6 of Chapter 2. n t A generalization Tk,s,t given by Tk,s,t = (µk (d))s , with application in d d|n the evaluation of certain infinite sums, can be found in B. C. Berndt [39]. For generalizations of Klee’s totient in the setting of A-convolutions, with many particular cases, see [184].

5

Nagell’s, Adler’s, Stevens’, Kesava Menon’s totients

Nagell’s totient function θ (n, r ) was introduced in 1923 by T. Nagell [307] (see also [81]) as the number of integers x (mod r ) such that (x, r ) = (n − x, r ) = 1. In 278

THE MANY FACETS OF EULER’S TOTIENT

fact, θ (n, r ) can be considered as the number of solutions of the congruence x+y≡n

(mod r )

(14)

under the restrictions (x, r ) = (y, r ) = 1. E. Cohen [81] proved that θ (n, r ) = ϕ(r )

µ(d) ϕ(d) d|r,(d,n)=1

θ (n, r ) is multiplicative in both n and r , and another identity is: (−1)ω(d) ϕ(r/d); θ (n, r ) =

(15)

(16)

d|r,(d,n)=1

see [421]. As a generalization of (14), let Ns (n, r ) be the number of solutions of the congruence x1 + x2 + · · · + xs ≡ n (mod r ) (17) under the restriction (x, r ) = 1, where x = (x1 , . . . , xs ). Then (see E. Cohen [82]), s r Js (g), with g = (n, r ) (18) Ns (n, r ) = g In 1948 H. L. Alder [4] defined a totient ϕ(n, r ) as the number of ordered pairs x, y such that x + y = n + r,

(x, r ) = (y, r ) = 1

Clearly ϕ(0, r ) = ϕ(r ). Now, ϕ(n, r ) = r

εn ( p) =

1 ≤ x ≤ r (n ≥ 0)

εn ( p) , 1− p

p|n

where

and

(19)

1, if p|n 2, otherwise

Remark that ϕ(1, r ) = S2 (r ), the Schemmel totient of order 2. Also, by (14), ϕ(n, r ) = θ (n, r ) for all n, r

(20)

For a generalization of Nagell’s totient function and Ramanujan’s sum, see [177]. For a recent generalization, with combinatorial applications, see [446]. An extension of Nagell’s (and Alder’s) totient has been introduced by H. Stevens [440]. Let F = { f 1 (x), . . . , f k (x)} (k ≥ 1) be a set of polynomials with integral 279

CHAPTER 3

coefficients, and let A be the set of all ordered k-tuples of integers a1 , . . . , ak such that 0 ≤ a1 , . . . , ak ≤ n. Let ϕ F,k (n) be the number of elements in A such that r a gcd( f 1 (a1 ), . . . , f k (ak ), n) = 1. If n = p j j is the prime factorization of n, then j=1

ϕ F,k (n) = n

k

r j=1

N1 j . . . Nk j 1− p kj

(21)

where Ni j is the number of incongruent solutions of f i (x) ≡ 0 (mod p j ). The Stevens totient is multiplicative, and contains, as special cases the Jordan totient Jk (n) ( f 1 (x) = · · · = f k (x) = x); the Schemmel totient St (n) (k = 1, f 1 (x) = x(x + 1) . . . (x + t − 1)); the Nagell totient θ (n, r ) = ϕ F,1 (r ) (k = 1, f 1 (x) = x(n − x)). Let now ϕl,k (n) with 0 ≤ l < k be the numbers of k-tuples a1 , . . . , ak with l ≤ ak ≤ n such that (al+1 , . . . , ak , n) = 1. This is the Cashwell-Everett totient (see B. Rizzi [353]. By letting f 1 (x) = · · · = fl (x) = nx, fl+1 (x) = · · · = f k (x) = x, in (21), one gets ϕl,k (n) = n

k

p|n

pl 1− k p

(22)

Clearly, ϕ0,k (n) = Jk (n), Jordan’s totient. For generalized Ramanujan sums connected with Stevens’ totient, see also L. Berardi and B. Rizzi [37]. For an asymptotic formula for ϕ F,k , see L. T´oth and J. S´andor [460]. In the case of a single polynomial f , with integral coefficients, in 1967 P. Kesava Menon [222] considered the number of integers ϕ f (n) of x (mod n), such that ( f (x), n) = 1. Let D(s, d, n) = {s, s + d, . . . , s + (n − 1)d} be an arithmetic progression with (s, d) = 1, and let ϕ(s, d, n) denote the number of elements x ∈ D(s, d, n) such that (x, n) = 1. This function has been introduced by P. G. Garcia and S. Ligh [145]. This is a special case of ϕ f (n) for f (x) = s + (x − 1)d. L. T´oth [457] obtained the asymptotic formula

ϕ(s, d, n) =

n≤x

An asymptotic formula for

3d 2 x 2 + O(x log x) π 2 J (d)

ϕ(s, d, n) was given by T. Maxsein [287].

d≤x

280

(23)

THE MANY FACETS OF EULER’S TOTIENT

A combination of generalizations from [184] and Kesava Menon’s totient ϕ f is given in [458], where asymptotic results more general than (23) are also deduced. D. Goldsmith [149] considered a generalization of ϕ f (n) as follows. For each divisor d of n, let ϕ f,d (n) be the number of 0 ≤ x < n such that ( f (x), n) = d. Clearly, ϕ f,1 (n) = ϕ f (n). Let (n, r ) = 1 and d|n, e|r . Then ϕ f,ed (nr ) = ϕ f,d (n)ϕ f,e (r ) For d = e = 1, this contains the multiplicative property of ϕ f .

6

Unitary, semi-unitary, bi-unitary totients

The unitary analogue of ϕ(n) was introduced by E. Cohen [83] as follows. Let (a, b)∗ denote the greatest divisor of a which is a unitary divisor of b (a divisor r of b = 1). If (a, b)∗ = 1, then a is said to be semi-prime to b is called unitary, if r, r b. Let ϕ ∗ (n) be the number of positive integers r ≤ n, semi-prime to n. In fact, dµ∗ (n/d) = ( p α − 1), (24) ϕ ∗ (n) = p α n

d|n

where µ∗ (m) = (−1)ω(n) is the unitary M¨obius function. For unitary divisors see also 1.9 of Chapter 1, 2.2.2 of Chapter 2, and 3.6.1 of Chapter 3. For the corresponding notions of bi-unitary divisors and convolution, see 1.9 of Chapter 1, and 2.2.3 of Chapter 2. A divisor d of n is called semi-unitary divisor (see J. Chidambaraswamy [73]) if d is semi-prime to n/d. For various sum evaluations on semi-unitary divisors, see [73]. The unitary analogue of Nagell’s totient has been introduced by J. Morgado [299]: θ ∗ (n, r ) is the number of integers x such that (x, r )∗ = (n − x, r )∗ = 1, 1 ≤ x ≤ r . θ ∗ is multiplicative in r , and is given by α αj ( pi i − 1) ( p j − 2) (25) θ ∗ (n, r ) = α

αj

pi i |n

p j n

where r = p1α1 . . . pmαm is the prime factorization of r . m 1 ∗ ∗ 1 − αi . If r |n, then θ (n, r ) = ϕ (r ) = r pi i=1 For identities involving θ ∗ , see [300]. The unitary analogue of Schemmel’s totient is the number Sk∗ (n) of k consecutive numbers x, x + 1, . . . , x + k − 1 with 1 ≤ x ≤ n and all semi-prime to n. This has been introduced by J. Morgado in [301]. 281

CHAPTER 3

Let n = p1α1 . . . pmαm , and p α = min{ p1α1 , . . . , pmαm }. If k ≥ p α , then Sk∗ (n) = 0. The function Sk∗ is multiplicative, and Sk∗ ( piαi ) = piαi − k, so  if n = 1  1, ∗ 0, if n > 1 and k ≥ p α Sk (n) =  α1 αm ( p1 − k) . . . ( pm − k), if n > 1 and 1 ≤ k < p α

(26)

For the unitary analogues of certain remarkable identities (e.g. of BrauerRademacher’s, Landau’s, or Anderson-Apostol’s), see P. J. McCarthy [72].

7

Alladi’s totient

The k-ary divisors have been considered e.g. in 2.3.5 of Chapter 2 and in relation with (11). Another notion of higher order divisors is due to K. Alladi [5]. A divisor d of n is called of first order, in notation d|1 n. Let (a, b)1 denote the largest divisor of a dividing b. When (a, b)1 = 1, we say that a is prime to b order 1. A divisor d of n n is said to be of second order, denote d|2 n, if , d = 1. Let (a, b)2 denote the d 1 prime to b order 2. A largest divisor c of a satisfying c|2 b. If (a, b)2 = 1 we say na is divisor d of n is a divisor of third order, denoted d|3 n, if , d = 1. Let (a, b)3 be d 2 b)3 = 1, we say a is prime to b order the largest divisor c of a that satisfies c|3 b. If (a, n 3. By generalizing, we say that d|r n if = 1 and (a, b)r = max{c|1 a, c|r b}. ,d d r −1 If (a, b)r = 1, then a is prime to b order r . Let ϕr (n, x) = number of 1 ≤ a ≤ x such that (a, n)r = 1 be Alladi’s totient, and put ϕr (n) = ϕr (n, n). Similarly, let ϕr (n, x) = number of 1 ≤ a ≤ x with Fr −1 x, when r (n, a)r = 1. Let ϕr (n) = ϕr (n, n). Let fr (x) be the least integer > Fr Fr −1 is odd, and ≥ x, when r is even, where (Fr ) is the Fibonacci sequence given Fr by F0 = 0, F1 = 1, Fn = Fn−1 + Fn−2 (n ≥ 2). Let n = p1α1 . . . pmαm be the prime factorization of n. Then ϕr (n) = n

m

1−

i=1

1

f (αi )

pi r

(27)

An interesting consequence is that ϕ1 (n) ≤ ϕ3 (n) ≤ ϕ5 (n) ≤ · · · ≤ ϕ6 (n) ≤ ϕ4 (n) ≤ ϕ2 (n) 282

(28)

THE MANY FACETS OF EULER’S TOTIENT

Let σr,k (n) =

d k , σr,k (n) =

d|r n

n k d|r n

n

d

. Then

σr,k (m) = cr,k n k+1 + O(n k+1/2 )

m=1 n

(29) σr,k (n) = cr,k n k+1 + O(n k+1/2 )

m=1

where cr,k and

8

cr,k

are certain constants depending only on r and k.

Legendre’s totient

A famous generalization of Euler’s totient is the Legendre totient function ϕ(x, n) = number of positive integers ≤ x, which are relatively prime to n, introduced by Legendre in 1794 (see [103], p. 115). Clearly, ϕ(n, n) = n. Legendre proved essentially that x (30) ϕ(x, n) = µ(d) d d|n Relation (30) immediately implies ϕ(x, n) = x

ϕ(n) + O(d(n)), n

(31)

where d(n) is the number of divisors of n. This gives rise to the function (x, n) = ϕ(x, n) −

xϕ(n) . n

In 1946 P. Erd¨os [120] proved that |(x, n)| > c · 2ω(n)/2 /(log ω(n))1/2 ,

(32)

while D. H. Lehmer [249] showed that |(x, n)| < 2ω(n)−1 ,

(33)

where ω(n) denotes the number of distinct prime factors of n. It can be conjectured that (see [120]) (34) |(x, n)| = O(2ω(n) ), if ω(n) → ∞ A related result is the following (see [251]): ω(n) µ(d) ≤ d|n,a≤d≤b [ω(n)/2] 283

(35)

CHAPTER 3

In 1974 D. Suryanarayana [246] obtained the following two inequalities: (x, n) + {x} − 1 ϕ(n) ≤ 1 (d(n) − ψ(n)/n); x ≥ 1, n ≥ 2, 2 n 2

(36)

ϕ(n) d(n) ψ(n) ψ(n) d(n) − ≤ (x, n) + {x} ≤ − + 1, for x ≥ 1, n ≥ 2, (37) n 2 n 2 n such that ([x], n) = 1. Here ψ(n) is the Dedekind arithmetical function (see [103], p. 123) given by 1 ( p prime). ψ(n) = n 1+ p p|n For Dedekind’s function, see e.g. E. Cohen [84], D. Suryanarayana [449], J. S´andor [369] and J. S´andor and R. Sivaramakrishnan [383]. A. Sivaramasarma [425] stated that if x ≥ 1, n ≥ 2, and m = (n, [x]), then (x, n) + {x} ϕ(n) − 1 1 ≤ d(n) + d(m) − md(m)ψ(n) , (38) n 2 m 2 2 nψ(m) where {x} = x − [x] is the fractional part of x. The unitary analogue of Legendre’s totient has been introduced by Cohen in [83] as the number ϕ ∗ (x, n) of positive integers a ≤ x such that (a, n)∗ = 1. In particular, ϕ ∗ (n, n) = ϕ ∗ (n). A more general function, considered by J. Chidambaraswamy [73] is ϕr∗ (x, n) = a r (r ≥ 0) a≤x,(a,n)∗ =1

He proved that ϕr∗ (x, n) =

x r +1 ϕ ∗ (n) · + O(x r d ∗ (n)), r +1 n

(39)

for any r ≥ 0, x ≥ 1, n ≥ 1; where d ∗ (n) denotes the number of unitary divisors of n. P. Haukkanen [186] obtained the unitary analogue of (38): ∗ ∗ ∗ (x, n) + {x} ϕ (n) − 1 1 ≤ d(n) + d(m) − md(m)ψ (n) (40) n 2 m 2 2 nψ ∗ (m) 1 ∗ ∗ ∗ where m = ([x], n) , ψ (n) = σ (n) = n 1 + a is the unitary analogue p pa n of Dedekind’s function (in fact, the sum of unitary divisors of n), and ∗ (x, n) = ϕ ∗ (x, n) − xϕ ∗ (n)/n. 284

THE MANY FACETS OF EULER’S TOTIENT

In fact, Haukkanen works in the context of A-convolutions, and (38), (40) are particular cases. Let S ⊂ N be a non-void set of positive integers. The Legendre totient of u arguments associated to S was introduced by E. Cohen [191] as ϕ S (x1 , . . . , xu ; n) = number of u-tuples a1 , . . . , au with a j (mod x j ) ( j = 1, u) such that ((a1 , . . . , au ), n) ∈ S. When S = {1}, and u = 1, this reduces to the classical Legendre function. Then x x 1 u ... , (41) ϕ S (x1 , . . . , xu ; n) = µ S (d) d d d|n where µ S is the M¨obius function of the set S (see 2.2.4 of Chapter 2, where the more general µ S,A is introduced, too). For certain arithmetical products, with asymptotic expansions, which involve also ϕ S , see J. S´andor and L. T´oth [386]. In [184] this function is generalized to Ak -convolutions, so (41) becomes µ S,Ak (d)[x1 /d k ] . . . [xu /d k ] (42) ϕ S,A,k (x1 , . . . , xu ; n) = d∈Ak (n)

For an application of Legendre’s totient function ϕ(x, n) in the study of primes in a real quadratic field, see A. Granville, R. A. Mollin and H. C. Williams [157], where it is proved e.g that if x ≥ 4.7 · 1015 , then 1 π(x) ≤ ϕ(x, n), 3 where n is the product of all primes ≤ 14 log x, and π(x) is the number of primes ≤ x.

9

Euler totients of meet semilattices and finite fields

Relation (35) of 3.6 gives a generalization of Euler’s totient to a finite meet semilattice, and a function f : P → C. In [191] P. Haukkanen and J. Wang considered set-valued maps f : P → A, where P is a meet-semilattice, S ⊂ P is a fixed set, and A is a collection of finite subsets of S. The mapping f is called a regular S-representation of P if f (x ∧ y) = f (x) ∩ f (y) for all x, y ∈ P. The Euler totient ϕ f (x) of P with respect to a regular S-representation f is defined as the number of elements z ∈ f (x) such that z ∈ f (y) for y < x (we assume that P is a meet-semilattice and that every principal order ideal of P is finite). Then ϕ f (x) = µ(y, x)| f (y)| (43) y≤x

285

CHAPTER 3

Examples. 1) When P is N, ordered by divisibility, and f (n) = Un , the set of all nth roots of unity, then with U = Un , f will be a regular U -representation of N. n≥1

Then ϕ f (n) = ϕ(n), the classical Euler totient. 2) Let f k (n) = Un k , U = Un . Then ϕ fk ≡ ϕk = Cohen’s totient function. n≥1

3) For gk (n) = (Un )k one obtains ϕgk = Jk = Jordan’s totient function. Let Fq denote a finite field of q elements, and let V be a vector space over Fq . Let A be the set of all finite-dimensional subspaces of V . Let f q : P → A be a regular V -representation of P, i.e. let f q (x ∧ y) = f q (x) ∩ f q (y), ∀ x, y ∈ P

(44)

The q-Euler function ϕ fq (x) is the number of 1-dimensional subspaces of f q (x) that are not contained in f q (y) for y < x. Then ϕ fq (x) =

µ(y, x)

y≤x

q dim fq (y) − 1 q −1

(45)

E.g., for example 1) above, when f q (n) = Fq (Un ), we get the q-analogue of the classical Euler totient: ϕq (n) = ϕ fq (n) =

µ(n/d)

d|n

qd − 1 q −1

(46)

The q-analogue of Jordan’s totient Jk (see example 3) above) will be ϕ J,q (n) = ϕgq,k (n) =

d|n

k

qd − 1 µ(n/d) q −1

(47)

Let f be a regular representation of P, and α : P → G, where (G, +) is an Abelian group. The Ramanujan sum of P with respect to f and α is defined by α(z), (48) C f,α (x) = z∈S f (x)

where S f (x) = {z ∈ f (x) : z ∈ f (y) for y < x}. One has C f,α (x) = µ(y, x) α(z),

(49)

z∈ f (y)

y≤x

and the q-analogue of the classical Ramanujan sum can be introduced (see [191]). 286

THE MANY FACETS OF EULER’S TOTIENT

10

Nonunitary, infinitary, exponential-totients

We now consider totient functions suggested by certain notions of Chapters 1, 2. In 1.9 of Chaper 1, as well as 2.2.6, 2.2.10 of Chapter 2 there are considered the nonunitary, infinitary, exponential divisors, and convolution products. See also 1.13 and 1.14. The nonunitary totient function ϕ # (n) is defined as the number of positive integers ≤ n which are not divisible by any of the nonunitary divisors of n. If n = n ·n # , where n is the squarefree, while n # the powerful part, of n, then ϕ # (n) = n · ϕ(n # ) One has also

(50)

ϕ # (d) = σ ∗ (n),

(51)

d|n

where σ ∗ (n) is the sum of unitary divisors of n (see [257]). Similarly,     ϕ # (d) = σ (n) σ ∗ (n # ) − ( p e − p e−1 + 1)   # e #

(52)

p n

d| n

It is conjectured that if

ϕ # (d) = n, then n # is even.

d|n

In part 2.2.10 of Chapter 2 there are introduced the infinitary divisors and M¨obius’ infinitary function. If n = p1 p2 . . . pt , where p1 < p2 < · · · < pt are the I components of n, then one defines the infinitary totient function by ϕ∞ (1) = 1,

ϕ∞ (n) = n

t j=1

1 1− pj

(n > 1)

(53)

The function ϕ∞ is I -multiplicative, i.e. (m, n)∞ = 1 (i.e. greatest common infinitary divisor is =1), then ϕ∞ (mn) = ϕ∞ (m)ϕ∞ (n). Similarly, ϕ∞ (d) = n (54) d|∞ n

For these results and what follow, see [87]. Let also ϕ∞ (x, n) be the infinitary Legendre totient defined by ϕ∞ (x, n) = number of a ≤ x such that (a, n)∞ = 1. One has the following asymptotic results: ϕ∞ (x, n) = k∞ (n)x + O(n ε x ε ) 287

(55)

CHAPTER 3

for any x > 0, n ≥ 1, (ε > 0), where the multiplicative constant implied by the big of notation depends only on ε, and k∞ is an arithmetic function defined by k∞ (1) = 1, t pj . Similarly, k∞ (n) = p +1 j=1 j

ϕ∞ (n) =

n≤x

x2 · A + O(x 1+ε log x) 2

for any x > 1, and any ε > 0. Here A = one has:

(56)

∞ 2 µ∞ (n)k∞ (n) . For the function k∞ (n) 2 n n=1

k∞ (n) = C x + O(x ε log x)

(57)

n≤x

1 1− . for all x > 1, where C = (P + 1)2 P∈I For the exponential divisors and convolutions, see 1.10 of Chapter 1, and 2.2.6 of Chapter 2. The exponential totient function is introduced in [270] of Chapter 2 (classical case), and in [486] of Chapter 2 (regular convolutions). Let ϕe (n) denote the number of all a ≤ n with (a, n)e = 1, and put ϕe (1) = 1. If n = p1a1 . . . prar is the prime factorization of n > 1, then ϕe (n) = ϕ(a1 ) . . . ϕ(ar ) ϕe is a multiplicative function, and (see [380]) ϕe (d) = a1 . . . ar

(58)

(59)

d|e n

Other properties of ϕe are: If n|e m, then ϕe (n)|ϕe (m); ϕe (n)de (n) ≥ a1 . . . ar ; if all ai (i = 1, r ) are odd, then ϕe (n)de (n) ≥ σ (a1 ) . . . σ (ar ), where de (n) is the sum of exponential divisors of n; lim sup(log ϕe (n)) n→∞

log log n log 4 = log n 5

(60)

Finally, note that the Golomb-Guerin totient function ϕ∇ is introduced in 2.2.8 of Chapter 2 (see [165]). For generalized Euler functions involving rational arithmetic functions, and the extensions of the Fleck-Popoviciu-Buschman-Hsu-Wang M¨obius function, see 2.2.1 and 2.2.4 of Chapter 2. For a Lucas-sequence based generalization of Euler’s totient, see 3.2.2 of Chapter 3. 288

THE MANY FACETS OF EULER’S TOTIENT

11

Thacker’s, Leudesdorf’s, Lehmer’s, Golubev’s totients. Square totient, core-reduced totient, M-void totient, additive totient

In 3.4.1 we have introduced Thacker’s totient function, as well as the generalized totient due to Leudesdorf. Lehmer’s totient function is considered in 3.4.2. In 3.2.5 there is a class F1 of generalized totient functions due to Siva Rama Prasad and Fonseca. Other functions are the Golubev totient functions. In 1953 V. A. Golubev [151], by studying twin primes and other special types of primes, defined ϕ2 (n) = number of pairs (a1 , a2 ) of positive integer such that a2 − a1 = 2, (a1 , n) = 1, a1 ≤ n. Then ϕ2 is multiplicative, and  2   n , if n is odd 1−   p   p|n (61) ϕ2 (n) =   n 2   , if n is even 1−   2 p p|n If ϕ2k (n) = number of pairs (a1 , a2 ) with a2 − a1 = 2k, (a1 , n) = 1, a1 ≤ n, then ϕ2k is multiplicative, and 1 2 (62) 1− 1− ϕ2k (n) = n q q|n,q 2k q q|n,q|2k More generally, let ϕ (m) (n; r1 , . . . , rm−1 ) = number of sets of positive integers a1 , . . . , am each relatively prime to n, with given differences ak − ak−1 = rk−1 (1 ≤ k ≤ m − 1), all of the r ’s being even, and a1 ≤ n. Then 1 1 ϕ (m) (n; r1 , . . . , rm−1 ) = n 1− ... 1 − (63) p1 pm p1 ,..., pm in which pk denotes the prime divisors of n such that the set {0, r1 , r1 + r2 , . . . , r1 + r2 +· · ·+rm−1 } contains at most j ( pk ; r1 , . . . rm−1 ) incongruent elements (mod pk ). (64) See also R. G. Buschman [58]. The square-totient of R. Sivaramakrishnan [422] is introduced as follows. Let n ≥ 1, and consider the set of integers a (mod n) such that (a, n) = a square ≥ 1, and contained in a complete residue system (mod n) is called a ”square-reduced residue system (mod n)”. The number of elements in a square-reduced residue system (mod n) is denoted by b(n). 289

CHAPTER 3

Let

ε(n) =

1, if n is a perfect square 0, otherwise

Then b(n) =

ε(d)ϕ

d|n

n

(65)

d

The function b is multiplicative, and satisfies certain identities, see e.g. [421]. Let γ (n) denote the product of distinct prime divisors of n > 1 (i.e. ”core of n”), and put γ (1) = 1. Let S denote a complete residue system (mod n), and let M be the set of integers c (mod n) such that (c, n) = γ (n), and contained in S. This subset M will be called as the ”core-reduced residue system (mod n)”. The core-reduced totient of J. S´andor and R. Sivaramakrishnan [383] is the cardinality m(n) of M. The function m(n) is multiplicative, and m(n) = ϕ(s),

where s =

n γ (n)

(66)

Let c ≡ 0 (mod n). If c (mod n) is such that (c, n) = γ (n), then ck ≡ 0 (mod n) for some integer k ≥ 1. Therefore, the elements of a core-reduced residue system (mod n) are nonzero nilpotent elements in the ring Zr . In fact, m(n) does not count all the nilpotent elements in Zr , but the number of nilpotents elements in n Zn is s − 1, where s = (see [383]). γ (n) Let M be a set of positive integers with min M = s ≥ 2. A positive integer r piai is said to be M-void, if ai ∈ M, i = 1, r . Let Q M denote the set of all n= i=1

M-void integers, and denote by q M its characteristic function. Let λ M be defined by q M (n) = λ M (d). It is clear that q M and λ M are multiplicative functions, and for d|n

all prime powers pa ,   −1, if a ∈ M a − 1 ∈ M a 1, if a ∈ M a − 1 ∈ M λM ( p ) =  0, otherwise

(67)

M. V. Subbarao and R. Sitaramachandrarao [444] define the M-void analogue of the Euler totient function by the number ϕ M (n) of integers x ≤ n such that (x, n) ∈ Q M . (This is a particular case of the Cohen totient ϕ S with S = Q M ). We have n ϕ M (n) = λ M (d) (68) d d|n 290

THE MANY FACETS OF EULER’S TOTIENT

and

ϕM ( p ) = a

if 1 ≤ a ≤ s − 1 pa a s a−s a p + λM ( p ) p + · · · + λ M ( p ) if a ≥ s

Let A(x, ϕ M ) be the number of positive integers n such that 1 ≤ ϕ M (n) ≤ x. Then, for each c > 0, as x → ∞, A(x, ϕ M ) = A(ϕ M )x + O(x(δ(x))c ) where A(ϕ M ) =

1− p

∞ + (1 − p ) (ϕ M ( pa ))−1

−s

−1

a=s

p prime

and

(69)

3

δ(x) = exp{−(log x) 8 −ε }.

For particular sets M one can reobtain the sets of r -free integers, (k, r )-integers, unitary r -free integers, semi-r -free integers, etc. By counting rational points on a projective space or on quadratic twists of an elliptic curve, A. Sato [389] introduced the following generalizations of Euler’s totient. Let q > 0 be a given integer, and put ϕq (x) = number of 0 < y ≤ x such that (x, qy) = 1; ψq (y) = number of 0 < x ≤ y such that (x, qy) = 1. Clearly, ϕ1 (x) = ϕ(x), ϕ1 (y) = ϕ(y). The following asymptotic formulae are due to Shigeki Akiyama (see [389]): ϕq (x) = Cq B 2 + O(B log B) x≤B

ψq (y) = Cq B 2 + O(B log B), as B → ∞

(70)

y≤B r 3 pi , with ( pi ) the distinct prime divisors of q. 2 π i=1 pi+1 Finally, we note that the additive analogue of Euler’s totient has been defined by r piai (prime factorization) and n > 1, let K. Atanassov [18] as follows: For n =

where Cq =

i=1

ρ(n) =

r

piai −1 ( pi − 1)

(71)

i=1

Put ρ(1) = 0. Then clearly ρ is an additive function, i.e. ρ(nm) = ρ(n) + ρ(m) for all (m, n) = 1. The function ρ is connected also to other functions, e.g. to the 291

CHAPTER 3

additive analogue of the σ (sum-of-divisors) function, defined by θ (n) =

r piai +1 − 1 , pi − 1 i=1

θ(1) = 0.

(72)

One has ρ(n) ≤ ϕ(n);

θ (n) ≤ n − 1 for all n ≥ 1

(73)

and ρ(n) ≥ r (ϕ(n))1/r ;

12

θ (n) ≥ r (σ (n))1/r for n > 1.

(74)

Euler totients of arithmetical semigroups, finite groups, algebraic number fields, semigroups, finite commutative rings, finite Dedekind domains

Generalized integers and arithmetical semigroups have been considered in 2.2.11 and 2.5.1 of Chapter 2. If G is an arithmetical semigroup and H an arithmetical subsemigroup of G, for xi ∈ G (1 ≤ i ≤ k) one can define the H -greatest common divisor of the xi ’s by (x1 , . . . , xk ) H = h if h is an element from H with the largest possible norm that divides each xi (1 ≤ i ≤ k). Let ϕk (H, x) denote the number of k-tuples x1 , . . . , xk of elements of G such that |xi | ≤ x (i = 1, k) and (x1 , . . . , xk ) H = 1. H. Wegman [479] called an arithmetical semigroup δ-regular, if its counting function G(x) satisfies G(x) = x δ L(x), where L(x) is defined for all x > 0, and L(ax) → 1 as x → ∞ for every a > 0. If G is δ-regular, then Wegman proved L(x) |n|−s converges for s > δ and diverges for s < δ. that its zeta functions ζG (s) = n∈G

t δ G(x/t) − 1, for x, t > 0. The following result is due to S. Porubsk´y Let r (x, t) = G(x) [339]. Let G be δ-regular such that r (x, t) is uniformly bounded for all x, t > 0. Let H be an arithmetical subsemigroup such that the series |h|−kδ = ζ H (kδ) h∈H

converges. Then

ϕk (H, x) =

1 + o(1) (G(x))k ζ H (kδ)

(75)

If G satisfies the axiom: there exist positive constants A and δ and a constant η with ≤ η < δ such that G(x) = Ax δ + O(x η ) as x → ∞ 292

(76)

THE MANY FACETS OF EULER’S TOTIENT

then for k ≥ 2, 1 x kδ + ϕk (H, x) = A ζ H (kδ)

k

O(x (k−1)δ+η ) if k > 2, or k = 2 and η > 0 O(x δ log x) if k = 2, η = 0

(77)

In 2.5.3 there were introduced M¨obius functions and Euler functions of finite groups, soluble groups, algebraic number fields, or algebraic function fields (in fact, having introduced a M¨obius function, the Euler functions follow at once). We note here that L. Weisner [477] computed the M¨obius and Euler functions of a finite pgroup as follows. Recall that the Frattini subgroup of a finite group G (denoted by Frat G) is the intersection of all maximal, proper subgroups of G. If G is a p-group, then Frat G = [G, G] · G p , by the Burnside basis theorem. Now, Weisner proved that the M¨obius function of a finite p-group G is given by (−1)d p d(d−1)/2 , where p d = [G : H ] if Frat G ≤ H, (78) µ(H ) = 0, if Frat G H Now set pr = |G| and p s = |G : Frat G|. The Euler function of G is then given by ϕn (G) = p (r −s)n

s−1 ( pn − pi )

(79)

i=0

For a recent account on the Euler functions of finite groups, see K. S. Brown [55]. √ Let K = Q( D) be an arbitrary (real or complex) quadratic field; let a denote an integral ideal in the ring R of algebraic integers in K (a = (0)) with norm N (a) (i.e. the number of residue classes of R modulo a). Let ϕ(a) be the number of those residue classes which are prime to a (see e.g. E. Hecke [196]). Then it is well known that ϕ(a) = N (a) (1 − N ( p)−1 ) ( p a prime ideal), and p|a

ϕ(a) = kx 2 + R0 (x),

(80)

ϕ(a) = 2kx + R1 (x), N (a) N (a)≤x

(81)

N (a)≤x

N (a)≤x

1 = αρ log x + λ + E 0 (x), ϕ(a)

N (a) = αρx + ρζk (0) log x + E 1 (x) ϕ(a) N (a)≤x

(82) (83)

where ρ is the residue of the Dedekind zeta-function ζk (s) at s = 1; k = ρ(2ζ K (2))−1 ; α = ζ K (2)ζ K (3)/ζ K (6); and λ is another constant. By generalizing 293

CHAPTER 3

results obtained in the classical case by Pillai and Chowla, W. G. Nowak [324] has obtained the following estimates:

R0 (n) =

n≤x

k 2 x + O(x 2 δ(x)), 2

R1 (x) = kx + O(xδ(x)),

(80 ) (81 )

n≤x

where δ(x) = exp{−c(log x)3/5 (log log x)−1/5 } and c > 0 is a constant depending on K . Similarly, by extending results on E 0 and E 1 from the classical case, Nowak has proved: 1 αρ − ζ K (0)ρ log x + O(1) E 0 (n) = (82 ) 2 n≤x E 1 (n) = ωx + O(x 13/14 ) (83 ) n≤x

where ω is a constant. We note that R0 (x) = O(x 4/3 ), R1 (x) = O(x 1/3 ), E 0 (x) = O(x −2/3 ), E 1 (x) = O(x 1/3 ) (84) but in the classical case the results are stronger (see [291], Chapter 1). For the Euler functions of semigroups, finite commutative rings, and finite Dedekind domains, see 3.2.2 of this Chapter.

294

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[463] E. Vantieghem, On a congruence only holding for primes, Indag. Math. N.S. 2(1991), no. 2, 253-255. [464] M. Vassilev, Three formulae for n-th prime and six for n-th term of twin primes, Notes Numb. Theory Discr. Math. 7(2001), no. 1, 15-20. [465] R. C. Vaughan, Some applications of Montgomery’s sieve, J. Number Theory 5(1973), 64-79. [466] R. C. Vaughan, Bounds for the coefficients of cyclotomic polynomials, Michigan Math. J. 21(1975), 289-295. [467] A. C. Vasu, A note on an extension of Klee’s ψ-functions, Math. Student 40(1972), 36-39. [468] C. S. Venkataraman, Math. Student 19(1951), 127-128. [469] T. Venkataraman, Perfect totient number, Math. Student 63(1975), 178. [470] A. Verjovsky, Discrete measures and the Riemann hypothesis, Kodai Math. J. 17(1994), no. 3, 596-608. [471] C. R. Wall, Analogs of Smith’s determinant, Fib. Quart. 25(1987), 343-345. [472] D. W. Wall, Conditions for φ(N ) to properly divide N − 1, in: A collection of manuscripts related to the Fibonacci sequence (Ed. V. E. Hogatt and M. V. E. Bicknell-Johnson), San Jose, CA, Fib. Association, pp. 205-208, 1980. [473] R. Wang, Congruence properties of Euler’s function ϕ(n) (Chinese), J. Math. Res. Exposition 22(2002), no. 3, 476-480. [474] H. Wang, Z. Hu and L. Gao, On the distribution of D. H. Lehmer number in the primitive roots modulo q (Chinese), Pure Appl. Math. 12(1996), no. 2, 84-88. [475] L. Weisner, Quadratic fields over which cyclotomic polynomials are reducible, Ann. Math. 29(1928), 377-381. [476] L. Weisner, Polynomials f (ϕ(x)) reducible in fields in which f (x) is reducible, Bull. Amer. Math. Soc. 34(1928). [477] L. Weisner, Some properties of prime-power groups, Trans. Amer. Math. Soc. 38(1935), 485-492. [478] E. Weisstein, http://mathworld.wolfram.com/CyclotomicPolinomial.html 326

THE MANY FACETS OF EULER’S TOTIENT

[479] H. Wegman, Beitr¨age zur Zahlentheorie auf freien Halbgruppen, I, J. Reine Angew. Math. 221(1966), 20-43. ´ Eyrolles, 1998, [480] D. Wells, Dictionnaire Penguin des Nombres Curieux, Ed. Paris (The French ed. of ”The Penguin Dictionary of Curios and Interesting Numbers”, Penguin Books, 1997). [481] J. Westlund, Proc. Indiana Acad. Sci. 1902, 78-79, see also [103]. [482] H. S. Wilf, Hadamard determinants, M¨obius functions, and the chromatic number of a graph, Bull. Amer. Math. Soc. 74(1968), 960-964. [483] J. Wolstenholme, On certain properties of prime numbers, Quart. J. Math. 5(1862), 35-39. [484] K. R. Wooldridge, Values taken many times by Euler’s phi-function, Proc. Amer. Math. Soc. 76(1979), 229-234. [485] M. Yorinaga, Numerical investigation of some equations involving Euler’s φfunction, Math. J. Okayama Univ. 20(1978), no. 1, 51-58. [486] M. Zhang, On a divisibility problem (Chinese) (English summary), J. Sichuan Univ., Nat. Sci. Ed. 32(1995), no. 3, 240-242. [487] M. Zhang, On the equation ϕ(x) = ϕ(y) (Chinese), J. Sichuan Univ., Nat. Sci. Ed. 32(1995), no. 6, 628-631. [488] M. Zhang, On a nontotients, J. Number Theory 43(1993), 168-172. [489] W. Zhang, On a problem of D. H. Lehmer and its generalization, Compos. Math. 86(1993), no. 3, 307-316. [490] W. Zhang, On a problem of D. H. Lehmer and its generalization (Chinese), J. Northwest Univ., Nat. Sci. 23(1993), no. 2, 103-108. [491] W. Zhang, On the distribution of inverses modulo n, J. Number Theory 61(1996), no. 2, 301-310. [492] W. Zhang, On the difference between an integer and its inverse modulo n, J. Number Theory 52(1995), no. 1, 1-6. [493] Z. Zheng, On generalized D. H. Lehmer’s problem, Adv. Math., Beijing 22(1993), no. 2, 164-165. [494] Z. Zheng, On generalized D. H. Lehmer’s problem, Chin. Sci. Bull. 38(1993), no. 16, 1340-1345.

327

Chapter 4

SPECIAL ARITHMETIC FUNCTIONS CONNECTED WITH THE DIVISORS, OR WITH THE DIGITS OF A NUMBER

4.1

Introduction

There are many particular arithmetic functions, numbers or sequences connected with some important notions or results which appear in Number theory, and in fact all Mathematics. Some of them are non standard functions (such as d, σ, ϕ, µ, ω, , p, P etc.), and were studied in [260] or in former chapters of this book. The aim of this chapter is the study of some other functions which at one part are not so well-known, and are scattered in various fields of study, or at another part have not been previously included. The former one includes arithmetic functions connected to the prime factorization of a number, or the consecutive divisors (prime or not). The functions related to the digits of a number written e.g. in a decimal (or binary) scale constitute also an important field of study, with many applications. The divisors and the digits of a number have a strong connection, it is sufficient to only mention the congruence property n ≡ s(n) (mod 9), where s(n) denotes the sumof-digits of n in decimal representation of n. 329

CHAPTER 4

4.2 1

Special arithmetic functions connected with the divisors of a number Maximum and minimum exponents

Let n = p1a1 . . . prar > 1 be the prime factorization of n, and put H (n) = max{a1 , . . . , ar },

h(n) = min{a1 , . . . , ar }

(1)

and H (1) = h(1) = 1. These functions (called as the maximum and minimum exponents in factoring) occured also in Chapter III, when extending the Euler divisibility theorem. In 1969 I. Niven [265] proved that H (n) = c0 x + R H (x) (2) n≤x ∞ where c0 = 1 + (1 − ζ (k)−1 ), and R H (x) = o(x) and that k=2

h(n) = x + Rh (x)

(3)

n≤x

√ √ where Rh (x) = c x + o( x), with c = ζ (3/2)/ζ (3) (where ζ is the Riemann zeta function) conjectured previously by P. Erd¨os (see [265]). We note here that the generating functions of max{n 1 , . . . , n k } and min{n 1 , . . . , n k } (where a1 , . . . , ak are arbitrarily nonnegative integers) are deduced by L. Carlitz [52]. For example ∞ n 1 ,...,n k =0

min{n 1 , . . . , n k }x1n 1 . . . xkn k =

x1 x2 . . . xk (1 − x1 ) . . . (1 − xk )(1 − x1 . . . xk )

(4)

D. Suryanarayana and R. Sitaramachandra Rao [322] improved Niven’s results to

and

R H (x) = O(x 1/2 exp(−c(log x)3/5 (log log x)−1/5 ))

(5)

√ √ √ √ Rh (x) = c x + c1 3 x + c2 4 x + c3 5 x +O(x 1/6 )

(6)

A(x)

where c > 0 and c1 , c2 , c3 are constants. H. Cao [49] rediscovered (5), and a weaker form of (6). In another paper [50] he improved the exponent 3/5 to 3/5 − ε. In [163] T. Gu and H. Cao assert without proof that Rh (x) = A(x) + C4 x 1/6 + O(x 1/6 exp(−D(log x)4/7 (log log x)−3/7 ) 330

(7)

SPECIAL ARITHMETIC FUNCTIONS

but, according to A. Ivi´c [191] this was formerly established. These results can be further improved by using the strongest known form for the asymptotic formula on the number of squarefull integers not exceeding x. ˇ Porubsk´y [278] has extended (2) and a By applying the ideas used by Niven, S. weaker form of (3), to arithmetical semigroups (see e.g. Section 2.5 of Chapter II). For example, if G is an arithmetical semigroup with δ-regular G(x) (where G(x) = 1), then n∈G,|n|≤x

1 h G (m) = 1 x→∞ G(x) |m|≤x,m∈G lim

(8)

where h G (m) = min{a1 , . . . , ar }, when m = p1a1 . . . prar ∈ G is the unique factorization of m into a product of generators. Let f : N → N and put M f = {n = 1, 2, · · · : f (n)|n}, i.e. the set of positive integers n such that f (n) divides n. The asymptotic density of the set M f is well known for some special functions, including d(n), ω(n), (n). In 1994 A. Schinzel ˇ at [301] have proved that and T. Sal´ d(Mh ) = 1,

(9)

and

∞ p α−or d p α+1 − 1 1 1 1 p α−or d p α − 1 d(M H ) = + − ζ (2) α=2 ζ (α + 1) p|α p α+1 − 1 ζ (α) p|α pα − 1 (10) where or d p α denotes the exponent of the prime p in the canonical representation of A(x) denotes the asymptotic density of the set A. α. Here d(A) = lim x→∞ x They proved also that the sequences (xn ) and (yn ), given by

H (n) h(n) , yn = (n = 2, 3, . . . ) (11) log n log n

1 are dense in the interval 0, . log 2 In 1951 H. Fast [121] defined the concept of statistical convergence. A sequence of real numbers (an )n≥1 is said to converge statistically to a ∈ R, denoted by limstat an = a, provided that for each ε > 0 we have xn =

d(Aε ) = 0, where Aε = {n : |an − a| ≥ ε}. 331

(12)

CHAPTER 4

There exists today a vast literature on statistical convergence of sequences, see e.g. J. Fridy [132], J. Fridy and C. Orhan [134], J. Connor [57], J. A. Fridy, M. I. ˇ at [301] prove that Miller and C. Orhan [133]. Schinzel and Sal´ limstat

H (n) h(n) = limstat =0 log n log n

(13)

It is interesting to note that there is a strong connection between the concepts of normal order of an arithmetic function and statistical convergence. Let f, F : N → R be two arithmetic functions. Then F is a normal order of f if and only if limstat

f (n) =1 F(n)

(14)

Indeed, suppose F is a normal order of f . If 0 < ε < 1, then there is a set Bε ⊂ N such that d(Bε ) = 1, F(n) > 0 for n ∈ Bε and (1 − ε)F(n) < f (n) < (1 + ε)F(n). From this we obtain f (n) F(n) − 1 < ε, n ∈ Bε . f (n) − 1 ≥ ε holds at most for all n ∈ N \ Bε = Therefore the inequality F(n) Aε , where d(Aε ) = 0. Thus (14) holds. Conversely, if (14) holds, then it follows immediately that (1 − ε)Fn < f (n) < (1 + ε)F(n) for almost all n ∈ N (see e.g. G. H. Hardy and E. M. Wright [177]). Clearly, the constant function F(n) = 1 (n ∈ N) is a normal order of h. (15) ˇ at [301] prove that H cannot have any non-decreasing normal Schinzel and Sal´ order, i.e. if F is any non-decreasing function on N, then F os not a normal order of H . (16)

2

The product of exponents Now, similarly to (1) define β(n) = a1 a2 . . . ar ,

β(1) = 1

In 1947 D. G. Kendall and R. A. Rankin [213] showed that 1 ζ (2)ζ (3) = 1.943 . . . β(n) = lim x→∞ x ζ (6) n≤x

(17)

(18)

In 1973 J. Knopfmacher [223] proved that for all ε > 0, 1

β(n) < e 3 (1+ε) log log n for almost all n (i.e. excepting a set of density zero). 332

(19)

SPECIAL ARITHMETIC FUNCTIONS

In fact, the Dirichlet series of β(n) is ∞ β(n) n=1

Let dk =

∞ gk (n)

ns

=

ζ (s)ζ (2s)ζ (3s) ζ (6s)

, where gk (n) =

µ

n . t

n t|n,β(t)=k Then E. Kr¨atzel [236] proved that 1 = (dk + o(1))h n=1

(20)

(21)

x
for h = x θ logα x, where θ=

581 = 0.331 . . . , 1744

α = 1 + ε for k ≡ 0 (mod 6)

θ=

1740 = 0.332 . . . , 5229

α=ε

105 = 0.257 . . . , θ= 407 θ=

for k ≡ ±2 (mod 6) (22)

α = 2 + ε for k ≡ 3 (mod 6)

109556 = 0.221 . . . , α = ε 494419

for k ≡ ±1 (mod 6)

The same result holds true, when β(n) is replaced with S(n) = number of nonisomorphic semisimple rings of n elements. By studying some unconventional problems, P. Erd¨os [96] showed that there are infinitely many positive integers n such that m + β(m) ≤ n for every m < n

(23)

In fact, the density of integers satisfying (23) is positive. More generally, let αi be the density of integers n for which max{m + β(m) : m < n} = n + i αi = 1. Then αi exists for every i, and

(24)

i≥0

Denote by Si the set of integers m for which the number of solutions of equation n + β(n) = m 333

(25)

CHAPTER 4

is equal to i. Erd¨os conjectures that the set Si has a density βi ≥ 0 and βi = 1. Here probably β0 > 0, but βi > 0 is not sure always. i≥0

The analogues questions for n +d(n), n +ϕ(n), n +σ (n), etc. are open problems. We note that G. E. Hardy and M. V. Subbarao [176] have introduced the so-called highly powerful numbers n as numbers satisfying β(n) > β(m) for all 1 ≤ m < n.

3

Arithmetic functions connected with the prime power factors

We now consider other arithmetical functions related to the prime power factors of n. Let p be a prime. Let a p (n) = or d p n be the arithmetical function defined by a p (1) = 0, for n > 1, pa p (n) n,

(26)

i.e. pa p (n) |n but pa p (n)+1 n. It is immediate that a p is completely additive, i.e. a p (nm) = a p (n) + a p (m) for all n, m ≥ 1. On the other hand, since log n log p

a p (n) ≤

(27)

it easily follows that the series a p (n) is convergent for s > 1 and divergent for s ≤ 1. ns n≥1

(28)

Since a p is completely additive, one has n

where bn =

a p (k) = a p (n!) =

[n/ p k ],

n≤bn

k=1

log n (Legendre’s theorem, see e.g. [177]); it follows easily that log p a p (1) + a p (2) + · · · + a p (n) 1 = n→∞ n p−1 lim

(29)

Let Tk = {n : a p (n) = k} and Tk (x) denote the number of elements of Tk which   x  pk  x   − are ≤ x. Since Tk (x) =  p , from simple estimates we get pk d(Tk ) =

p−1 pk + 1

334

(30)

SPECIAL ARITHMETIC FUNCTIONS

where d is the asymptotic density. Related to the asymptotic density of the set Ma p (defined in section 1 (see relation (9), (10)) one has the following result due to T. ˇ at [296]: Sal´ 1 1 d(Ma p ) = ( p − 1) + ( p − 1) , (31) k+1 k−s kp kp k +1 (k, p)=1 (k, p)>1 where p sk k. As a corollary (see [296]) one gets lim d(Ma p ) = 0

(32)

p→∞

a p (n) is dense in (0,1); The sequence log p log n n≥2 for a proof of O. Strauch, see [296]. For the statistical limit one can write:

limstat log p If n =

r

a p (n) =0 log n

(33)

(34)

piai is the prime factorization of n, define

i=1

γ j (n) =

j−1 1 log piαi , log p j i=1

2≤ j ≤r

(35)

Put P(n) = max γ j (n). Then P. Erd¨os [97] proved that 2≤ j≤r

P(n) = (1 + o(1))(log3 n)/(log4 n)

(36)

where logk denotes an iterated logarithm. A generalization of γ j and P above has been considered by J. M. De Koninck, I. K´atai and A. Mercier [229]. Let now f 1 (n) = p j , where j−1

piαi ≤ n 1/2 <

i=1

j

piαi

(37)

i=1

It is not difficult to prove that the density of integers n for which f 1 (n) < n α exists. Denote this density by g(α). Then g(0) = 0, g(1) = 1 and g(α) is continuous and strictly increasing (see P. Erd¨os [98], pp. 28-29). (38) In [98] the following open problem is stated: Is it true that for every ε > 0 there is a K ε so that for every k > K ε the density of integers n for which

k k 12 −ε k 12 +ε < f1 (n + i) < n (39) n i=1

is greater than 1 − ε? 335

CHAPTER 4

Another function introduced by P. Erd¨os in [98] is α f 2 (n) = pi i , piαi ≤ n < piαi +1

(40)

f 2 (n) → ∞ on a sequence of upper density 1. (41) n Erd¨os conjectures that the logarithmic density L(α) of integers n satisfying

Then

f 2 (n) ≥α n

(42)

exists, and is a continuous strictly decreasing function of α, L(0) = 1, L(∞) = 1. The ordinary density will not exist, and even the mean value of f 2 will not exist on base of (41). Another conjecture is that max f 2 (n) = (1 + o(1)) n<x

4

x log x log log x

(43)

Other functions; the derived sequence of a number If n = p1a1 . . . prar , and Bk (n) =

r

ai pik , then many properties of Bk can be

i=1

found in [260] (pp. 143-149). A similar function is g(n) = 1 +

r

ai ( pi − 1),

g(1) = 1,

(44)

i=1

introduced by Euler, and studied by H. W. Gould [149]. J. S´andor [299] introduced the function

a r k pi + 1 i , f k (1) = 1 f k (n) = 2 i=1 Put f (n) = f 1 (n) in (45). Some simple inequalities satisfied by f k (n) are n f (ϕ(n)) ≤ 2 with equality only for n = 1, 2, 3 (where ϕ (where ϕ is Euler’s totient), and f k+m (n) ≥ f k (n) f m (n) (k, m ≥ 1)

(45)

(46)

(47)

Let σk∗ (n) be the sum of kth powers of unitary divisors of n. Then the normal order of magnitude of σ ∗ (n) (48) log k f k (n) is log 2 log log n. 336

SPECIAL ARITHMETIC FUNCTIONS

One has also lim sup n→∞

= lim inf n→∞

1 log σk∗ (n)/ f k (n) = (n)

1 log σk∗ (n)/ f k (n) = log 2 ω(n)

(49)

Let D ∗ (n) = 2(n) . Then there exists a sequence (n k ) such that log f (n k )D ∗ (n k )/n k log n k

(50)

G. L. Cohen and D. E. Iannucci [56] define a multiplicative arithmetic function D such that (51) D( pa ) = apa−1 ( p prime) β(n) n, γ (n) where the function β appears in section 2 (see (17)), and γ (n) = p1 . . . pr is the ”core” of n (see [260]), i.e. the greatest squarefree divisor of n. Let D (0) (n) = n and D (k) (n) = D(D (k−1) (n)) be the iteration of order k of D. The sequence and D(1) = 1. Since D(n) = a1 . . . ar p1a1 −1 . . . prar −1 , we have D(n) =

D(n) = {n, D(n), D (2) (n), . . . }

(52)

is called the derived sequence of n. If D j+k (n) = D j (n), where j ≥ 0 and k ≥ 1 is the smallest integer with this property, then D j (n), . . . , D j+k−1 (n) is a derived k-cycle of n. The following result shows the existence of 5-cycles: Let s be such that (s, 2 · 3 · 5 · 47) = 1, and suppose (s1 , 5 · 23 · 47) = 1, where s1 = (3s − 1)/2. Put n = 23s · 345 · 55 · 23 · s1 . Then n, D(n), . . . , D (4) (n) is a 5-cycle. (53) For similar results on 6 or 8-cycles, see [56]. Cohen and Iannucci prove also that for all n < 1.5 · 1010 , the derived sequence D(n) is bounded; (54) but the density of integers n for which D(n) is unbounded, is less than 0.004 (55) It is not known the existence of unbounded derived sequences, but the authors conjecture this existence.

5

The consecutive prime divisors of a number

Let p1 (n) < p2 (n) < · · · < pr (n), where r = ω(n), be the consecutive prime factors of n ≥ 2. In 1946 P. Erd¨os [99] proved that for almost all n one has log log p j (n) = (1 + o(1)) j For a simple proof of (56), see [100]. A more precise estimate is log log p j (n) = j + O( j log log j) 337

(56)

(57)

CHAPTER 4

for almost all n, and all ξ(n) ≤ j ≤ ω(n), where ξ(n) → ∞ (see R. R. Hall and G. Tenenbaum [172]). Let λ j ( p) = d{n : p j (n) = p} (58) be the density of integers n whose j-th prime factor is p. By the sieve of Eratosthenes it follows that

1 1 λ j ( p) = 1− 1/m (59) p q 0 1, t =0

P. Erd¨os and G. Tenenbaum [114] have proved that

1 F(ρ, p)ρ 1−k 1+O λ j ( p) = p F(1, p)w(k − 1) R

(60)

uniformly, for all 1 ≤ j ≤ p 1−ε (where ε > 0 is given), where ρ is the solution of the equation ρ = j −1 (61) q< p q − 1 + ρ As corollaries of (60) and (61) one obtains the following:

M j−1 1 j −1 λ j ( p) = exp O p F(1, p) ( j − 1)! R where

and

(62)

M = log log p − log(1 + log+ ( j/L)),

log p L = log log( j + 1) log+ x = max{0, log x}

λ j+1 ( p) 1 M = 1+O , λ j ( p) j R

where in (62) and (63) one has 1 ≤ j ≤ p 1−ε . 338

(63)

SPECIAL ARITHMETIC FUNCTIONS

Other corollaries are: max λ j ( p) = j≥1

1 + O(1/ log log p) as p → ∞ p 2π log log p

(64)

and all values of j realizing this maximum satisfy j = log log p + O(1), 2 log2 j + 1 max λ j ( p) = exp − j log j − log2 j − 1 + + p log j

2(log2 j)2 − log2 j + O(1) + (log j)2

(65)

(66)

where log2 j = log log j. Moreover, all values of p realizing this maximum satisfy 2 log2 j j 2(log2 j)2 − 3 log2 j + O(1) 1+ (67) log p = + log j log j (log j)2 Erd¨os and Nicolas [110] have studied the functions r −1 pi (n) f (n) = p (n) i=1 i+1

and

F(n) =

r −1 i=1

pi (n) 1− pi+1 (n)

They proved, for example, that there exists a constant c = 1.70 . . . such that F(n) ≤ log n − c + o(1), for all n, and F(n) ≥ log n − c + o(1) for infinitely many n. The proofs use besides results on the distribution of primes, also classical theorems of Optimization theory. The following results are due to P. Erd¨os (see [101] and [100]). For almost all integers

∗ 1 1 = + o(1) log log log n (68) j 2 where ∗ indicates that log log p j (n) > j. Similarly, for almost all n,

∗∗ 1 1 = + o(1) log log log n j 2

(69)

where ∗∗ indicates that p j (n) > p j (n + 1) in the summation. For almost all n one has ∗∗∗ 1 (70) √ = (1 + o(1))c log log log n j 339

CHAPTER 4

where ∗ ∗ ∗ shows that the summation is extended over the j for which j < log log p j (n) < j + 1, and c is a constant. log log p j (n) − j has normal distribution, and if j1 /j2 → ∞, then The function √ j log log p j2 (n) − j2 log log p j1 (n) − j1 and are asymptotical independent. √ √ j1 j2

(71)

J. Galambos [137] defined a number n to be weakly-composite if r j=1

1 ≤2 p j (n)

(r = ω(n))

(72)

He proved that (by solving a conjecture of I. K´atai) for all sufficiently large integers n, there is a weakly composite number between n and n + log log log n On the other hand, the behaviour of 1 n≤x

p j (n)

= R j (x)

(73)

(74)

has been investigated by several authors for some specific values of j. In particular, for j = 1, or whenever j is preassigned, it is not difficult to prove that R j (x) ∼ c j x, with computable constants c j . When j = ω(n), the problem of finding good approximations to R j is more difficult. The case j = ω(n) − 1, or j = ω(n) − k with k fixed, shows no similarity to the above case. For asymptotic results (due to many authors, with refinements), see [260]. In order to obtain an average type of information on the magnitude of p j (n), when j is not fixed, or j = ω(n)−k, with k fixed, J. M. De Koninck and J. Galambos [230] have set up the following probabilistic approach. For every integer n ≤ x, pick one 1 p(n) ∈ { p1 (n), . . . , pr (n)} with equal probabilities , and consider the sums ω(n) 1 = R(x) (75) n≤x p(n) Assume that x is an integer. Then there are ω(2) . . . ω(x) sums of the type (75) and Rr (x) ≤ R(x) ≤ R1 (x). We shall say that a property holds for almost all sums in (75), if the number N (x) of the sums with the property in question satisfies N (x)/ω(2) . . . ω(x) → 1 as x → ∞. 340

SPECIAL ARITHMETIC FUNCTIONS

De Koninck and J. Galambos [230] prove that for almost all sums in (75) one has cx +O R(x) = log log x where c =

x , (log log x)2

(76)

1/ p 2 . The proof is based on a form of the Chebyshev inequality of

p prime

Probability theory. In another paper, De Koninck and Galambos [231] have studied the intermediate prime divisors of an integer. With density one, we can distinguish three types of prime divisors: we call p j ”small” if j is bounded as n → +∞; p j ”large” if ω(n)− j remains bounded as n → ∞; all others are ”intermediate”. For the investigation of small prime divisors, tools from elementary number theory are sufficient. Large prime divisors require special tools (due e.g. to Dickman, see e.g. [260] for asymptotic log p j formulas involving large prime divisors), and old results show that falls into log n the interval (a, b), where 0 ≤ a < b ≤ 1, with positive density, when j = ω(n). Extensions are known also for all large prime divisors, with similar results in nature. This perhaps explains why it was so important in probabilistic number thelog r ory to truncate additive functions at r = r (N ) with → 0; it simply cancels log N the effect of the large prime divisors (see P. D. T. A. Elliott [95]). The truncation methods already show that the intermediate prime divisors behave asymptotically as independent random variables. In 1976 J. Galambos [138] showed that for intermediate terms, log log p j+1 (n) − log log p j (n) are asymptotically unit exponential variables, (77) i.e. the density for which the above stated difference is smaller than 0 < z, equals 1 − e−z . The remarkable part of this result is that the density does not depend on j. H. Maier [246] extended this result by showing that a finite set of the above differences are asymptotically independent in the sense of probability theory. De Koninck and Galambos [231] further generalize these results, by proving the following statement: Let j = j (N ) be a positive integer valued function, j (N ) → +∞ as N → +∞. Assume that j is such that, with perhaps the exception of a set of density zero, p j (n) → +∞ with N , and (log p j (n))/ log N → 0 as N → +∞, where 1 ≤ n ≤ N . Then the points log log p j+k (k ≥ 1) form a Poisson process in limit as N → +∞. (78) In the proof, a result of A. R´enyi [284] on the Poisson process is applied. For another notion of asymptotic prime divisor, see S. McAdam [1]. 341

CHAPTER 4

6

The consecutive divisors of an integer

Let 1 = d1 (n) < d2 (n) < · · · < dk (n), k = d(n), be the consecutive divisors of n. For each n ≥ 2, pick at random one r (n) of the divisors d j (with equal probabilities) and consider the sum 1 = Q(x) n≤x r (n)

(79)

As in (75), De Koninck and Galambos [230] say that a property holds for almost all sums in (79) if it holds for M sums such that M/d(2) . . . d(x) → 1 as x → ∞ (where x is an integer, and d(k) denotes the number of distinct divisors of k). Then one has the following results (see [230]): For almost sums in (79),

c1 x x Q(x) = (80) +O (log x)3/2 log x Define similarly

r (n) = S(x)

(81)

n≤x

Then, for almost all sums in (81) one has c2 x 2 S(x) = +O log x

x2 (log x)3/2

(82)

Here c1 , c2 are positive constants. We note that similar results can be deduced for the more general sums h(r (n)), where h is a general arithmetic function, n≤x

satisfying certain conditions. In relation (58) we have defined a density function on the consecutive prime divisors of an integer. Define similarly j (k) = d{n : d j (n) = k},

k ≥ 1.

(83)

P. Erd¨os and G. Tenenbaum [114] have shown that the following two assertions are equivalent: (i) j (k) > 0; (ii) d(k) ≤ j ≤ k (84) Further, the following estimates are true: e−γ + O(1/ log k) 3 log3 k + O(1) ≤ max j (k) ≤ j k log k k log2 k 342

(k → ∞),

(85)

SPECIAL ARITHMETIC FUNCTIONS

where logm denotes an iterated logarithm, and γ is Euler’s constant. Moreover, for all j realizing this maximum, (86)

d(k) ≤ j ≤ d(k)(log k)e log 2+o(1) log j log2 j . Then Similarly, for j ≥ 3, let K := exp log 2 K −1 j −(log3 j+O(1))/ log 2 ≤ max j (k) ≤ K −1 j (log3 j+O(1))/ log 2 k

(87)

as j → ∞. For all k realizing the maximum, one has K α+o(1) ≤ k ≤ K 1+o(1)

(88)

where α = 0.293815 . . . is the unique solution of the equation (α + 1) log(α + 1) − α log α = log 2.

7

Functional limit theorems for the consecutive divisors

Let γx (. . . ) denote the uniform probability measure on the set {1, . . . , [x]}; put Lu = log max{u, e}, and L k = L(L k−1 ). Then one has (see [172])

max |L 2 dk (m) − log2 k| ≥ (1 + ε) 2(log2 k)L 3 k = 0 (89) lim lim sup γx n→∞

x→∞

n≤k≤d(m)

which is the counterpart of the following result on prime divisors:

max |L 2 pk (m) − k| ≥ (1 + ε) 2k L 2 k = 0 lim lim sup γx n→∞

x→∞

n≤k≤ω(m)

(90)

These results were extended into the Strassen functional form by E. Manstaviˇcius [249]. Let d(m, n) = car d{d ∈ N : d|m, d ≤ u}. G. Tenenbaum [323] (see also [172]) showed that γx (d(m, x t ) − d(m, x s )) < ud(m) ⇒ Fst (u),

0 ≤ s < t ≤ 1,

(91)

where ⇒ denotes weak convergence of distribution functions, Fst (u) is some purely discrete distribution function, and x → ∞. This relation can be interpreted as referring to the increments of the arithmetic process d(m, x t ) (92) Yx = Yx (m, t) = d(m) 343

CHAPTER 4

having trajectories in the space D = D[0, 1] of real-valued functions on [0, 1] which are right-continuous and have left-hand limits. The behaviour of the expectation of this process is due to J. M. Deshouillers, F. Dress and G. Tenenbaum [75]: √ 2 1 Yx (m, t) = arcsin t + o(1), x m≤x π

0≤t ≤1

(93)

Let ω(m, v) = car d{ p : p|m, p ≤ v} ( p = prime) and define the process ψx = ψx (m, t) := √

1 (ω(m, exp{(L x)t }) − t L 2 (x), L2x

0≤t ≤1

(94)

which ”simulates” the Brownian motion w = w(t) given on some probability space {, F, P}. Let ρ(·, ·) be the supremum metrics on the space D, and D be the Borel σ -algebra in D with respect to ρ, and γx ◦ ψx−1 stand for the measure on D defined by γx (X x ∈ A), where A ∈ D, and X x = X x (m, t) = √

1 (log2 d(m, exp{(L x)t }) − t L 2 x), L2x

0≤t ≤1

(95)

Put µ for the Wiener measure P ◦ W −1 . In 1970 P. Billingsley [27] proved that the sequence of measures γx ◦ ψx−1 (96) weakly converges to the Wiener measure µ as x → ∞ (see also [28]). Denote µx = γx ◦ X x−1 . In 1996 E. Manstaviˇcius [250] proved the similar result that the sequence of measures µx weakly converges to the Wiener measure µ as x → ∞. (97) As a corollary, we note that (97) implies u 1 y2 dy (98) exp − γx (X x (m, t) − X x (m, s) < u) ⇒ √ 2(t − s) 2π(t − s) −∞ Hence if s = 0 and t = 1, we have the central limit theorem for the additive function log2 d(m) belonging to the Kubilius class H (see [238]). But if 0 < t < 1, then log2 d(m, t) is only subadditive. This shows new direction of possible investigations, e.g. to extend the probability number theory to the class of subadditive functions. For a generalization of the Kubilius class, see I. Z. Ruzsa [288]. √ 2 Let As(u) = arcsin t for 0 ≤ t ≤ 1, and As(u) = 0 when u < 0; = 1, when π u > 1. Then another corollary of (97) is: γx (meas{0 ≤ t ≤ 1 : log2 d(m, exp{(L x)t }) > t L 2 x} < u) ⇒ As(u), where meas is the Lebesgue measure. 344

(99)

SPECIAL ARITHMETIC FUNCTIONS

In 1969 P. Erd¨os [101] proved that γx (car d{k ≤ ω(m) : L 2 pk (m) < k} 0}. Then we have 1 + γx (101) I (log2 k − L 2 dk (m)) < (L2)u L 2 x ⇒ As(u) k k≤d(m) For a survey of functional limit theorems in Probabilistic number theory, see the recent paper by E. Manstaviˇcius [251].

8

Miscellaneous arithmetic functions connected with divisors

In what follows we shall study various arithmetic functions connected with the divisors of a number. In 1973 P. Erd¨os and S. K. Zaremba [118], in connection with work on good lattice points modulo composite numbers, have studied the arithmetical function a(n) =

log d d|n

d

(102)

Obviously, lim inf a(n) = 0. On the other hand, one has n→∞

lim sup n→∞

a(n) = eγ , (log log n)2

(103)

where γ is Euler’s constant. The arithmetical function a(n) has a continuous purely singular distribution function. Let d + (n) denote the number of integers k, such that the interval [2k , 2k+1 ) contains at least a divisor d of n. (104) This function appears in Erd¨os [100]. In [115] Erd¨os and Tenenbaum proved that for all ε > 0 there is c(ε) > 0 such that for all 0 ≤ α ≤ 1, the upper density of all integers n satisfying d + (n) ≤ αd(n) (105) is less than c(ε)α 1−ε . This implies a conjecture of Erd¨os, namely that apart from a sequence of integers of density zero, d + (n) → 0 as n → ∞ (106) d(n) 345

CHAPTER 4

Various arithmetic functions have been studied by P. Ed¨os and J. L. Nicolas in [107], [108] and [109]. Let dn be the smallest divisor of n such that d≥n d|n,d≤dn

and put

n dn

f 1 (n) =

(107)

In [107] one can find that for ε > 0 and n > n 0 (ε), one has f 1 (n) ≤ exp(1 + ε)

log2 n log3 n , log4 n

(108)

log2 m log3 m log4 m

(109)

and for infinitely many m one has f 1 (m) ≥ exp(1 − ε) Another function introduced in [107] is f 2 (n) = max

1≤k≤n

1 d k d|n,d≤k

(110)

It is proved that cd(n)

≤ f 2 (n) ≤ c d(n) log n log log n

log log n , log n

(111)

with c, c > 0 constants. The function f 2 (n) is supermultiplicative, i.e. (m, n) = 1 ⇒ f 2 (mn) ≥ f 2 (m) f 2 (n)

(112)

Let k0 = k0 (n) the value of k in (110) for which the maximum is realized, i.e. f 2 (n) =

1 d. k0 (n) d|n,d≤k (n) 0

In [109], by probabilistic arguments it is shown that for all constants c1 ≤ 0.23 and sufficiently large n, we have k0 (n) ≥ exp(c1 log n/ log log n); 346

(113)

SPECIAL ARITHMETIC FUNCTIONS

and that there exists a constant c2 > 0 such that for infinitely many n, k0 (n) ≤ exp(c2 log n/ log log n)

(114)

Another result says that for almost all n one has k0 (n) < n

(115)

The function f 2 is connected with another function F defined by 1 F(n) = max t

via the relations

(116)

d|n, 2t
1 F(n) ≤ f 2 (n) ≤ 2F(n), 2

(117)

and this is exploited in [108]. A number n is called F-highly abundant if m < n ⇒ F(m) < F(n). Then, if n is F-highly abundant, one can find two positive constants c1 , c2 > 0 such that d(n) d(n) ≤ F(n) ≤ c2 c1 log n log log n log n log log n

(118)

In 1976 G. Tenenbaum [324] introduced the functions ρ1 (n) = greatest divisor of √ n less than or equal to n; and ρ2 (n) = smallest divisor of n greatest than or equal √ to n. (119) He showed that for all ε > 0 there is an absolute constant c and an x0 (ε) such that x 3/2 (log x)−α−ε < ρ1 (n) < x 3/2 (log x)−α (log log x)−1/2 , (120) n≤x

where α = 1 − Further,

1 log(e log 2), and x ≥ x0 (ε). log 2 n≤x

ρ2 (n) =

π2 x2 +O 12 log x

x2 log2 x

;

(121)

and the density of integers n such that (ρ1 (n), ρ2 (n)) = 1

(122)

is equal to 1. Certain arithmetic functions related to distinct divisors have been studied by P. Erd¨os and C. Pomerance [111] and [112]. 347

CHAPTER 4

Let f (n) denote the least integer so that in the interval (n, f (n)] there are distinct integers a1 , . . . , an with i|ai for i = 1, 2, . . . , n. For example, f (10) = 24, since a1 = 11, a2 = 22, a3 = 21, a4 = 16, a5 = 15, a6 = 12, a7 = 14, a8 = 24, a9 = 18, a10 = 20. More generally, if m is any positive integer, let f (m, n) denote the least integer so that in (m, m + f (n, m)] there are distinct integers a1 , . . . , an with i|ai , for i = 1, 2, . . . , n. (123) In [111] it is proved that for n ≥ 3,

2 (124) √ + o(1) n log n/ log log n ≤ f (n) ≤ (2 + o(1))n log n e The right side of (124) holds true also for n = 2. For f (n, m) the following upper bound is valid: √ f (n, m) ≤ 4n([ n] + 1), m, n ≥ 1

(125)

The upper bounds in (124) and (125) rely on a theorem of D. K¨onig [225] and P. Hall [167] (called also as the ”Marriage theorem”) on bipartite graphs. Let L(n) = [1, 2, . . . , n] be the least common multiple of 1, 2, . . . , n. Then for all sufficiently large n, L(n) 1 f (n, m) > n(log n)α L(n) m=1

(126)

where α = .08607 . . . is the constant of (120); and for all n L(n) 1 f (n, m) ≤ n exp((β + o(1)) log n/ log log n), L(n) m=1

(127)

where β = log(55/2 /2 · 33/2 ) = 1.6825 . . . The authors conjecture that f (n, m) ≤ n 1+o(1) , and that L(n) 1 f (n, m) n(log n)γ for some γ > 0 L(n) m=1

(128)

Related to Grimm’s conjecture, in [112] the following concepts are introduced. We say that a set of positive integers S has the distinct divisor property if for each s ∈ S we can find a divisor ds |s, 1 ≤ ds < s such that the ds are all distinct. If c > 0, let f (n, c) denote the cardinality of the largest subset of [n, cn] which has the distinct divisor property. 348

SPECIAL ARITHMETIC FUNCTIONS

By applying again the Marriage theorem, in [112] it is proved that for each c > 1 there is a constant δ(c) such that f (n; c) ∼ δ(c)n as n → ∞.

(129)

Further, δ is a continuous and strictly increasing function, and δ(c) δ(c) 1 1 δ(c) = 1, lim = , < < 1 for all c > 1 c→∞ c→1+ c − 1 c−1 2 2 c−1 lim

The following Open Problem is stated: is

9

(130)

δ(c) a monotonic function? c−1

Arithmetic functions of consecutive divisors

Let 1 = d1 < d2 < · · · < dk (n) (k = d(n)) be the consecutive divisors of n. In what follows we shall study certain arithmetic functions connected to these consecutive divisors. In 1948 P. Erd¨os [102] proved that the density of all integers n having two divisors d and d such that d < d ≤ 2d is positive. (131) He conjectured that this density is 1, or written equivalently di+1 : 1 ≤ i ≤ k → 1 a.e. (132) E(n) = min di (where a.e. - almost everywhere - means that the relation holds true on a sequence of density one). Conjecture (132) has been established in 1984 by H. Maier and G. Tenenbaum [247]. The arithmetical function f (n) = car d{1 ≤ i ≤ k : (di , di+1 ) = 1}

(133)

has been introduced by P. Erd¨os and R. R. Hall in [106], while its ”dual” is the Erd¨osMontgomery function from 1979 (see [116]), defined by g(n) = car d{1 ≤ i ≤ k : di |di+1 } A more general function than f is fr , introduced in [106], too, is max (d j , dk ) = 1 fr (n) = car d 1 ≤ i ≤ k − r + 1 : 1≤ j
Then f 2 (n) = f (n). 349

(134)

(135)

CHAPTER 4

By improving earlier results, Erd¨os and Tenenbaum [117] prove the following results: 3 log log(2d(n)) (n ≥ 2), (136) f (n) ≤ log d(n) f (n) ≤ 4(log(1 + (n))/ (n))1/3 d(n) f (n) d(n)1−c , On the other hand,

c=

(n ≥ 2)

5 − log 3/ log 2 = 0.0817 . . . 3

f (n) x(log x)α

(137) (138) (139)

n≤x

where α = 3 · 4−1/3 − 1 = 0.8898 . . . ; and f (n) x(log x)log 3−1+o(1) ,

x ≥3

(140)

n≤x

In fact, (140) is a corollary of a result on an arithmetic function h(n), introduced by Erd¨os and Tenenbaum ([117]) as h(n) =

k ∗ di , di+1 i=1

(141)

where ∗ indicates that in the sum one takes (di , di+1 ) = 1. The result on h(n) is the following: h(n) = (log n)log 3−1+o(1) a.e. (142) The following theorem improves essentially an earlier estimate from [106]: max f (n) ≥ exp{((log 2)2 + o(1)) log x/(log log x)2 } n≤x

The similar result for fr (n) is

1 max fr (n) ≥ exp + o(1) (log log x)2 / log log log x n≤x 2 In [325] G. Tenenbaum introduced the function di+1 1 max log ψ(n) = : 1≤i ≤k−1 log n di 350

(143)

(144)

(145)

SPECIAL ARITHMETIC FUNCTIONS

and proved that is has a continuous distribution function d(x) on [0, 1]. This functions satisfies a1 x ≤ d(x) ≤ a2 x log(2/x) (a1 , a2 - constants), see [327]. Another function, by Hall and Tenenbaum (see [115]) introduced in 1979 is d α (146) U (n, α) = car d d|n, d |n : (d, d ) = 1, log ≤ (log n) d In [172] it is proved that U (n, α) ∼ 1 + (log n)log 3−1+α+o(1) a.e.,

(147)

which for α ≤ 0 contains the Erd¨os conjecture (132), and which is used also in the proof of (142). Other properties of U (n, α) have been obtained by R. R. Hall [168]: U (n, α)/(2α + 2)ω(n) = C · x + O(x(log x)−γ ), n≤x

where C = C(α), γ = γ (α) > 0; U (n, 0)/2ω(n) x log log x; n≤x

5 3 ω(n) ·C ·x U (n, 1)/4 ∼ 4 4 n≤x

(147 )

as x → ∞ Erd¨os and Hall [106] considered the important function k(n) = car d{d : d(d + 1)|n} and showed that

√ e+o(1)

max k(n) > (log x) n≤x

(x → ∞),

(148) (149)

and which was strengthened to max k(n) > (log x)log3 x/(9 log4 x) n≤x

(x → ∞)

(150)

by A. Balog, O. Erd¨os and G. Tenenbaum [15]. A generalization of k is the function ks introduced by R. de la Bret`eche [39], [40]. Put (151) ks (n) = car d{d : (d, s) = 1, d(d + s)|n} Clearly k1 ≡ k. Let D(n) = 2log n/ log log n . Then (see [40]): ks (n) ≤ D(n)c1 +o(1) , as n → ∞ uniformly for all s ≥ 1, with c1 = 0.565 . . . 351

(152)

CHAPTER 4

In 1986 J. M. De Koninck and A. Ivi´c [232] have studied two functions, namely H1 (n) =

k−1 i=1

and H1 (n) =

r −1 i=1

1 , di+1 − di

k = d(n),

(153)

1 , pi+1 − pi

r = ω(n),

(154)

where p1 < p2 < · · · < pr are the consecutive prime divisors of n. They proved that

x log log x (155) H1 (n) = Ax + O log x where A = 0.299 . . . is a constant; and H1 (n) = B · x + O(x(log x)−1/3 )

(156)

n≤x

with another constant B = 1.77 . . . In [117] Erd¨os and Tenenbaum improved the error term in (156) to O(x(log x)−1 (log log x)3 )

(157)

They proved also that 1

H1 (n) d(n)(log d(n))− 3 +ε

(n ≥ 2)

(158)

(x → ∞)

(159)

for any fixed ε > 0, and that 1

max H1 (n) > exp{(log x) 2 +o(1) } n≤x

Further,

H1 (n) d(n)1−c2 log(1 + ω(n)) 5 log 3 = 0.08170 . . . whenever n is squarefree, where c2 = − 3 log 2 In [15] it is proved that

1−c2 +o(1) max H1 (n) ≤ max d(n) (x → ∞), n≤x

n≤x

(160)

(161)

where c2 is the same as in (160). It is conjectured that a bound of type H1 (n) d(n)1−δ with an absolute δ > 0 holds unconditionally. 352

(162)

SPECIAL ARITHMETIC FUNCTIONS

The function H1 has a distribution function (see [117]), and this is everywhere continuous on the real line (see [15]). In [116] the authors study also a general function of type

k−1 di , θ F(n, θ) = di+1 i=1

(163)

where θ : [0, 1] → R is an arbitrary function. They show that when θ is bounded, then the function F(n, θ) (164) d(n) has a distribution function. When θ is of class C 2 , then an asymptotic formula of type log log log x F(n, θ) = x log x θ (1) + O (165) (log x)δ log log x n≤x is valid, where δ = 0.086 . . . is an optimal constant, if one assumes that tθ (t) is a monotonic function. For the particular case θ (t) = t r a more precise estimation is true: F(n, t → t r ) = x(log x − K r (x) + O(1)) (166) n≤x

uniformly for r log x 1, where K r (x) satisfy the double inequality e−εr L 1 (log x) ε with

K r (x) (1 + r δ )L 2 ((r/(r + 1)) log x), (r log x)1−δ

(167)

L 1 (v) = exp{−c(ε) log v log log v},

L 2 (v) = (log v)−1/2 log log v

1+ε 1 −1 (ε > 0), M. D. Vose [345] constructed a In the case of θ (t) = t sequence (Nn )n≥1 such that F(Nn , θ) = Oε (1), solving a conjecture by Erd¨os. In fact, the above relation holds true also by replacing Nn with the sequence of factorials (n!) or with lcm[1, 2, . . . , n], see G. Tenenbaum [334], who studied the 1 more general case of θ (t) = h − 1 , where h : [0, ∞) → [0, ∞) belongs to a t certain class generalizing the one considered by Vose. 353

CHAPTER 4

For similar or related problems on the sequence of factorials, as well as the distribution of divisors of factorials, see M. D. Vose [346] and D. Berend and J. E. Harmse [21]. Another function of type (153) is Tenenbaum’s function ([326]) H2 (n) =

1≤i< j≤k

1 d j − di

(168)

Tenenbaum proves that H2 (n) d(n)1−c with an explicit positive constant c. He further shows that the inequality k ξi ξ j ξi2 (ξi ∈ R), ≤M 1≤i< j≤k d j − di i=1

(169)

(170)

which is a special case of Hilbert’s inequality, holds with a constant M = M(n) satisfying M(n) log log d(n) as n → ∞ and M(n i ) log log d(n i ) for a suitable sequence (n i ) with d(n i ) → ∞ as i → ∞. R. de la Bret`eche [39] shows that H2 (n) d(n)c3 log d(n) log log(2d(n)),

(171)

where c3 = 0.918 . . . is a constant; and in [40] that H2 (n) D(n)c1 +o(1) ,

(172)

where D(n) and c1 are as in (152). In [326] Tenenbaum has introduced a general class of arithmetical functions F. We say that an application gn , defined on the set of divisors of n with integer values, is regular if gn is injective, and for all divisors d, t of n one has ((d, gn (d)) = 1, (t, gn (t)) = 1, dgn (d) = tgn (t)|n) ⇒ d = t

(173)

Let F be the class of arithmetic functions F ∈ F defined by F(n) = car d{ f |n : gn (d)|n, (d, gn (d)) = 1}

(174)

f F (n) = max{F(n) : F ∈ F}

(175)

Let This function is a common generalization of the functions f and k defined in (133) and (151), respectively. 354

SPECIAL ARITHMETIC FUNCTIONS

The properties of f F have been recently investigated in [39], where the following are proved: f F is super-multiplicative, i.e. f F (mn) ≥ f F (m) f F (n) for all (m, n) = 1;

(176)

f F (n) ≤ d(n)c3 for all n,

(177)

where c3 is the constant of (171); lim sup d(n)→∞

log f F (n) = c3 ; log d(n)

f F (n) ≥ max(min{d(k), d(n/k)}) ≥ √ k n

(178) d(n)1/2 ; H (n) + 1

(179)

where k n means that k is a unitary divisor of n (i.e. k|n, (k, n/k) =1), and H (n) is the maximum of the exponents γ in the prime factorization n = p γ (i.e. the p γ n

function defined by (1) and studied in section 1). One has also 2√γ

f F (n) ≤ d(n) γ +1 p γ n

(180)

As a corollary, one obtains f F (n) = O

d(n) (n)

(181)

Let p(n) denote the least prime divisor of n. By studying certain sievetheoretical problems, in 1959 A. Schinzel and G. Szekeres [290] considered the function S(n) = max{d p(d) : d > 1, d|n} (n ≥ 2); S(1) = 1 (182) This function is connected with the set of consecutive divisors by the surprizing formula S(n) di+1 : 1 ≤ i ≤ k − 1 = E ∗ (n), (183) = max n di due to G. Tenenbaum [327]. Let D(x, y), A(x, y) be distribution functions defined by D(x, y) = car d{n ≤ x : S(n) ≤ yn}, A(x, y) = car d{n ≤ x : S(n) ≤ yx} 355

(184)

CHAPTER 4

where x ≥ 1, y > 0. By (183) one can say that D(x, y) is exactly the distribution function of E ∗ (n). For D and E the following estimates are true (see [327]):

log δ 5 1− = 4.20 . . . Let ε, λ real numbers such that ε > 5/3, λ > 3 δ √ 1+ 5 δ = log . Then for all x ≥ y ≥ 2 one has 2 x x L(x, y) y,λ D(x, y) ≤ A(x, y) log 2u, u u where u =

(185)

log x , and log y L(u, y) =

(log u)−λ , 2 ≤ y ≤ exp{(log log x)ε } 1, y > exp{(log log x)ε }

The following inequality for the Schinzel-Szekeres function is true: S(mn) ≤ max{S(m) · n, S(n)}

(m, n ≥ 1)

(186)

with equality if P(m) ≤ p(n) (where P(m) denotes the greatest prime factor of m). As a corollary, S(mp) ≤ pS(m) for all m ≥ 2, p ≤ S(m) ( p = prime),

(187)

which follows by (186), since S( p) = p 2 for a prime p. In analogy with (184) define D(x, y, z) = car d{n ≤ x : S(n) ≤ ny, p(n) > z} A(x, y, z) = car d{n ≤ x : S(n) ≤ x y, p(n) > z}

(188)

E. Saias [290] found that there exist three positive constants c4 , c5 and y0 such that under the conditions x ≥ y ≥ z + z 0.535 ≥ 3, x ≥ z 5 and y ≥ y0

(189)

we have c4 Let u =

x x log(y/z) log(y/z) · ≤ D(x, y, z) ≤ c5 · log x log z log x log z

log x log x and v = . log y log z 356

(190)

SPECIAL ARITHMETIC FUNCTIONS

For v > 1, the Buchstab function ω(v) is defined as the unique continuous solution to the difference-differential equation (vω(v)) = ω(v − 1) (v > 2) vω(v) = 1 (1 ≤ v ≤ 2)

(191)

Let ω(v) = 0 for v < 1, and define ω at 1 and ω at 1 and 2 by right continuity. The Buchstab function arises in the study of D(x, x, z). For an overview of results on these functions, see [260] (Chapter 4). For u > 1 we define the function d(u, v) as the unique continuous solution to the equation

∂ u(v − 1) 2u (192) (vd(u, v)) = d ,v − 1 v > u > 1, v > ∂v u+v u−1 with initial conditions

  ω(v), if 0 ≤ u < 1,   u 2u d(u, v) = ω(v) − ω(u), if 1 ≤ u < v ≤ ,  v u−1   0, if u = v ≥ 0

2u , u−1 v = 2. To extend the domain of d to all of R2 we let d(u, v) = 0 outside the region 0 ≤ u ≤ v. It is easily verified that  1 (v − 1)(u − 1) 2u 4u u   , if 3 ≤ ≤v≤ ω(v) − ω(u) − log   v v u+1 u−1 u−1 d(u, v) =   4u  ω(v) − u ω(u) − 1 log v − 1 , if 3 ≤ u ≤ v ≤ v v u−1 u−1 (193) In a recent paper, A. Weingartner [349] shows that uniformly for 3/2 ≤ z ≤ y ≤ x,

x y z x D(x, y, z) = d(u, v) + − +O log z log y log z log2 z (194)

√ xy x z x d(u, v) + +O A(x, y, z) = √ − log z log x y log z log2 z For the behaviour of d(u, v) Weingartner proves that for 1 ≤ u ≤ v we have

4u 0.9g(u, v) ≤ d(u, v) ≤ 1.7g(u, v) v ≥ (195) u−1 d(u, v) ε g(u, v) (v ≥ 2 + ε, (u, v) ∈ Wε ) Note that d also satisfies the above equation in the region 1 ≤ u < v <

357

CHAPTER 4

where

v
and g(u, v) =

u 2e−γ 1− , u+1 v

γ being the Euler constant (ε > 0 is given). In [328] G. Tenenbaum showed that for λ ∈ [0, 1] fixed, S(n) λ car d n ≤ x : ≤ n ∼ xs(λ) (x → ∞), n

(196)

where s is a continuous function. Put d(u) = s(1/u) Then for u ≥ 1 fixed D(x, y) := D(x, y, 1) ∼ xd(u) as x → ∞ A(x, y) := A(x, y, 1) ∼ xd(u)

(197)

In 1997 E. Saias [291] showed that there exist constants a1 and a2 such that a1 ≤ ud(u) ≤ a2 for u ≥ 1

(198)

In [349] it is shown that for 1 ≤ u ≤ v, d(u, v) = e−γ d(u) + O(1/v)

(199)

and using (194), (199) it follows that 3·4 1·8 ≤ d(u) ≤ , u+1 u+1

u≥1

(200)

which is a precision of (198). Similarly, if u ≥ 1 fixed, and v → ∞, from the above one gets

1 (x → ∞) D(x, y, z) ∼ D(x, y) 1− p p≤z (201)

1 A(x, y, z) ∼ A(x, y) 1− p p≤z

358

SPECIAL ARITHMETIC FUNCTIONS

Explicit formulas for d(u) and d(u, v) are given in [291], resp. [349]. These are:

1 1 − yk d x1 d x2 . . . d xk (202) d(u) = ρ xk x1 x2 . . . xk A

1 where A = 0 < xk ≤ xk−1 ≤ · · · ≤ x1 ≤ 1, max (x j + y j ) ≤ 1 + , and ρ is the 1≤ j≤k u Dickman function, defined to be the unique continuous solution to the equation vρ (v) + ρ(v − 1) = 0 (v > 1) ρ(v) = 1 (0 ≤ v ≤ 1), ρ(v) = 0 (v < 0), ρ (v)

(203)

extended to v = 1 by right continuity For properties of ρ, see [260] (Chapter 4). Here y j = x1 + x2 + · · · + x j and k = [u]. Similarly,

1 1 − yk 1 − yk , d x1 . . . d xk d(u, v) = σ xk 1/v x1 . . . xk

(204)

A

where A, k and y j are as above, and σ (u, v) is the unique solution, continuous for u > 1, to

∂σ v u (u, v) + σ (u − 1, v − v/u) = 0 u > , v>1 ∂u v−1 with initial conditions σ (u, v) = ω(v) (0 ≤ u < 1),

u . σ (u, v) = ω(v) − 1/v 1 ≤ u ≤ v ≤ u−1

(205)

This function has been studied by J. B. Friedlander [135], and E. Saias [292]. We note that results on D(x, y) and A(x, y) of (184) have been applied to the small sieve of Erd¨os and Ruzsa ([289]) and to practical numbers e.g. in [327] or [293]. E. Saias [291] improved result (185) to c6

x x ≤ D(x, y) ≤ A(x, y) ≤ c7 u u

where c6 , c7 > 0 are constants.

359

(x ≥ y ≥ 2),

(206)

CHAPTER 4

In a recent paper [350] A. Weingartner proves that for any ε > 0, for x ≥ 3 and x ≥ y ≥ exp{(log log x)5/3+ε }, we have

1 x = xd(u) 1 + O (207) A(x, y) = xd(u) + O log x log y and a similar result for D(x, y). Let B(x, y, z) be defined by B(x, y, z) = car d{n ≤ x : S(n) ≤ x z, P(n) ≤ y}

(208)

Let a function a(u, v) defined by a(u, v) = 0 if u < 0 or v < 0, = ρ(u) if u > v ≥ 0, = ρ(v) + A(u, v) for 0 ≤ u ≤ v, where ρ is the Dickman function, and v ds . A(u, v) =

a(s − 1, v(1 − 1/s)) 2v s max u, v+1 Then in [350] it is proved that for x ≥ 3, and x ≥ y ≥ z ≥ exp{(log log x)5/3+ε } we have

log u 1 + (209) B(x, y, z) = xa(u, v) 1 + O log z log y A similar result holds true when B(x, y, z) is replaced with E(x, y, z) = car d{n ≤ x : S(n) ≤ nz, P(n) ≤ y}

10

(210)

Hooley’s function

An important arithmetic function (with applications to many parts of Number theory) is the function by C. Hooley, introduced in 1979 [185]. For n ≥ 1 and all real x put (n, x) = car d{d|n : x < log d ≤ x + 1} and (211) (n) = max (n, x) x∈R

This function satisfies the elementary inequalities 1 ≤ (n) ≤ d(n) (n ≥ 1); (mn) ≤ (m)d(n) (m, n ≥ 1) Put S(x) =

(212)

(n).

n≤x

Hooley proved that 4

S(x) x(log x) π −1 360

(213)

SPECIAL ARITHMETIC FUNCTIONS

and that

S(x) → ∞ as x → ∞ x Hooley in [185] introduced also the more general function

(214)

r (n) = max car d{d1 , . . . , dr −1 : d1 . . . dr −1 |n, x1 ,...,xr −1

xi < log di ≤ xi+1 , 1 ≤ i < r } and with Sr (x) =

(215)

r (n)

n≤x

showed that

√ r −1

Sr (x) r x(log x)

(r ≥ 2)

(216)

4 In 1982 Hall and Tenenbaum [173] improved the exponent − 1 = 0.27323 . . . π in (213) to 0.23457 . . . , and with an interesting more elaborate method, to 0.23454. Furthermore, they proved that S(x) x log x

(217)

(n) < (log n) B for almost all n,

(218)

and log 2 + ε. log 3 In 1984 and 1985 Maier and Tenenbaum [247], [248] have obtained the strong improvements

where B = log 2 −

(log log n)γ < (n) < ξ(n) log log n for almost all n,

(219)

for all constants γ < 0.28754 . . . , and all functions ξ(n) → ∞ as n → ∞. The left side of (219) implies (n) > 1 for almost all n,

(220)

and this contains essentially the famous conjecture of Erd¨os (see also (132)). In [172] it is proved that √

S(x) ε x L(log x)

2+ε

where L(u) = exp{ log u log log u}, (u ≥ 3), while for Sr (x) Maier and Tenenbaum deduced (related to (216)), x log log x Sr (x) ε,r x L(log x)αr +ε 361

(221)

(221 )

CHAPTER 4

for all r ≥ 2, where the constants αr satisfy " αr ≤ (r − 1)

1 r (r + 2) 2

(r ≥ 2)

(222)

For certain applications, it is important to introduce also the pondered sums r (n)y ω(n) (223) Sr (x, y) = n≤x

In 1985, Hall and Tenenbaum (see [172]) showed that

Sr (x, y) r x(log x)

Cr log r (log3 x 2 ) exp 1−y

for 0 < y < 1,

(234)

Sr (x, y) r,y,ε x(log x)r (y−1) L(log x)αr +ε for y ≥ 1, ε > 0,

(235)

y−1

and where the constants αr satisfy (222). In 1986 they obtained the asymptotic majorization √ (n)t y ω(n) t,y x(log x)β(t,y)−1 L(log x)2 t+o(1)

(236)

n≤x

for all t, y satisfying t ≥ 1, y ≥ t/(2t − 1), where β(t, y) = 2t y − t, and L is defined as above. In 1990 G. Tenenbaum [329] proved a partial generalization of (236): Let F(X ) be an irreducible polynomial over Z[X ] of degree > 1. Then for all t ≥ 1 one has

√

(F(n))t t x(log x)β(t)−1 L(log x)

2t+o(1)

,

x →∞

(237)

n≤x

where β(t) = 2t − t. Let M(x) denote the number of integers n ≤ x such that (n) = 1. A quantitative version of (n) = o(n) is the following (see [330]): M(x) ε x(log log x)−β+ε , where β = 1 − (1 + log log 3)/ log 3 = 0.00415 . . . 362

(238)

SPECIAL ARITHMETIC FUNCTIONS

11

Extensions of the Erd¨os conjecture (theorem) Let Bλ (n) = {m : there exists d|n and d |m such that d < d ≤ (1 + (log n)−λ )d},

where λ ≥ 0. Then the Erd¨os conjecture can be written as dB0 (n) = 1 + o(1) for almost all n

(239)

where d denotes asymptotic density. In 1995 A. Rouj [283] proved a generalization of (239) in a strong effective form. In part I he proved that √ 1 − e−cλ log log n ≤ dBλ (n) ≤ 1 − (log n)−Q(β)+o(1) (240) for almost all n, where β = ((1 + λ)/ log 2) − 1 and Q(β) = β log β − β + 1. In part II, it is shown that for almost all n we have dBλ (n) = (log n)−F(λ)+o(1) ,

(241)

where F(λ) = 0 for λ ≤ log 4 − 1, F(λ) = Q(β) for log 4 − 1 < λ ≤ log 8 − 1, F(λ) = λ − log 2 for λ > log 8 − 1. An analogue of the Erd¨os conjecture result is due to M. Car [51]. Let Fq [X ] denote the finite field of order q. Then almost all (monic) polynomials of degree n in Fq [X ] possess two distinct divisors of the same degree. (242) Let t (n) be the number of exceptional polynomials. Then for any ε > 0, t (n) = Oε (q n (log n)−b+ε ),

(243)

1 where b = (1 − ((1 + log log 3)/ log 3) = 0.00103 . . . 4

12

The divisors in residue classes and in intervals

We now state certain results on the distribution of divisors in residue classes and in intervals. Let k and l be integers satisfying 0 < l < k, (l, k) = 1. Denote by f (x, k, l) the number of integers n < x which have a divisor t satisfying t ≡ l (mod k); and F(x, k) denote the number of integers n < x which have a divisor ≡ l (mod k) for every l. Clearly F(x, k) ≤ f (x, k, l). It is easy to see that for fixed k, F(x, k) = x + o(x). If x tends to infinity with k, the questions become much difficult. In 1964 363

CHAPTER 4

Erd¨os [103] proved that for any fixed ε > 0 and k < 2(1−ε) log log x , we have, uniformly in k, F(x, k) = x + o(x) (244) In other words, if k < 2(1−ε) log log x , then almost all numbers have a divisor in every residue class l (mod k). Similarly, let k > 2(1+ε) log log x . Then uniformly in k and l, x (245) f (x, k, l) = + o(x) l 1

Finally, let k < 2 2 (1−ε) log log x and denote by d(n, k, l) the number of divisors of n which are ≡ l (mod k). Then for every η > 0, we have for every l1 and l2 , for all but o(x) integers n < x, 1−η <

2

d(n, k, l1 ) <1+η d(n, k, l2 )

(246)

Erd¨os and √ Hall [169] proved that for any fixed real c, by assuming that k = log log x , one has

∞ 1 x (247) exp − y 2 dy as x → ∞ F(x, k) ∼ 2π c 2

log log x+(c+o(1))

Let A(k, α) denote the density of integers having a divisor t ≡ 1 (mod k) in the range 1 < t < exp(k α ). In [100] Erd¨os conjectured the existence of α0 > 1 with the property that A(k, α) → 0 or 1 as k → ∞, depending on whether the fixed value of α satisfies α < α0 or α > α0 . In 1992 R. R. Hall [169] prove this conjecture with α0 = 1/ log 2, and makes further progress by allowing α to depend on k, and 1 − c log3 k/ log k, where letting α → α0 , as k → ∞. For example, if α < log 2 c > 2/ log 2, then A(k, α) → 0 as k → ∞ (248) Let A∗ (k, α) be the density of integers n such that for every (l, k) = 1, the number n has a divisor t ≡ l (mod k) in the interval exp exp( log k) < t < exp(k α ). Then 1 ξ(k) , where ξ(k) → ∞ (k → ∞), then if α = + log 2 log k A∗ (k, α) → 1 as k → ∞

(249)

The proofs use the law of iterated logarithms of Probability theory, as well as results from Probabilistic group theory. Let H (x, y, z) denote the number of integers n ≤ x, which have at least one divisor d satisfying y < d ≤ z. The first results on H (x, y, z) are due to S. Besicovitch, 364

SPECIAL ARITHMETIC FUNCTIONS

these were sharpened and extended by Erd¨os in 1952 and Tenenbaum in 1981 (see e.g. [329] and the References therein). Their result is: 1 lim H (x, y, 2y) = (log y)−δ+o(1) , (y ≤ x 1−ε ), (250) x→∞ x where δ = 1 − log(e log 2)/ log 2 = 0.08607 . . . and ε > 0 is fixed. More generally, let F(x) be an irreducible polynomial with integer coefficients and of degree > 1. If HF (x, y, z) is the number of n ≤ x such that F(n) has at least a divisor d such that y < d ≤ z, then HF (x, y, 2y) = x(log y)−δ+o(1) (251) when x, y → ∞ and y ≤ x 1−ε . In [329] it is shown also that for any η > log 4 − 1, (252) HF (x, y, 2y) > x(log x)−η 1 when x, y → ∞ in the domain y ≤ x. 2 If 1 < y ≤ z < x, then in [331] it is proved that uniformly in x, y, z we have H (x, y, z) = x(1 + O(log y/ log z))

(253)

Further, for every ε > 0 and y ε z < x, for y > y0 (ε) we have x − H (x, y, z) εx(log y/ log z) (254) √ When 3 ≤ y ≤ z ≤ x, in 1984 Tenenbaum proved (see [332]) that if z = y + y(log y)−β with a fixed β ≥ 0, then H (x, y, z) = x(log x)−G(β)+o(1) ,

(255)

where



1+β  1+β log − 1 + 1, if β < log 4 − 1 G(β) = log 2 log 2  β, if β ≥ log 4 − 1. When β = log 4 − 1 + ξ/ log log y, Hall and Tenenbaum [174] were able to show that H (x, y, z) x(log y)−G(β) (1 + max{−ξ, 0})−1 (256)

uniformly in ξ ≥ −C(log log y)1/6 for any positive constant C. When y = eu , z = eu+1 one has a connection with Hooley’s function ; in this case one has (see [332]): 1 Uniformly for 3 ≤ u ≤ log u, 2 H (x, y, z) = H (x, eu , eu+1 ) = xu −δ exp(O log u log log u) (257) where δ = 0.08607 . . . 365

CHAPTER 4

13

Divisor density and distribution (mod 1) on divisors

We shall consider the notions of divisor density of a sequence, due to R. R. Hall [170], [171], as well as that of distribution (mod 1) on divisors, due to I. K´atai [205]. A sequence A has divisor density D A = z, if there is a sequence of integers S of asymptotic density 1 such that for d(n, A) = car d{d|n : d ∈ A} one has d(n, A) ∼ zd(n) as n → ∞ for n ∈ S (258) (χ A (n)−z)/4(n) ·n, where χ A is the characteristic function Let gk (x, A) = n<x,k|n

of the set A, and (n) is the number of prime divisors of n, counted with multiplicity. G. Tenenbaum [333] has shown that a necessary and sufficient condition for a sequence A to have divisor density D A = z is that |gk (x, A)| = o( log x) as x → ∞ (259) k<x

For example, the set A z,α = {d : (log d)α ≤ z (mod 1)} has divisor density z for all real α > 0 and all 0 < z < 1; i.e. the function f (x) = (log x)α is equidistributed (mod 1) with respect to divisors, conjectured by Hall. In fact, it was K´atai who first studied equidistribution on divisors of an additive arithmetic function satisfying for all 0 ≤ z ≤ 1, car d{d|n : f (d) ≤ z

(mod 1)} = (z + o(1))d(n).

K´atai proved that an additive function f satisfies this property iff ν f ( p) 2 / p = +∞ (ν = 1, 2, . . . )

(260)

p prime

where x denotes the distance of x to the nearest integer. For an arbitrary arithmetic function f , (259) has the following consequence: e(ν f (n)) = o( log x) (ν = 1, 2, . . . ) (261) n4(n) n<x k<x n≡0

(mod k)

where e(x) = exp(2πi x), is the necessary and sufficient condition for a function f to be equidistributed on divisors. For the function f (x) = (log log x)α , with α > 1, see [170], for f (x) = θ x (θ ∈ R \ Q), see [94]. Another function, f (x) = x α for α ∈ R+ \ N was studied by Hall and Tenenbaum. 366

SPECIAL ARITHMETIC FUNCTIONS

14

The fractal structure of divisors

When d runs through the divisors of a typical integer n, the numbers log d/ log n exhibit a fractal-like behaviour on the unit interval. This has been studied in 1993 by M. Mend`es France and G. Tenenbaum [258]. More generally, consider a dynamical system of points X = (X n )n≥1 , where for each n, X n = {0 = x0(n) < x1(n) < · · · < xk(n) = 1} with kn → ∞ as n → ∞. n Let ε = (εn )n≥1 such that εn → 0 as n → ∞, and put s(ε) = s(ε, X ) = lim sup n→∞

log(1/εn ) log kn

Let ε(α) = ε(α, X ) = (kn−α )n≥1 for α > 0, and consider # (n) 1 1 (n) x j − εn , x j + εn ∩ [0, 1] µn = µn (εn ) = meas 2 2 0≤ j≤kn where meas is the Lebesgue measure on R. The degree of resolution de X via ε is defined by g(ε, X ) = lim sup n→∞

log(µn /εn ) , log kn

while the resolution function on R+ is defined by ρ(α) = ρ(α, X ) = π(ε(α), X ) = where π(ε, X ) = lim sup n→∞

1 g(ε(α), X ), α

(262)

log(µn /εn ) log(1/εn )

Since µn ≤ min{1, εn kn }, one has

1 g(ε, X ) ≤ min{1, s(ε)}, ρ(α, X ) ≤ min 1, α

(α > 0)

(263)

When the function ρ(α) is constant on an interval (0, α0 ], then the value of ρ(α) for 0 < α ≤ α0 , will be called as the fractal dimension of X . By (263), clearly, 0 ≤ dim f X ≤ 1 367

CHAPTER 4

On the other hand, in order to introduce another dimension, for α ∈ [0, 1] put (n) α Hn (α, X n ) = (x j+1 − x (n) j ) 0≤ j
and define the Hausdorff dimension of X by log H (α, X n ) dim X = inf α ≥ 0 : lim sup =0 log kn n→∞

(264)

A connection between the Hausdorff dimension and the resolution degree is given by: g(ε, X ) ≤ s(ε, X ) dim X

(265)

for all X and ε. Moreover, sup ρ(α, X ) = dim X α>0

Let now Dn =

log d : d|n log n

(n = 2, 3, . . . )

(266)

A subsequence {Dn : n ∈ A} of (266) will give a system D = D(A) Then Mend`es France and Tenenbaum prove that there exists a sequence A of density 1 such that for all α > 0, 1 ρ(α, D(A)) = min log 2, (267) α so there exists a sequence A of density 1 such that dim D(A) = log 2

(268)

Particularly, from (267), (268) it results that for a suitable sequence A of density 1 dim f D(A) = dim D(A) = log 2

(269)

If the distribution of the prime divisors of a typical integer n were sufficiently regular, then this dimension would be 1; the fact that this dimension is strictly less than 1 is therefore evidence of some degree of irregularity in the distribution of the prime divisors. 368

SPECIAL ARITHMETIC FUNCTIONS

15

The divisor graphs

In what follows we will study graphs connected to the set of divisors of a number. Given a number n one can generate a graph D(n) that reflects the structure of divisors of n as follows. The vertices of the graph represent all the divisors of n, each vertex is labelled by a certain divisor. If r and s are two divisors of n and r > s, then there is an edge between the vertices s and r iff s divides r and the ratio r/s is a prime number (see K. R. Bhutani and A. B. Levin [26]). As in the theory of graceful graphs (see J. A. Gallian [140] for a survey on various graph labelings), we label such an edge by the difference r − s of the labels of its vertices. Let S D(n) and S D(n) denote the sum of the labels of all edges, and all edges except the ones terminating at n, respectively. It is immediate that

n (270) n− S D(n) = S D(n) − p p|n where p denotes a prime. If n = p1a1 . . . prar is the prime factorization of n, then r S D(n) = ( piai − 1)

i=1

1≤ j≤k, j =i

a +1

pj j

−1

pj − 1

(271)

A number n is called graceful if S D(n) = n

(272)

In [26] it is shown that n is graceful iff n = 4q, where q is an odd prime. (273) As a corollary, the only perfect number (see Chapter 1), which is graceful is 28. In a series of papers, beginning from 1995, E. Saias has studied another graph, called as the divisorial graph the graph of a relation R f defined on the positive integers by (274) a R f b ⇔ a|b or b|a Let R xf be the restriction of R f to the integers ≤ x. For such integers, another relation Rgx will be defined as a Rgx b ⇔ [a, b] ≤ x

(275)

where [a, b] denotes the l.c.m. of a and b. If n i R xf n i+1 for i = 1, 2, . . . , k − 1 and n i = n j for i = j, then n 1 , . . . , n k is said to be a chain of length k for the relation R xf . Let f (x) denote the maximum 369

CHAPTER 4

value of k, and define similarly a quantity for the relation Rgx . In [293] Saies proved that cx/ log x ≤ f (x) ≤ g(x) ≤ c x/ log x, x ≥ 2, (276) for certain positive constant c and c . In [113] Erd¨os and Saias considered the minimum number φ f (x) of chains for the relation R xf that are required in order that every positive integer ≤ x belongs to at least one such chain, and the corresponding number φg (x) for the relation Rgx . The analogues quantity when the chains for R xf , Rgx are pairwise distinct is denoted by φ ∗f (x), φg∗ (x), respectively. The following results hold true: c2 x c1 x ≤ φg (x) ≤ φ f (x) ≤ , log x log x

x ≥2

(277)

x , x ≥2 (278) 2 Let R(x) denote the maximum number of integers r ≥ 2 which can be written as n i+1 ni r= or r = for R xf . Then in [294] the double-inequality ni n i+1 √ √ (279) (1 − o(1)) 8x ≤ R(x) < 8x c3 x ≤ φg∗ (x) ≤ φ ∗f (x) ≤

is proved. Let f (x, y) (resp. g(x, y)) denote the maximum number of a union of y chains of R xf (resp. Rgx ). In a recent paper, Saias [295] shows that c3 x

log 2y log 2y ≤ f (x, y) ≤ g(x, y) ≤ c4 x for x ≥ y ≥ 1, log 2x log 2x

(280)

with c3 , c4 > 0 constants. For the graph Rgx , a t-chain of integers ≤ x of length k is a k-tuplet (a1 , a2 , . . . , ak ) of integers ≤ x, ai = a j for i = j, such that for all 1 ≤ i < k one has [ai , ai+1 ] ≤ xt. Let h(x, t) be the maximum length of a t-chain of integers ≤ x. Then c5 x

log 2t log 2t ≤ h(x, t) ≤ c6 x for x ≥ t ≥ 1, log 2x log 2x

(281)

where c5 , c6 are positive constants. This improves a result of C. Pomerance [277] to the effect that lim+ lim sup

a→0

x→∞

h(x, x a ) =0 x

370

(282)

SPECIAL ARITHMETIC FUNCTIONS

We note that Erd¨os, Freud and Hegyv´ari [105] have shown that for all function o(1) one has h(x, x 1−o(1) ) ∼ x as x → ∞ (283)

4.3 1

Arithmetic functions associated to the digits of a number The average order of the sum-of-digits function

Let q > 1 be a fixed integer and denote by sq (n) the sum of digits of n written in base q, i.e. m sq (n) = ai (1) i=0

where n =

m

ai q i for some integer m and 0 ≤ ai < q, am > 0. We will write

i=0

s(n) = s10 (n) for the decimal expansion (i.e. when q = 10). This function occurs in many problems of Number theory or Mathematics. It is sufficient to mention the well known Legendre formula giving the exponent e p (n!) of a prime p in n! (see e.g. [177]): e p (n!) =

n − s p (n) p−1

(2)

For a similar result, which can be attributed to Kummer (see J. W. Sander [298])

we note that 2n = s2 (n), (3) e2 n

2n where denotes a binomial coefficient. (For prime factors of binomial coeffin cients or consecutive integers, see [260], Chapter 12). It was L. E. Bush [46] who first showed that q −1 x log x as x → ∞ (4) sq (n) ∼ 2 log q n≤x In 1949 L. Mirsky [259] proved that q −1 x log x + O(x), sq (n) = 2 log q n≤x

(5)

which is best possible as can be seen by taking x = (1 − q)q l . A weaker error term, namely O(x log log x) was discovered a year earlier by R. Bellman and H. N. 371

CHAPTER 4

Shapiro [20]. P. Cheo and S. Yien [54] have rediscovered (5), and further showed that if bm (x) = car d{y ≤ x : sq (y) = m}, then

1 log x m bm (x) ∼ as x → ∞ (6) m! log k M. P. Drazin and J. S. Griffith [76] have studied more generally the sum of kth powers of digits of n written in base q: m

sq,k (n) =

(7)

aik

i=0

for a positive integer k, and by putting Aq,k (n) =

n−1

sq,k (m),

δq,k =

m=1

n−1 1 mk , q m=1

Fq,k (n) − Aq,k (n) n log n , q,k (n) = log q nδq,k they proved that for all q, k, n one has: Fq,k (n) = δq,k

q,k (n) ≥ 0,

2,k (n) < 1,

(8)

and for all q > 3,

q − 1 log(q − 1) · (9) q −2 log q For q = 2 (when one obtains the most precise results), these have been rediscovered by M. D. Mc Ilroy [186] as well as by I. Shiokawa [308]. For results on Aq,k (n) when q = 2 and k ≥ 0 see also K. B. Stolarsky [319]. In 1968 J. R. Trollope [344] discovered the following result. Let g(x) be periodic of period one and defined on [0, 1] by  1 1   x, if 0 ≤ x ≤   2 2 g(x) =     1 (1 − x), if 1 < x ≤ 1 2 2 q,k (n) <

and put f (x) =

∞ 1 g(2i x) i 2 i=1

(10)

Now, if n = 2m (1 + x), where 0 ≤ x < 1, one has n−1 k=0

s2 (k) =

n log n − E 2 (n), 2 log 2 372

(11)

SPECIAL ARITHMETIC FUNCTIONS

where

E 2 (n) = 2

m−1

log(1 + x) − 2x . 2 f (x) + (1 + x) log 2

In 1975 H. Delange [73] has extended by another method Trollope’s results as follows: There exists a function F : R → R of period one, which is continuous and nowhere differentiable, such that

1 m−1 q −1 log m sq (n) = log m + F m n=0 2 log q log 2 for all m ≥ 1. In fact, F(x) =

(12)

q −1 (1 + [x] − x) + q 1+[x]−x h(q x−[x]+1 ), 2

where h(x) =

∞ g(q r x) r =0

and

g(x) = 0

x

qr

q −1 dt [qt] − q[t] − 2

Delange determined also the explicit Fourier series of F. More generally, for all real numbers x ≥ 1 we have

q −1 log x − h(x) + (1 + [x] − x)sq ([x]), sq (n) = x log x + x F 2 log q log q n≤x

(13)

where [x] denotes the greatest integer ≤ x. We note that for q = 2, relation (12) implies Trollope’s result (11). C. Cooper and R. E. Kennedy [58] have applied Trollope’s method to extend (11) to the general case: 1 1 and on 0, be equal to the piecewise Let g(x) be periodic of period q − 1 q − 1

a (q − a)a linear function connecting the points , where a is a nonnegative 2 q −q 2q integer and 0 ≤ a ≤ q. Let f (x) =

∞ 1 g(q i x) i q i=0

373

CHAPTER 4

and let n = q m (1 + (q − 1)x), with 0 ≤ x < 1. Then

n−1

sq (k) =

k=0

where

q − 1 n log n · − E q (n), 2 log q

(14)

q −1 (1 + (q − 1)x) logq (1 + (q − 1)x)− 2

(am − 1)am , −am (1 − am + (q − 1)x) − 2 m where n = q m (1 + (q − 1)x) = ai q i = am q m + n m−1 is the base q representation E q (n) = q

m

f (x) +

i=0

of n and logq denotes the base q logarithm (in fact logq t =

log t ). log q

In 1986 P. Kirschenhofer [220] showed that the variance 2 1 2 1 s (n) − s2 (n) Vx = x n<x 2 x n<x satisfies

1 log x log x Vx = +δ 4 log 2 log 2

(15)

where δ(u) is a continuous, periodic function of period 1. More general results have been obtained by P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy [153], as well as by J. M. Dumont and A. Thomas [93]. Let σ be a primitive substitution on a finite alphabet A, whose largest eigenvalue γ γ f fulfills θ > 1, and s (n) denotes f (m i ) (where for the integer n, |σ i−1 (m i )| i=1

i=1

is the unique admissible representation of n, and f an application from A∗ to R). Dumont and Thomas prove that there exist a real number α and the functions G k,h with 1 f (s (n) − αl)2k = (2k − 1)(2k − 3) . . . 1 · β k l k + x n<x l h G k,h (l) + η(x), (16) + h
where l = logθ x, lim η(x) = 0, β is an explicit constant; and such that the x→∞ functions G k,h are 1-periodic and continuous. 374

SPECIAL ARITHMETIC FUNCTIONS

For the odd moments (i.e. 2k − 1 in place of 2k), the first term of the right side of (16) disappears. The functions G k,h are nowhere differentiable if β = 0 and α = f (ω), with ω being the empty word. Result (16) has applications to some models in Theoretical physics. In 1986 J. Coquet [66] established (by using the Delange method) the formula

1 log x k log x h k s2 (n) = + Fk,h (l) x n<x 2 log 2 log 2 h
(17)

where k ≥ 1 is an integer and Fk,h are some periodic bounded functions. Dumont and Thomas [93] prove that Fk,h are continuous and nowhere differentiable. T. Okada, T. Sekiguchi and Y. Shiota [267] have studied the functions Fk,h by the application of binomial measure, giving further explicit formulas. Moreover, for primitive substitutions holds true 1 f (s (n))k = α k l k + l h Fk,h (l) + ε(x) x n<x h
(18)

where ε(x) → 0 as x → ∞ (see the notations of (16)), α ∈ R, and Fk,h are 1-periodic continuous functions. For example, when k = 2 one obtains:

1 log x log x q − 1 2 log x 2 log x 2 η1 + η2 , (sq (n)) = + x n<x 2 log q log q log q log q

(19)

where η1 , η2 are continuous and nowhere differentiable functions of period 1. Let s(n) = s10 (n) be the sum of the base 10 digits of the positive integer n. R. E. Kennedy and C. Cooper [59] have shown that for any integer k, k 9 1 1 k (s(n)) = logk x + O(logk− 3 x) x n≤x 2

(20)

They conjecture the better error term O(logk−1 x), and in [60] this is established in the special case when x is a power of 10 (i.e. x = 10a ). The conjecture is proved to be true by T. C. Brown [43] in the more general case provided x is restricted to the set of those positive integers having at most m nonzero digits in their base 10 representations. $q % or more) of n in base q. Let L q (n) denote the number of ”large” digits (i.e. 2 Recently Mirsky’s theorem (5), as well as Trollope’s (11) and (14) have been proved 375

CHAPTER 4

to the function L q (n) by C. Cooper and R. E. Kennedy [61]: $q % x log x L q (n) = 2 + O(x) q log q n≤x n−1

(21)

$q % L q (k) =

k=0

n log n 2 − E q∗ (n), q log q

(22)

where E q∗ (n) is defined similarly to that in (14).

2

Bounds on the sum-of-digits function

Relations (8) and (9) give certain lower and upper bounds for expressions related to the sum of digits function. As in (8) and (9), let Aq (n) = Aq,1 (n) =

n−1

sq (m),

m=1

and put

1 log n n , S(q, n) = 2 Aq (n) − (q − 1) 2 log q

where [·] denotes the integer part. D. M. E. Foster [127] obtained the upper bound S(q, n) < q − 1, n

(q ≥ 2)

(23)

which is best possible. For q = 2, 3, 4, 5, 7 one has the best possible lower bounds (see [128]) 2 S(2, n) − < ; 3 n

2 S(3, n) − < ; 7 n

S(5, n) 7 ; − < 13 n

−

9 S(4, n) < ; 23 n

(24)

6 S(7, n) − < 19 n

Clearly every positive integer n ≡ 0 (mod q) is of the form n = n m , where n m = a0 q t0 + a1 q t0 +t1 + a2 q t0 +t1 +t2 + · · · + am q t0 +t1 +···+tm

(25)

for some m ≥ 0, t0 = 0, positive integers t1 , . . . , tm and non-zero coefficients a0 , a1 , . . . , am ∈ {1, 2, . . . , q − 1}. 376

SPECIAL ARITHMETIC FUNCTIONS

For convenience of notation, given such an integer n, introduce n −1 = 0, n 0 = a0 and n i = a0 + a1 q t1 + · · · + ai q t1 +···+ti for 1 ≤ i ≤ m. Then S(q, n m ) has a simple form: S(q, n m ) =

m

ar (ar − 1)q t0 +···+tr +

r =0

m

{2ar − (q − 1)tr }n r −1

(26)

r =0

It is easily verified that for s, a positive integer, S(q, n m ) S(q, q s n m ) = , s q nm nm so that we may assume that n is of the form (25) for some integer m ≥ 0. The inequality m+1 S(q, n m ) ≤ (q − 1) 1 − m+1 nm q −1

(27)

appears in [127]. Let βq be the unique odd integer satisfying 3q −

8q 2 − 9q + 1 ≤ βq ≤ 3q − 8q 2 − 9q + 1 + 2,

unless the even integer 8q 2 − 9q + 1 is a perfect square. Put q 2 − 3q − (βq − 1)(βq − 5) hq = 2(3q − βq + 1) In [128] Foster shows that for all odd q ≥ 9, S(q, n m ) > −h q nm

(28)

For q = 3, 5, or 7 there is a more precise result, namely S(q, n m ) ≥ −h q (m), nm

(29)

where h 3 (m) =

6(3m − 1) , 7 · 3m+1 − 5

h q (m) =

(q 2 − 3q + 4)q m−1 − 1 2{(3q − 2)q m−1 − 1}

377

(q = 5, 7)

CHAPTER 4

Let 1 1 n ∗m = n m−l + (q − 1)(q m−l+1 + q m−l+2 + · · · + q m−2 ) + (q − βq )q m−1 + q m , 2 2 where n m−l =

m−l

ar q t0 +···+tr and q is odd.

r =0

If m − l remains fixed, then (see [128]) S(q, n ∗m ) → −h q n ∗m

(m → ∞)

(30)

Similar results when q is even are proved by Foster in [129]. By considering the change of the sum of digits by multiplication, put S(x) = car d{n ≤ x : s2 (kn) ≥ s2 (n) for all k ≥ 1}, K. B. Stolarsky [320] showed e.g. that lim inf x→∞

1 S(x) ≤ x 2

(31)

Let 3 (n) = s2 (3n) − s2 (n) and put K N (x) = car d{n ∈ {0, 1, . . . , 2 N − 1} such that 31/2 3 (n)N −1/2 < x}. Then I. K´atai [206] proved that for all x ∈ R x 1 lim 2−N K N (x) = exp(−u 2 /2)du (32) N →∞ 2π −∞ More generally (32) can be extended to k (n) = s2 (kn) − s2 (n), where k is not a power of two, and with k (n) < xβk N 1/2 (for certain explicit βk ) in the definition of K N (x); see L. Dringo and I. K´atai [77]. K. B. Stolarsky [321] considered the sequence (yn ) defined by y1 = given positive integer; yn+1 = yn + s2 (yn ) (n ≥ 1). Then the following is true: n yn = log n + O(n(log n log log n)1/2 ) 2 I. Shiokawa [308] proved that 1 x log x − lim inf s2 (n) = 0 x→∞ x 2 log 2 n≤x lim sup x→∞

1 x

x log x log 3 s2 (n) = 1 − − 2 log 2 n≤x 2 log 2 378

(33)

(34)

SPECIAL ARITHMETIC FUNCTIONS

while A. Stein and I. E. Stux [316] defined s2 (s2 (n)), A2 (x) = n≤x

c(x) = A2 (x) proving that 1 +O 2

log log log x log log x

2 log 2 , x log log x

3 +O 2

≤ c(x) ≤

log log log x log log x

(35)

The exponential sum of digit counting function was defined in 1899 by J. W. L. Glaisher [145] as φz (x) = z s2 (m) for z = 2. m<x

Glaisher showed that φ2 (n) represents the number of odd integers in the first n rows of Pascal’s triangle. (36) log(1 + z) φz (x) and define ψz (x) = Let t = tz = , where z > 0, z = 1 is a real log 2 xt number. For z = 2, upper and lower bounds for ψz were obtained by Stolarsky [319] and H. Harborth [175]. In the general case, A. H. Stein [314] proved that 0 < lim inf ψz (x) ≤ 1 ≤ lim sup ψz (x) < 2 x→∞

x→∞

lim inf ψz (x) < lim sup ψz (x) x→∞

(37)

x→∞

For z = 2 if we let qn = min{ψ2 (x) : 2n < x < 2n+1 }, then lim inf ψ2 (x) ≥ qn − x→∞

1 log 3 · 2n log 2

(38)

For properties of the more general exponential sum with s2 (m) replaced to sq (m), see A. H. Stein [315]. We note that the case z < 0, particularly z = −1 (involving the Thue-Morse sequence defined by tn = (−1)s2 (n) (n ≥ 1)) will be studied later.

3

The sum of digits of primes

In what follows we will study the sum of digits of primes. In 1961 W. Sierpinski [310] proved that for each q > 1, lim sup sq ( pn ) = +∞, n→∞

379

(39)

CHAPTER 4

where pn denotes the nth prime number. (39) implies that s1 ( pn+1 ) > sq ( pn ) for infinitely many n

(40)

In 1962 P. Erd¨os [104] solves an open question by Sierpinski, namely sq ( pn ) > sq ( pn+1 ) for infinitely many n

(41)

By assuming Schinzel’s H -Hypothesis, in [310] it is shown also that sq ( pn ) > sq ( pn+1 ) > sq ( pn+2 ) for infinitely many n;

(42)

sq ( pn ) = sq ( pn+1 ) = sq ( pn+2 ) for infinitely many n

(43)

and A. Schinzel has shown (see [310]) that accepting the H -hypothesis, for each given positive integer m, there exist positive integers n 1 , n 2 and n 3 such that s1 ( pn 1 +1 ) < s1 ( pn 1 +2 ) < · · · < sq ( pn 1 +m ) sq ( pn 2 +1 ) = sq ( pn 2 +2 ) = · · · = sq ( pn 2 +m ) sq ( pn 3 +1 ) > sq ( pn 3 +2 ) > · · · > sq ( pn 3 +m )

(44)

It is interesting to note that the proof of (41) is based on a strong theorem on the distribution of primes (Hoheisel-Ingham theorem on π(x), see [260]), and an elementary proof is missing. I. Shiokawa [309] proved that for all q > 1, 1 x sq ( p) = (q − 1) (45) + O(x(log log x/(log x))1/2 ) 2 log q p≤x where p runs through the primes. I. K´atai [207] was able to show that k sq ( p) − q − 1 log x x(log x) k2 −1 , 2 log q p≤x

(46)

where k is a positive integer. For k = 1 this improves relation (45). E. Heppner [180] considered, more generally the values of sq (n) for n ∈ B, where B ⊂ N is a subset of positive integers. Let B(x) be the counting function of B, and assume that log(x/B(x)) = o(log x). Then  

1/2  x       log log x + log      q − 1 log x B(x)    sq (n) = · B(x) 1 + O       2 log q log x   n≤x,n∈B     (47) 380

SPECIAL ARITHMETIC FUNCTIONS

For B = { p1 , p2 , . . . , pn , . . . } (i.e. the sequence of primes), all conditions are satisfied and (47) gives (45). For the prime values taken by sq (n), R. Warlimont [347] proved that car d{n ≤ x : sq (n) = prime}

4

x . log log x

Niven numbers An integer n is called a Niven number if (48)

s(n)|n,

i.e. if it is divisible by its (base 10) digital sum. These numbers were first considered by I. Niven [266] at a Conference on number theory. In fact this concept was introduced and investigated by R. Kennedy, T. Goodman and C. Best [217] and R. Kennedy [214]. Some examples of Niven numbers are 8, 12, 180, 4050. The set N of these numbers is clearly infinite since 10k ∈ N for any k = 1, 2, . . . Let N (x) be the number of elements of N not exceeding x. The natural density of Niven numbers is zero; that is N (x) =0 (49) lim x→∞ x This was shown in 1984 by R. E. Kennedy and C. N. Cooper [215]. Another proof by them can be found in [62], where general conditions are given for a set to have natural density equal to zero. Let Nk = {n ∈ N : s(n) = k}. Then (see C. N. Cooper and R. E. Kennedy [63]) Nk (x) ∼ C(k)(log x)k

(x → ∞)

(50)

which was a partial answer to the determination of the asymptotic formula for N (x). For similar particular results, see [64]. The first nontrivial bounds for N (x), by elementary methods are due to J.-M. De Koninck and N. Doyon [226]: For given ε > 0, x 1−ε N (x)

x log log x log x

(51)

Later, using complex variables as well as probabilistic arguments, De Koninck, Doyon and K´atai [228] have deduced that x (52) N (x) = (c + o(1)) log x where c =

14 log 10 = 1.1939 . . . 27 381

CHAPTER 4

Let T (x) denote the number of Niven number n ≤ x such that n + 1 is also a Niven number. Recently De Koninck and Doyon [227] showed that T (x)

x log log x (log x)2

(53)

Let T = {n ∈ N : n + 1 ∈ N }. Then (52) implies 1

< +∞,

(54)

1 = (c + o(1)) log log x, n

(55)

n∈T

n

while (52) implies that n≤x,n∈N

where c is the same as in (52). Consecutive Niven numbers have been first studied by Kennedy [214] where it was shown that no more than 21 consecutive Niven numbers is possible. Later, Cooper and Kennedy [65] replaced 21 with 20, and showed in fact the existence of infinitely many such sequences. This result is best possible. (56) Let q > 1. In 1994 H. G. Grundman [160] defined a q-Niven number n as satisfying sq (n)|n (57) For q = 10 one obtains the usual Niven numbers. Grundman proved that the length of a sequence of consecutive q-Niven numbers is at most 2q. (58) He found also the smallest sequence of 20 consecutive Niven numbers, and conjectured that there exists a sequence of consecutive q-Niven numbers of length 2q for each q > 1. (59) For q = 2, 3, T. Cai [47] showed that there exists an infinite family of sequences of consecutive q-Niven numbers of length 2n. (60) De Koninck and Doyon [227] denote by n k the smallest positive integer n such that the interval [n, n + k − 1] does not contain any Niven number (k ≥ 1 given), and prove: n k < (100(k + 2))k+3 (61) For each l ∈ [2, 20] let Tl (x) = car d{n ≤ x : n + 1, n + 2, . . . , n + l − 1 are all Niven numbers}. De Koninck and Doyon conjecture that Tl (x) 382

x loglx

(62)

SPECIAL ARITHMETIC FUNCTIONS

A number n is called all-Niven (see E. Weisstein [351]) if sq (n)|n for all q > 1. Then A. Kertesz proved that the only all-Niven numbers are 1, 2, 4 and 6. (63) Another special Niven numbers are the so-called sum-product numbers n for which n = s(n) · a(n), where a(n) = product of digits of n. E.G. 135 = (1 + 3 + 5) · (1 · 3 · 5). D. Wilson (see [312] (sequence A038369), [352]) proved that there are only three such numbers, namely 1, 135 and 144. (64)

5

Smith numbers

Let 1 < n = p1a1 . . . prar (r ≥ 2) be the prime factorization of the composite number n. A. Wilansky [354] defined n to be a Smith number if s(n) = a1 s( p1 ) + · · · + ar s( pr ) = S p (n)

(65)

i.e. numbers whose digit sum is equal to the sum of the digits of its prime factors (counted with multiplicity). For example, 493775 = 3 · 52 · 65837 is a Smith number, since 4 + 9 + 3 + 7 + 7 + 5 = 3 + 2 · 5 + 6 + 5 + 8 + 3 + 7(= 42). The first few such numbers are 4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, . . . (see [312], sequence A006753). Wilansky found 360 Smith numbers n ≤ 105 , and S. Oltikar and K. Wayland [268] have noted that relatively large Smith numbers are easily generated from primes whose digits are all 0’s or 1’s, but only a small number of such primes are known. In 1987 W. L. Mc Daniel [70] proved the existence of infinitely many Smith numbers. (66) Mc Daniel introduced also the concept of a k-Smith number which satisfies S p (n) = k · s(n),

(67)

where S p (n) is defined in (65). For example, n = 104 = 23 · 13 is a 2-Smith number as S p (104) = 10 = 2 · s(104). The number n = 402 = 2 · 3 · 67 is 3-Smith, and there are 21 2-Smith numbers, three 3-Smith numbers, one 7-Smith number, etc. for n ≤ 1000. There are infinitely many k-Smith numbers (see [70]). (68) For tables on k-Smith numbers, see [312] (e.g. sequences A050224 for k = 2 and A050225 for k = 3). Let Rn be the nth repunit number. It seems that the largest Smith number is (see [353]) (69) 9 · R1031 · (104594 + 3 · 102297 + 1)1476 · 103913210 For special sets of Smith numbers, see S. Yates [356], [357], Mc Daniel [71], Mc Daniel and Yates [72] (where one can find extensions to base q > 1, too). 383

CHAPTER 4

R. L. Bishop [29] says that n is a Smith-Jones number, if s(n) ≡ S p (n) (mod 9) p1a1

(70)

Let p1 and q1 be such numbers that s( p1 ) = s(q1 ), and suppose that n = . . . prar is a Smith-Jones number. Then m = q1 p2a2 . . . prar is also a Smith-Jones number. (71) If a composite number n satisfies s(n) = s( p1 ) + · · · + s( pr )

(72)

(in place of (65)), then it is called a hoax number, see [312], sequence A019506. The first few hoax numbers are 22, 58, 84, 85, 94, 136, 160, . . . Are there infinitely many? Consecutive Smith numbers (n, n + 1) are also called Smith brothers. The first few Smith brothers are (728, 729), (2964, 2965), (3864, 3865), (4959, 4960), . . . It is not know if there exist infinitely many such pairs. By making investigations with a computer, J. L. James [193] conjectures that there do not exist Smith numbers which are perfect numbers. Monica sets and Suzanne sets are defined in M. Smith [313]. For example, let Sn = {m composite : n|s(m) and n|S p (m)},

(73)

where S p is defined by (65). Then Sn is called the nth Suzanne set. Every Suzanne set has an infinite number of elements. (74) Similarly, let Mn = {m composite : n|s(m) − S p (m)} Then Mn has an infinite number of elements, Mn ⊂ Sn , and if m is a Smith (74 ) number, then m ∈ Mn for all n. If m is a k-Smith number, then m ∈ Mk−1 .

6

Self numbers

In 1963 D. R. Kaprekar [202] introduced the concept of a self-number. Let q > 1 be fixed integer. Then a positive integer n is called to be a q-self-number if the equation n = m + sq (m) (75) has no solution in positive integers m. Otherwise, it is called a q-generated number. In 1973 V. S. Joshi [200] proved that if q is odd, then n is a q-self number iff n is odd. Every even number in an odd base is a generated number. (76) 384

SPECIAL ARITHMETIC FUNCTIONS

This property appears also in Solution of Problem E2408, see [282]. The set of q-self numbers is infinite, and its asymptotic density exists, and is positive (when q 1 is odd, by (76) this density is ). (77) 2 More general results are treated by R. Guaraldo [164]. In part II of [164], the above density is explicitly calculated for all bases q > 1. For example, when q = 10, the density is approximately 0.9022222 . . . (78) When q = 2, a more precise result has been obtained by U. Zannier [359]: car d{n ≤ x, n = 2 − self-number} = L x + O((log x)s2 ([x]))

(79)

where L > 0; car d{n ≤ x, n = 2 − generated} = L x + O(log2 x)

(80)

More general results for the number of solutions of m + sq (m) ≤ x are proved by M. B. S. Laporta and E. Laserra [240]. For example, when b is odd, a is arbitrary, car d{m + sq (m) ≤ x, m + sq (m) ≡ a

(mod b)} =

L x + O(x µ ), b

(81)

where L is as in (80), and 0 < µ < 1. R. B. Patel [271] proved that n = 4q + 2 is a q-self-number iff 2|q, q ≥ 4; 2q is a q-self-number in every even base q ≥ 4; q 2 + 2q + 1 is a q-self-number in an even base q ≥ 4; 3q + 1 is a q-self-number iff 2|k, k ≥ 4; q 2 + 3q + 2 is a q-self-number iff 2|q, q ≥ 4. (82) For other classes, see also T. Cai [48]. An integer n is said to be universal generated if it is q-generated for every base q ≥ 2. E.g. 2, 10, 14, 22, 38, 60, . . . are such numbers. Let G(x) be the counting function of universal generated numbers. Then (see T. Cai [48]) √ G(x) ≤ 2 x (83) A generalization of q-self-numbers can be found in [164], part I. Let n =

m i=0

ai q i

be the unique representation of n in base q. Let f (a, i) be a nonnegative, integervalued function of the digit a, and the place where the digit occurs. m f (ai , i), and let Put s f (n) = i=0

T f (n) = n + s f (n),

R f = {n : n = T f (x) for some x},

C f = {n : n = T f (x) for any x}. 385

CHAPTER 4

A number n is f -self-number iff n ∈ C f , and it is a f -generated number, iff n ∈ Rf. Let f (a, i) satisfy the following conditions: f (0, i) = 0

(i = 0, 1, 2 . . . )

f (a, i) = o(q i ) for 1 ≤ a ≤ q − 1

(84) (85)

Then the density of f -generated numbers exists. When f depends only on a (1 ≤ a ≤ q − 1), and f (0) = 0, then the density L f exists, and L f < 1 if and only if the application T f is not one-one. (86) It is interesting to note that if one assumes only the second condition of (84), then the density may not exist. Let e.g. f (a, i) = 0 if i is even; = q i if i is odd. Then the density of R f doesn’t exist. Now, a result of different nature is the following ([164], I): If f (a, i) = O(q i /i 2 log2 i) for all a, (87) (88) then the density of R f exists. Similar results hold true, when considering ”factorial base” in place of the q-adic expansion. It is known that (see e.g. G. Faber [120]) n may be uniquely represented in the form n = a1 · 1! + a2 · 2! + · · · + am · m! (0 ≤ ai ≤ i). Let f (a, i) be a nonnegative integer-valued function of i for each ”digit” a, 0 ≤ a ≤ i (i = 1, 2, . . . ) and define s f (n), T f (n), R f , C f as above. Now, in place of (84) assume that (see [164], III) f (0, i) = 0

(i = 1, 2, . . . )

f (a, i) = o(i!) uniformly in i(i.e. sup{ f (a, i), 0 ≤ a ≤ i} = o(i!)).

(89)

Then the density of R f exists. (90) As a corollary, one obtains that if f (a, i) = F(a) depends only on a, and F(0) = 0, F(a) = o(i!) uniformly in i for all ”digits” a, then the density of R f exists, and (91) this density is < 1 iff T f is not one-one. In the case F(a) = a (when T f (n) = n+ sum of ”digits” of n), the density of R f is 0.879888 . . . (92) Similarly, assume that f (a, i) = O(i!/i 2 log2 i) uniformly in i Then the density of R f exists. More general base expansions will be considered in further sections. 386

(93) (94)

SPECIAL ARITHMETIC FUNCTIONS

7

The sum-of-digits function in residue classes

The distribution of sq (n) in residue classes was first studied by A. O. Gelfond [141], who for N ∈ N and r ∈ Z considered the sets Sr,m (N ) = {n < N : sq (n) ≡ r

(mod m)}

where q, m ∈ N are fixed positive integers, q ≥ 2, (m, q − 1) = 1. Then 1 1 car d Sr,m = (95) lim N →∞ N m for all r , i.e. Sr,m is equidistributed in residue classes mod r . See also M. N. J. Fine [125]. Gelfond conjectured that the joint distribution of sum of digit functions with different bases satisfies N + O(N λ ) m1m2 (96) for a certain λ < 1, where (m 1 , q1 − 1) = 1, (m 2 , q2 − 1) = 1 and r1 , r2 ∈ Z. In 1972 J. B´esineau [25] showed that the main term in (96) is true (i.e. the left N as N → ∞). side ∼ m1m2 In 1999 D.-H. Kim [218] proved completely Gelfond’s conjecture in the general context of q-additive functions. A function f is called q-additive if f (0) = 0 and f (aq k + b) = f (a) + f (b) for all integers a ≥ 1, k ≥ 1 and 0 ≤ b < q k . This notion is due to Gelfond [141]. Let q = (q1 , . . . , ql ) and m = (m 1 , . . . , m l ) be tuples of integers with q j , m j ≥ 2 and (qi , q j ) = 1 for i = j. For each j ∈ {1, . . . , l} let f j be a q j -additive function with integer values and let f = ( f 1 , . . . fl ). Further, let F = (F1 , . . . , Fl ) and d = (d1 , . . . , dl ) be defined by F j = f j (1) and car d{n < N : sq1 (n) ≡ r1 (mod m 1 ), sq2 (n) ≡ r2 (mod m 2 )} =

d j = gcd(m j , (q j − 1)F j , f j (r ) − r F j (2 ≤ r ≤ q j − 1)) Let f (n) = b (mod m) mean that f j (n) = b j (mod m j ) for each j. We call a tuple b of integers admissible with respect to q, m and f if the system of congruences Fn ≡ b

(mod d)

has a solution. Now, Kim’s result can be stated as follows: N /A + O(N λ ), if b is admissible, car d{n < N : f (n) ≡ b (mod m)} = 0, otherwise (97) 387

CHAPTER 4

as N → ∞, where λ = 1 − 1/(120l 2 (max q j )3 (max m j )2 ) and A = car d{b : 0 ≤ b j < m j , b admissible}. The implied constant depends only on l and q. In 1996 C. Mauduit and A. S´ark¨ozy [254] proved an Erd¨os-Kac theorem for the distribution of sq (n) in residue classes: 1 car d S (N ) car d{n ∈ Sr,m (N ) : ω(n) − log log N ≤ x log log N } − φ(x) = r,m =O

log log log N log log N

,

(98)

where φ(x) denotes the normal law. Other versions can be found in [255] and in J. M. Thuswaldner [336]. For systems of q-additive functions, J. M. Thuswaldner and R. F. Tichy [341] defined q, b, m and f as in (97). Let M(N ) = {n < N : f (n) ≡ b

(mod m)},

and assume that d = (1, . . . , 1). Then (98) holds for Sr,m (N ) replaced with M(N ), uniformly for all real x and N ≥ 8. (99) For the distribution in residue classes of the sum of digits function in number fields, see J. M. Thuswaldner [337]. The function sq (n) in number fields will be studied later. The theory of q-additive functions of Gelfond has been extended by J. Coquet [67]. Let Q = {Q j } j≥0 be a sequence of strictly increasing positive integers with Q 0 = 1 such that Q j |Q j+1 for all j. Note that the sequence Q is uniquely determined by the factors q j = Q j+1 /Q j . Is it easily seen that each nonnegative integer n has a unique ”base-Q” representation of the form n = a j (n)Q j , where the j≥0

”digits” a j (n) satisfy 0 ≤ a j (n) < q j . For example, for Q j = ( j + 1)! one obtains the factorial representation (see (89)), while for Q j = q j one has the ordinary base-q representation. For other particular cases, see e.g. A. Fraenkel [131] and the references therein. This representation is also called Cantor representation, since Cantor considered essentially the case Q j = a(0)a(1) . . . a( j), where a(0) = 1, a(i) ≥ 2 for i ≥ 2 are positive integers. For results on the sum-of-digits function in this case, see P. Kirschenhofer, H. Prodinger and A. F. Tichy [221]. 388

SPECIAL ARITHMETIC FUNCTIONS

Given a system Q, Coquet defined a Q-additive function f : N → C to be a function of the form f (n) = f j (a j (n)) where n = a j (n)Q j is the basej≥0

j≥0

Q representation of n, and the component functions f j are functions defined on {0, 1, . . . , q j − 1} and satisfying f j (0) = 0. Set µ j = 1 − max r

σ j = max

1≤N ≤q j

1 car d{0 ≤ n < q j : f j (n) ≡ r qj

(mod m)} and (100)

1 car d{0 ≤ n < N : f j (n) ≡ 0 (mod m)} N

We call f j uniformly distributed mod m if, for every integer r , 1 car d{0 ≤ n < q j : f j (n) ≡ r qj

(mod m)} =

1 m

We say that an arithmetical function f has a limit distribution mod m, if for each integer r , the limit lim

N →∞

1 car d{0 ≤ n < N : f (n) ≡ r N

(mod m)}

(101)

exists. If each of the limits (101) is 1/m, we say that f has a uniform limit distribution mod m (or that the set in (101) is equidistributed in residue classes mod r (see (95)). The following theorem is due to A. Hoit [183]. Let f be an integer-valued Qadditive function, and let m be a prime. Then the function f has a uniform limit distribution mod m iff at least one of the following conditions holds: (i) For some j, f j is uniformly distributed mod m; ∞ µ j = +∞ (ii)

(102)

j=0

The function has a (non-uniform) limit distribution mod m iff the following three conditions all hold: (i’) There is no j for which f j is uniformly distributed mod m; ∞ µ j < ∞; (ii’) j=0

(iii’)

lim σ j = 0,

j→∞

where µ j and σ j are defined by (100). 389

(103)

CHAPTER 4

For example, let f (n) = sum of digits with prime indices in the base-Q representation of n. Then f has a uniform limit distribution mod n. Let f (n) = sum of prime digits in the factorial representation of n. This function has also a uniform limit distribution mod m. If Q j = (( j + 1)!)2 , and f (n) = number of digits 1 in this representation, then f has no limit distribution mod n. (In all examples, m = prime.) E. Fouvry and C. Mauduit [130], by studying the problem of existence of almost primes in sets of integers generated by finite automata, have proved that the sets {n : s2 (n) ≡ r

(mod m)}

x below x) with at log x most two prime factors. (104) If M is a set of residue classes (mod m), and car d M ≥ 0.722q, then there are (105) infinitely many primes p such that the residue of s2 ( p) (mod n) is in M. (m ≥ 1, r integers) contain infinitely many numbers (even

8

Thue-Morse and Rudin-Shapiro sequences

In 1906 A. Thue [335] (see also J. Berstel [22]) initiated the study of what is now called combinatorics on words with his results on repetitions in words. We say a nonempty word w is a square if it can be written in the form x x for some word x. (For example: murmur in English, chercher in French, or mama in Hungarian). We say that w is an overlap if it can be written in the form axaxa for some word x and single symbol a. Thue explicitly constructed an infinite word on two symbols that is overlap-free, that is, contains no subword that is an overlap. He also constructed an infinite word on three symbols that is square-free, i.e. contains no subword that is a square. Thue’s constructions are based on what is now called the Thue-Morse sequence (in terms of its blocks) t = t (0)t (1)t (2) · · · = 0110100110010110 . . .

(106)

where t (n) = s2 (n) (mod 2) for all n ≥ 0. Thue proved that t is overlap-free. Since Thue’s pioneering work, many other investigators have studied overlap-free words or their generalizations. Thue’s construction was rediscovered by M. Morse [261], who used it in a construction in differential geometry. The chess master M. Euwe [119] rediscovered Thue’s construction in connection with a problem about infinite chess games. In 1980 E. D. Fife [124] described all infinite overlap-free binary sequences. P. S´ee´ bold [304] proved the remarkable result that t is essentially 390

SPECIAL ARITHMETIC FUNCTIONS

the only infinite overlap-free binary sequence which is generated by iterating a morphism. (107) J. P. Allouche and J. Shallit [9] proved that the sequence tq,m = (sq (n) (mod m))n≥0 over the alphabet m = {0, 1, . . . , m − 1} is overlap-free if and only if m ≥ q (where q ≥ 2, m ≥ 1 are integers). (108) A palindrome is a word that is equal to its reversal (e.g. radar). Allouche and Shallit prove also that tq,m always contains arbitrarily long squares. It contains arbitrary long palindromes iff m ≤ 2. (109) The sequence tq,m is ultimately periodic, iff m|(q − 1), (110) a result due to P. Morton and W. J. Mourant [262]. Overlaps, squares and palindromes in sequences have many applications in other fields. For example, in number theory they aid in proving the transcendence of real numbers whose base q expansion or continued fraction expansion have repetitions (see e.g. S. Ferenczi and C. Mauduit [123], M. Queff´elec [281]). In statistical physics they are useful for studying the spectrum of certain discrete Schr¨odinger operators (see e.g. A. Bovier and J.-M. Ghez [33], or A. Hof, O. Knill and B. Simon [182]). For Thur-Morse like sequences, see also J. Grytczuk [162]. Closely related to the Thue-Morse sequence is a sequence c defined to be lexicographically least sequence of positive integers satisfying n ∈ c ⇒ 2n ∈ c. Equivalently, c is defined inductively by c0 = 1 and ck + 1, if (ck + 1)/2 ∈ c, (111) ck+1 = ck + 2, otherwise This sequence appeared in a problem by C. Kimberling [219]. Define the sequence (dk ) by dk = ck − ck−1 for k ≥ 0 (c−1 = 0) Writing the Thue-Morse sequence in terms of its blocks

t = 011010011 · · · = 0d0 1d1 0d2 1d3 . . . , one obtains a sequence (dk ). Then the following surprizing fact is true (see J. P. Alloche et al. [6]): dk = dk for all k ≥ 0 (112) The generating function of the sequence c is k≥0

ck x k =

1 (1 − x 2e j )/(1 − x e j ) 1 − x j≥1 391

(113)

CHAPTER 4

where e1 = 1,

e j+1 =

2e j + 1, if j is even 2e j − 1, otherwise

(answering a problem by S. Plouffe and P. Zimmermann [275]). In n ∈ c, let r (n) be its rank, i.e. satisfying cr (n) = n Then r (n) =

2n + O(log n), 3

(114)

2n infinitely often. and r (n) takes the value 3 As a corollary, one has (see [6]): ck =

3k + O(log k), 2

(115)

3k and ck = infinitely often. 2 For a recent new generalization of the sequence t, see U. Astudillo [12]. Let now tn = (−1)s2 (n) , and for any positive integer k and i ∈ Z denote Sk,i (n) =

t j

0≤ j
In 1969 D. J. Newman [264] proved a remarkable conjecture of L. Moser saying that for all n ≥ 1, S3,0 (n) > 0 (116) More precisely, he proved that S3,0 (n) 3α log 3 < < 5 · 3α with α = α 20 n log 4 In 1983 J. Coquet [68] provided an explicit precise formula for S3,0 (n) by the use of a continuous, nowhere differentiable function ψ3 (x), of period 1:

ε(n) log n α , (117) S3,0 (n) = n ψ3 − log 4 3 where α =

log 3 , and ε(n) ∈ {−1, 0, +1}. log 4 392

SPECIAL ARITHMETIC FUNCTIONS

The general case has been investigated by J. M. Dumont [91], and S. Goldstein, K. A. Kelly and E. R. Speer [146]. For example, when k = p (prime), in [146] it is proved that by denoting by Sk (n) the column vector with entries Sk,i (n), one has

log n α + ε(n), (118) Sk (n) = n φ r s log 2 log λ1 , where λ1 is the s log 2 largest eigenvalue of a certain matrix M satisfying Sk (2s n) = M Sk (n) with s being the multiplicative order of 2 (mod p). Moreover, ε(n) = O(n β ) with β < α. By using the method of Goldstein, Kelly and Speer, in 1999 M. Drmota and M. Skalba [86] have studied the sum Sk,i (y, n) = y s2 ( j) where φ : R → R p is continuous of period 1, and α =

j
where n ≥ 0 and y is a complex number. Let k be odd, and let s be the order of 2 (mod k). Further, let Sk (y, n) = (Sk,0 (y, n), . . . , Sk,k−1 (y, n))t denote the vector of Sk,i (y, n). Let e0 , . . . , ek−1 denote the canonical basis of the kdimensional vector space Ck , and let T denote the matrix defined by T ei = ei+1 (ek = e0 ). The identity matrix is denoted by I . Let M(y) be defined by M(y) =

s−1

m

(I + yT 2 )

m=0

Then one has Sq (y, 2s n) = M(y)Sq (y, n)

(119)

The eingenvalues of T are exactly the k-th roots of unity ζkl (0 ≤ l < k) with k−1 corresponding eigenvectors vl = ζk−il ei , which are orthogonal. Since M(y) is a i=0

polynomial in T , the eigenvalues of M(y) are given by λl (y) =

s−1

m

(1 + yζkl2 )

m=0

Let l = l2 be the multiplicative subgroup of (Z/kZ)∗ of order or d(k/(k,l)) (2). It is clear that λl (y) = λl (y) iff l 2 = l2. We write λl (y) instead of λl (y) if 393

CHAPTER 4

l ∈ l. Let L denote the system of equivalence classes l = l2. Then a basis of the eigenspace Vl corresponding to λl (y), l ∈ L, is given by vl , l ∈ l. All these eigenspaces are orthogonal. Let Pl (l ∈ L) denote the orthogonal projection on Vl . Furthermore, let V (0) denote the eigenspace corresponding to the eigenvalue 0 (if 0 is an eigenvalue), V (s) the subspace corresponding to eigenvalues of modulus < 1, V (1) the subspace corresponding to those of modulus 1, V (u) corresponding to those with modulus > 1, and V (m) that corresponding to those eigenvalues with maximal modulus. Furthermore, let P (0) , P (s) , P (1) , P (u) and P (m) denote the orthogonal projections on V (0) , V (s) , V (1) , V (u) and V (m) , respectively. Then there exists a continuous function F(y, ·) : R+ → V (u) satisfying F(y, 2s x) = M(y)F(y, x) (x > 0), and Pu Sq (y, n) = F(y, n). Consequently Sq (y, n) = F(y, n) +

O(1) if V (1) = {0}, O(log n) if V (1) = {0}

Let |λl (y)| > 1. Then G l (y, t) = λl (y)−t Pl F(y, 2st ) is a continuous function log λl (y) G l (y, ·) : R → V1 which satisfies G l (y, t + 1) = G l (y, t). With αl (y) = s log 2 we obtain the fractal representation for Sq (y, n):

log n + O(log n) (120) Sq (y, n) = n αl (y) G l y, s log 2 |λ (y)|>1 l

Let Ak,i,r,m (n) = car d{ j < n : j ≡ i

(mod k), s2 ( j) ≡ m

and Ar,m (n) = car d{ j < n : s2 ( j) ≡ m Then

(mod r )}

1 Ar,m (n) + Rk,i,r,m (n) k Newman’s theorem (116) can be rewritten as Ak,i,r,m (n) =

A3,0,2,0 (n) > A3,0,2,1 (n) 394

(mod r )}

SPECIAL ARITHMETIC FUNCTIONS

Dumont proved also that (see [91]) A3,1,2,0 (n) < A3,1,2,1 (n) for almost all n ≥ 0 Drmota and Skalba [86] discuss the following two kinds of positivity phenomena: (N1) Ak,0,r,0 (n) > max Ak,0,r,m (n) for almost all n ≥ 0; 0<m
(N2) Rk,0,r,0 (n) > 0 for almost all n ≥ 0. They prove the following results: Let k be an odd multiple of 7, and suppose that r = 3. Then (N1) and (N2) hold. (121) Let k, r be positive integers such that k is odd, and r ≥ 2. If s = or dk (2) and r are coprime or if there exists an integer r > 0 such that λl (ζrm )r < 0 for those λl (ζrm ), 0 < l < k, 0 < m < r , with minimal modulus, then (N1) and (N2) fail. (122) Let Pt (t ≥ 1) denote the set of those primes p where the order or d p (2) of 2 is p−1 equal to . t For any r > 1 there exists a constant Cr > 0 such that for any t ≥ 1 primes q ∈ Pt satisfying (N1) or (N2) are bounded by q ≤ Cr t 4 log4 t

(123)

A2r −1,0,r,0 (n) > max A2r −1,0,r,m (n) for almost all n

(124)

The inequality 0<m
holds exactly for 2 ≤ r ≤ 6. There are infinitely many r ≥ 2 such that R2r −1,0,r,0 (n) < 0 for infinitely many n ≥ 0, but R127,0,7,0 (n) > 0 for almost all n ≥ 0

(125)

For the positivity of Sk,0 (n), we have: Suppose k is divisible by 3, or k = 4 N + 1. Then Sk,0 (n) > 0 for almost all n. The only primes p ≤ 1000 satisfying S p,0 (n) > 0 for almost all n are p = 3, 5, 17, 43, 257, 683. There exists a constant C > 0 such that for any t ≥ 1 and primes p ∈ Pt satisfying S p,0 (n) > 0 for almost all n are bounded by p ≤ Ct 2 log2 t. Furthermore, the total number of primes p ≤ x with S p,0 (n) > 0 for almost all n is o(x/ log x) as x → ∞. (126) 395

CHAPTER 4

n+1 s2 (n) n + 1 , where is a For results on the twisted sum (−1) p p Legendre symbol, see another paper by Drmota and Skalba [87]. P. J. Grabner, T. Herendi and R. F. Tichy [150] study the case p = 17, and use these functions to generate a special code: the analysis of this code shows that it is exponential, extremely redundant and thus strongly error correcting with a linear decoding procedure. The Rudin-Schapiro sequence (a(n)) is defined by a(k) = (−1)e(k) where e(k) =

s−1

εi εi+1 and k =

s

i=0

εi ei with εi ∈ {0, 1} (i.e. the number of con-

i=0

secutive 11-s in the binary expansion of k). In fact, a(2k) = a(k), a(2k + 1) = (−1)k a(k) for k ≥ 0, where a(0) = 1. J. Brillhart and P. Morton [42] have studied the sequences A(n) =

n

B(n) =

a(n),

k=0

(−1)k a(n).

k=0

They proved that " A(n) √ 3 < √ < 6, 5 n

n

B(n) √ 0≤ √ < 3 n

(n ≥ 1)

(127)

0 /"

3 √ B(n) A(n) , √ are dense in the intervals , 6 , resp. √ 5 n n

and that the sequences √ [0, 3]. (128) J. Brillhart, P. Erd¨os and P. Morton [41] have defined the continuous versions by A(x) =

[x]

B(x) =

a(k),

k=0

[x] (−1)k a(k) k=0

They defined also the functions A(4k x) λ(x) = lim √ , k→∞ 4k x

B(4k x) µ(x) = lim √ k→∞ 4k x

which are defined for x > 0. These limits exist for all x > 0, and one has µ(x) = √ 2λ(2x) − λ(x) and µ(4x) = µ(x). (129) The functions λ and µ are continuous on (0, ∞), √and in√fact the function λ maps both intervals (0, ∞) and [1, 4] continuously onto [ 3/5, 6]. 396

SPECIAL ARITHMETIC FUNCTIONS

√ The function µ maps (0, ∞) and [1, 4] continuously onto [0, 3]. A real number x0 > 0 is called normal (to the base 4) iff the sequence (4n x0 )n≥0 is uniformly distributed modulo 1. If x0 > 0 is normal, then λ is not differentiable at x0 . Therefore, λ is nondifferentiable almost everywhere. (130) The cumulative distribution function is defined as follows. Let (u n ) be a sequence of real numbers contained in an interval J , and let α ∈ J . If D(x, α) denotes 1 the number of n ≤ x for which u n ≤ α, and if the limit lim D(x, α) = D(α) exx→∞ x ists, then the sequence (u n ) is said to have the distribution D(α) at α. D(α) is called the cumulative distribution function of (u n ).

A(n) For the sequence √ the following is true: n

A(n) doesn’t exist at any point of The cumulative distribution function of √ n "

√ 3 √ B(n) , 6 . Similar result is true for √ (131) on (0, 3). 5 n The logarithmic distribution function is defined similarly by introducing 1/n L(x, α) = 1≤n≤x,u n ≤α

and considering 1 L(x, α) = L(α). x→∞ log x lim

If this limit exists, then it is called the logarithmic distribution of the sequence (u n ). Now, the /"following 0 can be proved (see [41]):

√ A(n) 3 If α ∈ exists at α, and , 6 , then the logarithmic distribution of √ 5 n has the value 1 1 d x, L(α) = log u Eα x where E α = {1 ≤ x ≤ 4 : λ(α) ≤ α}. Similar results holds true for α ∈ [0,

√ 3] for

1 L (α) = log 4 ∗

where E α∗ = {1 ≤ x ≤ 4 : µ(x) ≤ α}.

E α∗

B(n) : √ n

1 d x, x (132)

397

CHAPTER 4

Finally we mention that the function λ(x) has the logarithmic Fourier series expansion ∞ cn eπin log x/ log 2 (x > 0), λ(x) = n=−∞

where 1 cn = log 4

4 1

λ(x) x 1/2+γn

,

γn =

π ni 1 + , 2 log 2

and where the infinite series converges in the (C, 1) sense for all x > 0 ∞ 1 (i.e. cn eπin log x/ log 2 = lim (σ0 + σ1 + · · · + σk ) with k→∞ k + 1 n=−∞ k σk = cn eπin log x/ log 2 ).

(133)

n=−k

In [7], Allouche, Cohem, Mend`es France and Shallit have proved that for all x = 1, |x| ≤ 1 (x ∈ C),

x e(n) log

n≥0

If T (n) =

(2n + 1)2 − log 2 = . (n + 1)(4n + 1) 1−x

x e(k) , then on a nonreal, positive ray of the unit circle, one has

0≤k≤n

uniformly

T (n) = O(n α ) for 0 < α < 1.

For example, if x = −1 one obtains the infinite product result: ∞ n=0

(2n + 1)2 (n + 1)(4n + 1)

(−1)e(n)

= 2−1/2 .

See also 15. for related identities. In 1972 E. Zeckendorf [360] proved that any non-negative integer n can be uniquely represented as the sum of distinct Fibonacci numbers n=

z(n)

Fk j

(134)

j=1

where k1 ≥ 2 and k j+1 > k j + 1 ( j ≥ 1). Here F0 = 0, F1 = 1, Fk = Fk−1 + Fk−2 (k ≥ 2). The number z(n) of summands in the Zeckendorf representation will be called Zeckendorf sum-of-digits function; and the sequence (−1)z(n) will be the Zeckendorf 398

SPECIAL ARITHMETIC FUNCTIONS

Thue-Morse sequence. Let S(N ) = (−1)z(n)

and

Sk,i (N ) =

(−1)z(n) .

n
n
The first result (due to M. Drmota and M. Skalba [88]) shows that, although S(N ) = O(log N ) (and which is best possible), the values S(n), n < N , attain a Gaussian limit law with zero mean and variance of order log N : For every ε > 0, x 1 1 2 e−t /2 dt+ car d{n < N : S(n) ≤ x c log N } = √ N 2π −∞

1 log N − 2 +ε (135) +O log γ √ √ 5+3 5 1+ 5 and c = . uniformly for all x ∈ R (as N → ∞), where γ = 2 30 Similarly,

N −m 2 − 12 +ε car d{n < N : S(n) = m} = + O((lg N ) exp ) , 2c lg N 2πc lg n (136) log N . where lg N = log γ Furthermore,

1 1 1 1 S( N and S(n) = G lg N − +O N n
2 1 lg N 1 lg N + O , S(n)2 = c lg N + H N n
j

For the sums Sk,i one has: Let k = Fs , where s > 3 is odd and not divisible by 3. Then

1 αk lg N + O(N βk ) Sk,i (N ) = N G k,i 4k

(138)

as N → ∞, where 0 < αk < 1, βk < αk , and G k,i (x) is a real continuous periodic function with period 1. 399

CHAPTER 4

Drmota and Skalba conjecture that G k,i (x + h) − G k,i (x) = (|h|αk ) as h → 0,

(139)

which would imply that G k,i is nowhere differentiable. When k = 3, a stronger relation than (138) is valid:

1 1 α lg N + S(N ) + O(1) (140) S3,i (N ) = N G 3,i 8 3 √ log(4 + 15) and G 3,i (x) are real continuous functions as N → ∞, where α = 8 log γ with period 1. Moreover, G 3,0 (x) > 0 for x ∈ [0, 1), and S3,0 (N ) ≥ 0 for all N ≥ 0

(141)

This is an analogue of Newman’s theorem (116). For Zeckendorf expansion of polynomial sequences, see M. Drmota and W. Steiner [89]. J. Coquet and P. Van den Bosch [69] considered the unique representation n= a j (n)F j , where a j (n) ∈ {0, 1}, j≥1

and a j (n)a j+1 (n) = 0. a j (n) and S(n) = s F (n), then If S F (n) = n
j≥1

log N 3−α N log N + N G S(n) = + O(log N ), 5 log α log α √ 1+ 5 , and G : R → R is a continuous, nowhere differentiable function where α = 2 of period 1. Related to the Fibonacci numbers and to the Thue-Morse sequence t we mention the following combinatorial problem. Let k and n be positive integers and let Fk (n) be the set of binary sequences of length n such that there are at least k − 1 zeros between any two ones. For u ∈ Fk (n), let wk (u) be the total number of ones in these sequences. In 1994 P. Kiss and B. Zay [222] have shown that for any k ≥ 1, lim wk (u) /(n car d Fk (n)) = αk , (142) n→∞

n∈Fk (n)

400

SPECIAL ARITHMETIC FUNCTIONS

where αk = α k + k − 1, α being the greatest positive root of the polynomial √ P(x) = 1 5− 5 x k − x k−1 − 1. For k = 1, α1 = ; while for k = 2 one has α2 = , a result 2 10 due to P. H. St. John [196]. J. M. Dumont [92], by applying the theory of substitutive numeration systems, has obtained another proof of (142). He showed also that lim (wk (u) − nα2 )2 /(n car d Fk (n)) = βk (143) n→∞

u∈Fk (n)

√ αk2 · α k 5− 5 where α2 = and βk = . 10 k(α − 1) + 1

9 q-additive and q-multiplicative functions We now study more closely the q-additive and q-multiplicative functions. (For some results, see also 7.) Recall that a real-valued function f , defined on the non-negative integers, is said to be q-additive if f (0) = 0 and f (aq, j (n)q j ) for n = aq, j (n)q j , f (n) = j≥0

j≥0

where aq, j (n) ∈ E q = {0, 1, . . . , q − 1}. A special q-additive function is sq (n). The most general result concerning the mean value of q-additive functions is due to E. Manstaviˇcius [252]: Let m k,q =

1 f (cq k ), q c∈Eq

and

m 22,k,q =

[logq x]

Mq (x) =

which implies

[logq x]

m k,q ,

Bq2 (x)

k=0

Then

1 2 k f (cq ) q c∈Eq

=

m 22,k,q

k=0

1 ( f (n) − Mq (x))2 ≤ cBq2 (x), x n≤x

(144)

1 f (n) = Mq (x) + O(Bq (x)) x n≤x

There also exist distributional results for q-additive functions. In 1972 H. Delange [74] proved an analogue of the Erd¨os-Wintner theorem: 401

CHAPTER 4

There exists a distribution function F(y) such that, as x → ∞ 1 car d{n < x : f (n) < y} → F(y) x m k,q and m 22,k,q converge. if and only if the two series k≥0

(145)

k≥0

This theorem is generalized by I. K´atai [208] who proved that there exists a distribution function F(y) such that, as x → ∞ 1 car d{n < x : f (n) − Mq (x) < y} → F(y) x m 22,k,q converges. if and only if the series

(146)

k≥0

The most general known theorem concerning a central limit theorem is again due to E. Manstaviˇcius [252]. Suppose that, as x → ∞, max | f (cq j )| = o(Bq (x))

cq j <x

and that Dq (x) → ∞, where

[logq x]

Dq2 (x)

=

2 σk,q

k=0

and 2 σk,q =

1 2 k f (cq ) − m 2k,q . q c∈Eq

Then, as x → ∞,

f (n) − Mq (x) 1 car d n < x : < y → φ(y), x Dq (x)

(147)

where φ is the normal distribution function. Furthermore, N. L. Bassily and I. K´atai [19] studied the distribution of q-additive functions on polynomial sequences: Let f be a q-additive function such that f (cq j ) = O(1) as j → ∞ and c ∈ E q . Dq (x) → ∞ as x → ∞ for some η > 0 and let P(x) be a polynomial Assume that (log x)η with integer coefficients, of degree r and positive leading term. Then, as x → ∞ f (P(n)) − Mq (x r ) 1 car d n < x : < y → φ(y) (148) x Dq (x r ) 402

SPECIAL ARITHMETIC FUNCTIONS

and

f (P( p)) − Mq (x r ) 1 car d p < x : < y → φ(y) π(x) Dq (x r )

(149)

(here p denotes a prime). Let q1 , q2 , . . . , qd > 1be pairwisely coprime integers, and consider the ql additive functions fl (n) (l = 1, 2, . . . , d). For the joint distribution of these functions A. J. Hildebrand (see [78]) proved that 1 car d{n < x : fl (n) < yl , 1 ≤ l ≤ d} → F1 (y) . . . Fd (y) x

(150)

if fl satisfy (145) for all l = 1, . . . , d; and there is a joint central limit theorem of the form fl (n) − Mql (x) 1 car d n < x : < yl , 1 ≤ l ≤ d → φ(y1 ) . . . φ(yd ), (151) x Dql (x) if Bql → ∞ and Bql (x η ) ∼ Bql (x) for every η > 0, as x → ∞. (Note that the function sq (n) is not covered by this result.) M. Drmota [78] has extended the Bassily-K´atai theorem to the joint distribution case: Let q1 , . . . , qd > 1 be pairwisely coprime integers and let fl (1 ≤ l ≤ d) be j ql -additive functions such that fl (cql ) = O(1) as j → ∞ and c ∈ El . Assume that Dql (x)/(log x)η → ∞ as x → ∞ (1 ≤ l ≤ d), for some η > 0, and let Pl (x) be polynomials with integer coefficients of different degrees rl and positive leading term (1 ≤ l ≤ d). Then as x → ∞, fl (Pl (n)) − Mql (x rl ) 1 , 1 ≤ l ≤ d → φ(y1 ) . . . φ(yd ), car d n < x : < y l x Dql (x rl ) (152) and fl (Pl ( p)) − Mql (x rl ) 1 car d p < x : < yl , 1 ≤ l ≤ d → φ(y1 ) . . . φ(yd ) π(x) Dql (x rl ) (153) This theorem contains the condition that all polynomials Pl (x) have different degrees. It seems that this condition is not necessary. For the case d = 2 and linear polynomials Pl (x) = Al x + Bl , the following result holds true (see [78]): j Let q1 , q2 > 1 be coprime, fl be ql -additive (l = 1, 2), with fl (cql ) = O(1) as j → ∞ and c ∈ El . Assume further that Dql (x)/(log x)η → ∞ as x → ∞ (l = 1, 2) for some η > 0. Let Pl (x) = Al x + Bl (l = 1, 2) be arbitrary linear 403

CHAPTER 4

polynomials with integer coefficients and positive leading terms (Al , ql ) = 1. Then as x → ∞, fl (Pl (n)) − Mql (x) 1 car d n < x : < yl , l = 1, 2 → φ(y1 )φ(y2 ) (154) x Dql (x) For the sum-of-digits function sq (n), a local version of (154) holds true: Let (q1 , q2 ) = 1 and set d = gcd(q1 − 1, q2 − 1). Then, as x → ∞, 1 car d{n < x : sq1 (n) = k1 , sq2 (n) = k2 } = x 



q −1 logql x kl − 2   1 2   =d exp − "   q2 − 1 ql2 − 1 l=1 logql x 2· l logql x 2π · 12 12

2 

 1   + o  log x (155)

uniformly for all integers k1 , k2 ≥ 0 with k1 ≡ k2 (mod d). Note that sql (n) ≡ n (mod (ql − 1)), so we always have sq1 ≡ sq2 (mod d), and consequently car d{n < x : sq1 (n) = k1 , sq2 = k2 } = 0, if k1 ≡ k2

(mod d).

(156)

For digital expansion with respect to different bases see also M. Drmota and J. Schoissengeier [85]. For the joint distribution of Q-additive functions on polynomials over finite fields, see M. Drmota and G. Gutenbrunner [83]. Recently B. Gittenberger and J. M. Thuswaldner [143] have extended the BassilyK´atai theorem to canonical number systems in the ring of Gaussian integers Z[i]. Let b ∈ Z[i] and N = {0, 1, . . . , |b|2 − 1}. A pair (b, N ) is called a canonical number system if any γ ∈ Z[i] has a representation of the form γ = c0 + c1 b + · · · + ch bh ,

ch = 0 if h = 0,

where h ≥ 0 is a nonnegative integer and c j ∈ N for j = 0, . . . , h. Here b is called base and N is called set of digits of (b, N ). Let (b, N ) be a canonical number system in Z[i]. A function f is called badditive if f (0) = 0 and f (a j (γ )b j ) for γ = a j (γ )b j (a j (γ ) ∈ N ) f (γ ) = j≥0

j≥0

404

SPECIAL ARITHMETIC FUNCTIONS

The Bassily-K´atai theorem for b-additive functions can be stated as follows: Let f be a b-additive function such that f (cb j ) = O(1) for j ∈ N and c ∈ N . Furthermore, let 1 1 2 k mk = 2 f (cbk ), σk2 = 2 f (cb ) − m 2k , |b| c∈N |b| c∈N M(N ) =

L

mk ,

k=0

D 2 (N ) =

L

σk2 , with L = [log|b| N ].

k=0

Assume that D(N )/(log N )1/3 → ∞ as N → ∞, and let P(z) = pr z r + · · · + p1 z + p0 be a polynomial with coefficients in Z[i]. Then, as N → ∞, f (P(z)) − M(N r ) 1 2 car d |z| < N : < y → φ(y) (157) car d{z : |z|2 < N } D(N r ) where φ is the normal distribution function and z runs over the Gaussian integers. Let sb (z) = c0 + c1 + · · · + ch be the sum-of-digits function in Z[i]. As a corollary of (157), one has: sb (P(z)) − M(N r ) 1 2 car d |z| < N : < y → φ(y) (158) car d{z : |z|2 < N } D(N r ) The proof is based among others on certain results on exponential sums over number fields, as well as a two-dimensional version of the Erd¨os-Tur´an-Koksma inequality of discrepancy theory. We note that the notion of canonical number system in Z[i] is due to I. K´atai and J. Szab´o [212] who showed that the only bases are given by b = −n ± i, where n = 1, 2, . . . I. K´atai and B. Kov´acs [211] extended the notion to imaginary quadratic fields, while in 1992 I. K´atai and I. K¨ornyei [210] considered the general case of algebraic number fields. A function f is called q-multiplicative if f (0) = 1 and f (n) = f (aq, j (n)q j ) for n = aq, j (n)q j , j≥1

j≥0

where aq, j (n) ∈ E q . For q-multiplicative function f satisfying | f (n)| = 1, K. H. Indlekofer and I. K´atai [188] obtained the following result: There exists a constant c1 such that if f is q-multiplicative, with | f (n)| = 1 (n ≥ 0) and f ( p) = constant for every large prime p, then f k (nq) = 1 for every n ≥ 0, with a suitable integer k ∈ [1, c1 ]. (159)

405

CHAPTER 4

The result remains true also if we assume the fulfilment of f ( p) = constant with the possible exception of primes of a set of relative density zero. q-multiplicative functions of modulus at most 1 have been first studied (in the context of Q-multiplicative functions, see 7.) by J. Coquet [67]. Let Q = (Q j ) j≥0 be a sequence of strictly increasing positive integers with Q 0 = 1 such that Q j |Q j+1 for all j. Put q j = Q j+1 /Q j . Then every nonnegative integer n has a unique baseQ representation n = a j (n)Q j , where 0 ≤ a j (n) < q j . A function f will j≥0 f j (a j (n)), where n = a j (n)Q j , and be called Q-multiplicative if f (n) = j≥0

j≥0

the component functions f j are functions defined on {0, 1, . . . , q j − 1} satisfying f j (0) = 1. The mean value of f is defined by M( f ) = lim

N →∞

1 f (n) N 0≤n
provided this limit exists. We set σ j ( f ) = max

0
1 (1 − Re( f j (m))), n 0≤m
and µj( f ) =

1 f j (n) q j 0≤n
Coquet [67] proved the following theorem: Let f be a Q-multiplicative function satisfying | f (n)| ≤ 1. Then the mean value of f exists and is equal to zero if and only if at least one of the following two conditions holds: (i) For some j ≥ 0, µ j ( f ) = 0, (1 − |µ j ( f )|) diverges (ii) The series

(160)

j≥0

The case of Q-multiplicative functions of modulus at most 1, having a non-zero mean value, has been settled recently by A. Hoit [184]: Let f be a Q-multiplicative function satisfying | f (n)| ≤ 1. Then the mean value of f exists and is non-zero if and only if the following three conditions all hold: (a) For each j ≥ 0, µ j ( f ) = 0, (b) (1 − µ j ( f )) converges, j≥0

(c)

lim σ j ( f ) = 0

j→∞

406

(161)

SPECIAL ARITHMETIC FUNCTIONS

As a corollary of (160) and (161) one can deduce: Let f be as above. Then the value of f exists if and only if at least one of the following three conditions holds: (i);

(ii); (b)

and

(c)

(162)

The mean value is zero if either condition (i) or (ii) holds. Another generalization of q-multiplicative functions is due to J. Feh´er [122]. Let R0 , R1 , . . . be a sequence of subsets of nonnegative integers. We say that it is an R-system, if the following conditions hold: 1) 0 ∈ Ri and 1 < car dRi < ∞ (i = 0, 1, 2, . . . ), 2) For 0 ≤ i < j, the smallest positive element of Ri is smaller than the smallest positive element of R j , 3) Each n ≥ 0 can be uniquely written as n=

s

rj

(r j ∈ R j , s ≥ 0)

j=0

The system is called monotonic, if in addition one has 4) For each 0 ≤ i < j, the largest element of Ri is smaller than the smallest positive element of R j . Similarly, the system is called bounded, if in addition to 1)-3) the following holds true: 5) car dR j is bounded ( j ≥ 0). For example, let ki (i = 0, 1, . . . ) be a sequence of integers, ki ≥ 2, and furthermore let d0 = 1, di = di−1 ki−1 (i ≥ 1), Ni = {0, 1, . . . , ki − 1}, Ri = di Ni = {0, di , . . . , (ki − 1)di } (i ≥ 0). Then Ri (i ≥ 0) is an R-system. This is the so-called ”britannic number system” of N. G. De Bruijn [44]. For ki = q ≥ 2 for all i ≥ 0, one reobtains the q-ary number system. It is easy to see that the monotonic R-systems are exactly the system considered by De Bruijn. A function f defined on nonnegative integers, with complex arguments is called R-multiplicative (with respect to a given R-system), if f (0) = 1

and

f (n) =

s

f (r j ).

j=0

For example, f (n) = a n (0 = a ∈ C) is R-multiplicative, for all R. Let P(z) = ak z k + · · · + a1 z + a0 be a polynomial of complex coefficients, and denote by P(E) f (n) = ak f (n + k) + · · · + a1 f (n + 1) + a0 f (n) 407

CHAPTER 4

Let us consider the following conditions: 1 |P(E) f (n)| = 0, (1) lim n→∞ x n≤x (2) |P(E) f (n)| = o(x) as x → ∞, n≤x

(3) P(E) f (n) = 0 for all n ≥ 0, 1 (4) lim inf | f (n)| = 0, x→∞ x n≤x | f (n)| = o(x) as x → ∞. (5) n≤x

J. Feh´er [122] proves the following theorem: If (1) holds, then either (3) or (4) hold. There exists such an f for which (4) holds, but (5) doesn’t hold. Assuming that the R-system is monotonic and bounded, then (4) and | f (n)| ≤ 1 (n ≥ 0) imply relation (5). (163) For the R-multiplicative solutions of the recurrence P(E) f (n) = 0 the following result is valid: Let R0 , R1 , . . . be such an R-system for which R0 = {0, 1, . . . , d −1} (d ≥ 2). Assume that P is of degree k (1 ≤ k ≤ d), and that P(0) = 0. Then P(E) f (n) = 0 iff s f (n) = α j ρ nj (164) j=1

where

s

α j = 1, and ρ j ( j = 1, s) are distinct roots of P with

j=1

ρ1d = ρ2d = · · · = ρsd . G. Barat and P. J. Grabner [16] have introduced and studied the blockm multiplicative functions as follows. Let a j = qq, j (n) be the digits of n = ajq j j=0

in the base q expansion. For a positive integer l, an arithmetic function f is called l-block-multiplicative if there exists a function g defined on {0, . . . , q1 }l such that m m j f ajq = g(a j , a j+1 , . . . , a j+l−1 ) j=0

j=0

with a j = 0 for j > m, and g(0, 0, . . . , 0) = 1. Let Bs denote the set of all digital blocks of length s. For B ∈ Bs , the expression Bn indicates the concatenation of the block B with the digits of n. For a given lblock-multiplicative function f , define the summatory function F(N ) = f (n), n
408

SPECIAL ARITHMETIC FUNCTIONS

and for fixed s introduce the functions f (n) = ( f (Bn)) B∈Bs , FB (N ) =

f (Bn),

n
F(N ) = (FB (N )) B∈Bs Then for fixed s ≥ max{1, l − 1} and every nonnegative integer r , the following hold: f (B) f (ABC) = f (AB) f (BC), B ∈ Bs , A, C blocks of arbitrary length; F(B N ) = FC (N ) + f (C N ), C∈Bs

C
F(q r +s ) = U F(q r ), where the matrix U is given by

u B,C

  f (BC) = f (C)  0

for f (C) = 0

(165)

otherwise

and ”<” is the natural ordering for Bs . Let g ∞ = max |g(B)|. B∈Bs

The following result is true: Let f be an l-block-multiplicative function and assume that there exist an s ≥ max{1, l − 1} such that the corresponding matrix U (defined by (165)) satisfies the following condition: there exists a unique dominating eigenvalue λ of U with |λ| > g s∞ . Then there exist continuous periodic functions ψ j ( j = 0, . . . , m − 1) of period s such that n
f (n) = N δ eiα logq N

m−1

(logq N ) j ψ j (logq N ) + o(N µ )

(166)

j=0

1 for any µ > max logq g ∞ , logq t , where t = max{|γ | : γ ∈ spec(U ) \ {λ}}, s 1 1 δ = logq |λ|, α = arg λ, and m is the dimension of the largest Jordan block s s associated to λ. (Here spec denotes the set of eigenvalues.) 409

CHAPTER 4

Under the stated assumptions, for a positive-valued function f , we have f (n) = N δ ψ(logq N ) + o(N µ ), (167) n
where ψ is a periodic function of period 1. Another corollary of theorem (166) gives the asymptotic behaviour of summatory functions of products of a block-multiplicative function with a number of block-additive functions (defined analogously, but with a sum in the right side, i.e. m m j ajq = h(a j , a j+1 , . . . , a j+l−1 ), h(0, 0, . . . , 0) = 0): f j=0

j=0

Let f be a positive-valued block-multiplicative function satisfying the assumption of theorem (166), and let f 1 , . . . , f k be arbitrary real-valued block-additive functions. Then

f (n) f 1 (n) . . . f k (n) = N δ

n
k (logq N )i ψi (logq N ) + o(N µ )

(168)

i=0

where the functions ψi are continuous and 1-periodic functions, and µ and δ are given as in (166). We note that the block-additive functions have been introduced by E. Cateland [53], who obtained also Delange type theorems. For digital blocks, see also [17]. N. L. Bassily [18] has proved that if P(x) = cr x r + · · · + c0 is a polynomial with integer coefficients taking on positive values for x > 0, such that (cr , q) = 1, then 



 1  f (P(n)) −  qk n≤x



2

 log x  h(b0 , . . . , bl−1 ) r  x log x log q 0≤bν ≤q−1 0≤ν≤l−1

and a similar result, when n is replaced by a prime p, when the right side becomes π(x) log x.

10

Uniform - and well - distributions of αsq (n)

A real sequence (xn )n≥1 is called uniformly distributed mod 1 (for short u.d. mod 1) if N 1 χ I ({xn }) = λ(I ) lim N →∞ N n=1 for all intervals I ⊂ [0, 1], where χ I denotes the characteristic function of I , {x} = x − [x] is the fractional part of x, and λ(I ) is the length of the interval. Equivalently, 410

SPECIAL ARITHMETIC FUNCTIONS

the discrepancy

N 1 χ I ({xn }) − λ(I ) D N (xn ) = sup I ⊂[0,1] N n=1

satisfies lim D N (xn ) = 0.

N →∞

The sequence (xn ) is called well-distributed mod 1 (for short, w.d. mod 1) if the uniform discrepancy 1N (xn ) = sup D N (xn+k ) D k≥0

satisfies

1N (xn ) = 0. lim D

N →∞

Clearly, every w.d. sequence is u.d. However, the converse is not true, e.g. xn = n is u.d. but not w.d. Similar definitions are valid for higher dimensional sequences. In 1967 M. Mend`es France [257] discovered that the sequence (α · sq (n)) is u.d. mod 1 (α ∈ R \ Q). (169) In 1980 J. Coquet showed that this sequence is in fact w.d. mod 1. More generally, if a sequence (xn ) is w.d. mod 1, then the sequence (xsq (n) )n is w.d. mod 1, too. (170) P. J. Grabner and R. F. Tichy [157] have extended (169) to (α · sG (n)), where sG is the sum of digits function with respect to a linear recurrence G (see the next section for such expansions). For similar result for s Q (n) of base-Q representations, see A. Hoit [184]. R. F. Tichy and G. Turnwald [343] derived estimates for the discrepancy of (αsq (n)), and M. Drmota [79] generalized and sharpened their results. An arithmetical function f is called strongly q-additive if √

f (a + qb) = f (a) + f (b) (0 ≤ a < q, b ≥ 0) Let α be an irrational number and suppose that for every ε > 0 there exists a c(α, ε) constant c(α, ε) > 0 such that hα ≥ η+ε for all positive integers h (where x h denotes the nearest distance of x to integers). Let f be a strongly q-additive function which attains only non-negative integers such that there exists 1 ≤ b ≤ q − 1 with f (b) > 0. Then for every ε > 0, 1N (α f (b)) D

1 (log N )1/(2η)−ε

for all N > 1, where the constant implied by depends on q, α, ε and f . 411

(171)

CHAPTER 4

It is also possible to show (see [343] and [241]) that for every irrational α there exists a constant c (q, α, f ) such that D N (α f (n)) >

c (q, α, f ) (log N )1/2

(N ≥ 2)

(172)

A generalization of (170) is proved also in [79]: Suppose that f (n) is strongly qadditive which attains only non-negative integers such that gcd{0 < j < q : f ( j) > 0} = 1, Then, if a sequence (xn ) is w.d. mod 1, then (x f (n) ) is w.d. mod 1, too. (173) For the higher dimensional analogue of (169), M. Drmota and G. Larcher [84] have proved that if q1 , . . . , qd are pairwisely coprime integers > 1, then the ddimensional sequence (α1 sq1 (n), . . . , αd sqd (n)) (174) is u.d. mod 1 if and only if α1 , . . . , αd are all irrational. The same result for w.d. mod 1 has been proved by P. J. Grabner, P. Liardet and R. F. Tichy (see [155]) via ergodic theory, even for Cantor representations. See also Drmota and Tichy [90]. For applications of ergodic theory to digital problems see also e.g. P. Liardet [243], T. Kamae [201], U. Krengel [237], P. J. Grabner and P. Liardet [154]. For the discrepancy of the above d-dimensional sequence, Drmota and Larcher prove: Let α1 , . . . , αd be irrational numbers. Suppose that there exists η > 0 and a constant c1 > 0 such that for all integers h > 0 we have hα j ≥ c1 h −η for all 1 ≤ j ≤ d. Then the discrepancy D N of the sequence (α1 sq1 (n), . . . , αd sqd (n)) is bounded by

log log N 1/2η (175) D N ≤ c(c1 , η, q) log N with some constant c(c1 , η, q) depending on c1 , η, and q = (q1 , . . . , qd ). Conversely, if for some η ≥ 1 and some constant c2 > 0 there exists j such that hα j ≤ c2 h −η for infinitely many integers h > 0, then D N ≥ c (c2 , η, q) ·

1 (log N )1/2η

(176)

for infinitely many N ≥ 0, where c (c2 , η, q) depends on c2 , η and q.

4 An analogue of (169) is due to C. Mauduit and J. Rivat [253]: Let 1 ≤ c ≤ be 3 a real number and α an irrational number. Then the sequence (α · sq ([n c ])) 412

(177)

SPECIAL ARITHMETIC FUNCTIONS

is u.d. mod 1. This result follows from a more general theorem on q-multiplicative functions f : 1 f ([n c ]) = f (m)m c−1 + O(x 1−ε ) (178) c 1≤n≤x 1≤m≤x c as x → ∞ (where ε > 0). For q-multiplicative functions with modulus 1, and a kind of uniform distribution, (namely uniform summability), Indlekofer and K´atai [189] have recently obtained a representation theorem. A function g is called uniformly summable (a notion due to Indlekofer [187]) if sup x≥1

1 x

| f (n)| → 0 as K → ∞.

(179)

n≤x,| f (n)|≥K

Let g be uniformly summable. We say that α is in the Bohr-Fourier spectrum of g, if 1 (180) lim sup f (n)e(−nα) > 0 x x→∞ n≤x (where e(x) = exp(2πi x)). Assume now that f is q-multiplicative with modulus 1, and that g is uniformly summable multiplicative function (in ordinary sense), and that 1 lim sup f (n)g(n) > 0. (181) x→∞ x n≤x Then f (n) can be written as f (n) = e

r D

h(n)

r with a suitable rational number and with a function h, which is q-multiplicative D with modulus 1, for which q−1 ∞

Re(1 − h(cq j )) < ∞

(182)

j=0 c=0

holds. If the Bohr-Fourier spectrum of g is empty, then 1 f (n)g(n) → 0 x n≤x

(x → ∞)

for each q-multiplicative function f of modulus 1. 413

(183)

CHAPTER 4

11

The G-ary digital expansion of a number

Let G = (G j ) j≥0 be a strictly increasing sequence of integers with G 0 = 1. Then every nonnegative integer n has a unique proper G-ary digital expansion n= a j (n)G j j≥0

with integer digits a j (n) ≥ 0 provided that a j (n)G j < G k for all k ≥ 0. j
This has been proved by A. Peth¨o and R. F. Tichy [272] in case of linear recurrences, and by P. Grabner and R. F. Tichy [158] in the general case. See also A. S. Fraenkel [131]. These authors have obtained also asymptotic estimates for the mean value of the sum-of-digit function sG (n) given by sG (n) = a j (n) j≥0

The distribution of sG (n) has been studied by M. Drmota and J. Gajdosik [82]. Let (X N ) be a sequence of discrete random variables X N , (N ≥ 0) defined by Pr [X N = k] =

car d{n < N : sG (n) = k} N

The expected value and variance of X N are given by EX N =

1 sG (n), N n<M

VX n =

1 (sG (n) − EX n )2 N n
In the case of finite recurrences we will make the following assumptions: 1) There exist nonnegative integers gi (1 ≤ i ≤ r ) such that for j ≥ r, r gi G j−i Gj = i=1

2) gcd{i ≥ 1 : gi = 0} = 1 3) For all j > r and 1 ≤ k < r we have r G j−k ≥ gi G j−i

(184)

i=k+1

We note that the above assumptions imply that the characteristic polynomial r gi u r −i has a unique root α of maximal modulus which is real P(u) = u r − i=1

414

SPECIAL ARITHMETIC FUNCTIONS

and positive, and that G j ∼ Cα j as j → ∞ where C > 0 is a constant. When g1 ≥ g2 ≥ · · · ≥ gr ≥ 0 and G j =

j

g j G j−1 + 1 for j < r , then all

i=1

assumptions are satisfied. Furthermore, P(u) is irreducible and α is a Pisot number, see [159]. The fact that α is a Pisot number, is essentially due to A. Brauer [38]. For Pisot (or Pisot-Vijayaraghavan) and Salem numbers, see e.g. the monograph [24] by M. J. Bertin, et al. For more recent results, see D. W. Boyd [35], D. W. Boyd and R. D. Mauldin [37], or M. J. Bertin and D. W. Boyd [23]. For connections to Fourier analysis, see R. Salem [297], and to Harmonic analysis, see Y. Meyer [256]. For infinite recurrences our starting point is Parry’s α-expansion of 1 (see [270]) 1=

g1 g2 g3 + 2 + 3 + ... α α α

(185)

where α > 1 is a real number and gi (i ≥ 1) are positive integers. (In the case of ambiguity we take the infinite representation of 1.) The sequence G = (G j ) j≥0 is now defined by G 0 = 1,

Gj =

j

g j G j−1 + 1

( j ≥ 1)

(186)

i=1

If we further set A(u) =

gi u i ,

G(u) =

i≥1

G ju j,

j≥0

then G(u) =

1 (1 − u)(1 − A(u))

1 > 0 is the only singularity on the circle of convergence and it follows that z 0 = α |z| = z 0 , which is a simple pole. Hence G j ∼ C α j as j → ∞ and so we are in similar situation as above. The following theorem is a collection of results from [272], [158], [159] etc. (see [82]): 415

CHAPTER 4

Suppose that G = (G j ) satisfies a finite or infinite linear recurrence of the above types (see (184), (185), (186)). Set −1 j g G(z, u) = z l z g1 +···+g j−1 u j j≥1

l=0

and let 1/α(z) denote the analytic solution u = 1/α(z) of the equation G(u, z) = 1 for z in a sufficiently small (complex) neighbourhood of z 0 = 1 such that α(1) = α. Then log N EX N = µ + O(1), (187) log α and log N + O(1), (188) VX N = σ 2 log α α (1) α (1) and σ 2 = + µ − µ2 . where µ = α α For the asymptotic normality in the weak sense we have: 1 |car d{n < N : SG (N ) < EX N + xVX N }| = N (189) = φ(x) + O((log N )−1/2+ε ), uniformly for all real x, as N → ∞ (φ is the normal law). A local limit law is provided by car d{n < N : sG (n) = k} =

(k − EX N )2 N −1/2+ε exp − + O((log N ) ) , (190) =√ 2VX N 2π VX N uniformly for all nonnegative integers k, as N → ∞. By assuming that all digits are uniformly bounded by some integer bound K (i.e. the sequence of fractions G j+1 /G j is bounded), and letting F : {0, 1, . . . , K } K +1 → R (L ≥ 0) with F(0, . . . , 0) = 1, one can define a general sum-of-digits function s F (n) = F(a j+L (n), a j+L−1 (n), . . . , a j (n)) (191)

j≥0

In the case of q-ary expansions (i.e. G j = q j ) and finite recurrences (see (184)), similar results to (189) and (190) have been proved in M. Drmota [80]. It is an open problem if corresponding theorems hold for infinite recurrences. For summatory formulae in the case of q-ary expansions, see also G. Barat, R. F. Tichy and R. Tijdeman [17]. For functional limit theorems for digital expansions, see M. Drmota, M. Fuchs and E. Manstaviˇcius [81]. 416

SPECIAL ARITHMETIC FUNCTIONS

12

The sum-of-digits function for negative integer bases

The number systems with negative integer bases were first introduced by V. Gr¨unwald in 1885, in a little known journal [161]. Later the idea was rediscovered by A. J. Kempner in 1936, Z. Pawlak and A. Wakulicz in 1957, I. Wadel in 1957, D. E. Knuth in 1955 (see the book [224] by D. E. Knuth). Knuth studied also numerical systems with complex bases. It is easy to prove that for an integer q ≥ 2, every n ∈ Z has a unique representation of the form n = c0 + c1 (−q) + c2 (−q)2 + · · · + ch (−q)h

(192)

with ci ∈ {0, 1, . . . , q − 1}, 0 ≤ i ≤ h, and c j = 0 for h = 0. Expression (192) is called (−q)-adic representation of n. The sum of digits function of n now is defined by s−q (n) = c0 + c1 + · · · + ch For the mean value of s−q (n): s−q (n) = (q − 1)N logq N + N F(logq 2 N ),

(193)

|n|
see P. Flajolet and L. Ramshaw [126], and J. M. Thuswaldner [338]. Here F is a continuous function of period 1. A treatment of the higher moments (even for number fields) has been done by Gittenberger and Thuswaldner [144]. P. J. Granbner and J. M. Thuswaldner [156] have studied the properties of s−q (n) − s−q (−n). The following results are valid: (s−q (n) − s−q (−n)) = N G(logq 2 N ) + O((log N )2 ), (194) n

(s−q (n) − s−q (−n))2 = 2r1 (q)N logq N + N H (logq N ) + O((log N )3 ), (195)

n
"

|s−q (n) − s−q (−n)| =

n
2 r1 (q) N N · N · logq N + ψ(logq N ) + O =2 π (log N )3/2 logq N

(196)

where G(x), H (x) are continuous, nowhere differentiable functions of period 1, and r1 (q) =

1 (q − 1)(q 2 − 4q + 7) · 6 (q + 1) 417

CHAPTER 4

r2 (q) =

1 (q − 1)(q 6 − 11q 5 + 82q 4 − 322q 3 + 577q 2 − 371q + 76) , · 180 (q + 1)3 1 r2 (q) 1 ψ(x) = √ · H (x) + , − √ 2 2 πr1 (q) 2 πr1 (q)3 3Q πr1 (q)

and Q = 2 if q is even; and = 1, if q is odd. As a corollary one obtains that

q −1 N logq N − min{s−q (n), s−q (−n)} = 2 n
"

1 1 N + F(logq 2 N )N − ψ(logq N ) +O 2 2 (logq N )1/2

r1 (q) N (logq N )1/2 + π

N , (log N )3/2

(197)

and

q −1 max{s−q (n), s−q (−n)} = N logq N + 2 n
"

1 1 N + F(logq 2 N )N + ψ(logq N ) +O 2 2 (logq N )1/2

r1 (q) N (logq N )1/2 + π

N , (log N )3/2

(198)

where the periodic functions F and ψ are as in (193), and (196), respectively. The quantity s−q (n) − sq (−n) is asymptotically normally distributed with mean 0 and variance 2r1 (q) logq N , i.e.: x s−q (n) − s−q (n) 1 1 1 −t 2 /2 <x =√ e dt + O car d n < N : N 2r1 (q) logq N log N 2π −∞ (199)

13

The sum-of-digits function in algebraic number fields

H. G. Senge and E. G. Straus [305] in 1973 have discovered the following interesting fact. If (q1 , q2 ) = 1 and c > 0 is a constant, then there are only finitely many n ≥ 0 such that sq1 (n) ≤ c and sq2 (n) ≤ c (200) This result was later generalized and sharpened by C. L. Stewart [318], H. P. Schlickewei [302], [303], and A. Peth¨o and R. F. Tichy [272], [274]. The proofs use A. Baker’s method on linear forms of logarithms and the p-adic version of W. Schmidt’s subspace theorem by Schlickewei applied to S-unit equations. In [274] the sum of digit function with respect to canonical number system in an algebraic 418

SPECIAL ARITHMETIC FUNCTIONS

number field is considered. In the case of Gaussian integers, these system have been studied in section 9. In the most general case, these were introduced by K´atai and K¨ornyei [210]. Let K be a number field of degree [K ; Q] = n, and let Z K denote its ring of integers. Then it is well known (see e.g. [31]) that there exist β1 , . . . , βk ∈ K such that any γ ∈ Z K has a representation γ = b1 β1 + · · · + bn βn

(bi ∈ Z)

The vector (β1 , . . . , βn ) is called an integral basis of Z K . For α ∈ Z K we denote the norm of α by N (α). If we have α ∈ Z K , and N = {0, . . . , |N (α)| − 1}, then the pair (α, N ) is called a canonical number system if any γ ∈ Z K has a unique finite digital representation of the form γ = c0 + c1 α + · · · + ch α h ,

c j ∈ N ( j = 0, . . . , h)

(201)

Here α is called the base of the number system, N its set of digits, while (201) is called the α-adic representation of γ . The length of γ is defined by L α (γ ) = h In case of Z[i] the bases have been characterized by K´atai and Szab´o [212], in the more general case of quadratic number fields by K´atai and Kov´acs [211]. Any quadratic number field contains bases that form canonical number systems. We note that this is not true in number fields of higher degree. B. Kov´acs [234] in 1981 gave a criterion for a number field to contain canonical number systems: In Z K there exists a canonical number system iff there exists an α ∈ Z K , such that {1, α, . . . , α n−1 } is an integral bases. (202) The complete characterization was established in 1991 by Kov´acs and Peth¨o [235]: There exist α1 , . . . , αt ∈ Z K ; n 1 , . . . , n t ∈ Z, and N1 , . . . , Nt finite subsets of Z, which are effectively computable, such that (α, N ) is a canonical number system in Z K if and only if α = αi − h for some integers i, h with 1 ≤ i ≤ t and either h ≥ n i or h ∈ Ni . (203) By using this result, and an algorithm due to K. Gy¨ory [166], Kov´acs and Peth¨o were able to develop an algorithm for finding the canonical number systems of the ring of integers of a number field. The sum of digits function sb (γ ) of a number γ with b-adic representation γ = c0 + c1 b + · · · + ch bh will be sb (γ ) = c0 + · · · + ch . 419

CHAPTER 4

A general result for the summatory function of sb has been obtained by J. M. Thuswaldner [338] in 1998. Let θ be the primitive element of the number field K , and let θ1 , . . . , θ2 be its conjugates. Then any α ∈ K has a representation γ = r1 + r2 θ + · · · + rn θ n−1 , with ri ∈ Q (i = 1, n). Define the mappings σ1 , . . . , σs , σs+1 , σ s+1 , . . . , σs+t , σ s+t which map K = Q(θ ) onto the equivalent field Q(θi ) and fulfill σi |Q = idQ . Here, s is the number of real roots, t is the number of pairs of conjugate complex roots of the minimal polynomial of θ, hence the degree of K is n = s + 2t. The pair (s, t) is called the signature of K . With the help of these mappings we can transform each element γ ∈ K into the vector x(γ ) = (σ1 (γ ), . . . , σs (γ ), σs+1 (γ ), . . . , σs+t (γ )) The vectors are elements of a vector space of dimension s + 2t over R. One can show that the elements of Z K form a lattice in this vector space. The fundamental domain of this lattice, and any translate of it, is a parallelotope. If we assume that Z K contains a canonical number system, then by theorem (202) there exists an integral basis of the form {1, α, . . . , α n−1 }. So the vectors x(1), . . . , x(α n−1 ) generate the lattice induced by Z K . By using Gram’s determinant, the volume of the fundamental domain will be: 2 I (α) = det(x(α i ), x(α j )), i, j = 0, n − 1 We consider regions of the following form: |z 1 | < l1 , . . . , |z s | < ls , z s+1 z s+1 < ls+1 , . . . , z s+t z s+t < ls+t

(204)

where z 1 , . . . , z s , z s+1 , z s+1 , . . . , z s+t , z s+t are the conjugates of z ∈ K , and l1 , . . . , ls+t are certain numbers. Let D(l1 , . . . , ls ; ls+1 , . . . , ls+t ) denote the set of all z ∈ K satisfying (204). The following theorem holds true: Let K be a number field of degree n and signature (s, t), and Z K its ring of integers with integral bases {1, α, . . . , α n−1 }. Let b be the base of a canonical number system, and let M denote the absolute value of the norm of b over Q. Let b1 , . . . , bs ; bs+1 , bs+1 , . . . , bs+t , bs+t be the conjugates of b; let 1 < x < |b1 | (and 1 < x < |b1 |2 if s = 0), and x1 = x, xi (x) = ai x + ci with ai =

|bi | − 1 |b1 | − |bi | , ci = , (i = 2, . . . , s) |b1 | − 1 |b1 | − 1 420

SPECIAL ARITHMETIC FUNCTIONS

and xi (x) = ai x + ci with ai =

|bi |2 − 1 |b1 | − |bi |2 (i = s + 1, . . . , s + t) , ci = |b1 |1 |b1 | − 1

Furthermore, put D(b, k, x) = D(|b1 |k x1 (x), . . . , |bs |k xs (x); |bs+1 |2k xs+1 (x), . . . , |bs+t |2k xs+t (x)) and N = M k x1 . . . xs+t . Then we have z∈D (b,k,x)

sb (z) =

2s π t M − 1 n−1 · N log M N + N φ(log M N )+ O(N n log M N ) (205) I (α) 2

where the sum runs over all algebraic integers in D(b, k, x), and φ is a continuous periodic function of period 1. The special case of K = Q[i], b = −n ± i, s = 0, t = 1, n = 2 contains a result by P. J. Grabner, P. Kirschenhofer and H. Prodinger [152]: √ sb (z) = 2π N log N (b) N + N φ(log N (b) N ) + O( N log N (b) N ) (206) |z|2
where φ is a continuous periodic fluctuation of period 1. For the distribution of sb in residue classes, see [337]. The so-called ”fundamental domains” of a canonical number system (e.g. the ”twin dragon” from [224], as well as the fractal properties of number systems have been studied e.g. by W. J. Gilbert [142], S. Ito [192], I. K´atai and I. K¨ornyei [210], K.-H. Indlekofer, I. K´atai and P. Racsk´o [190], I. K´atai [209], J. M. Thuswaldner [339], S. Akiyama and T. Sadahiro [2], S. Akiyama and J. M. Thuswaldner [3], W. M¨uller, J. M. Thuswaldner and R. F. Tichy [263]. For digital version of Weyl’s lemma on exponential sums, with application to Waring’s problem with digital restrictions, see J. M. Thuswaldner and R. F. Tichy [342].

14

The symmetric signed digital expansion

If we write the word 1111010000100100000 (which is 106 in the binary system), the sublock 100 is contained three times. A generalization of the number of occurences (i.e. frequency) of a digit, one can consider the frequency of a block, i.e. 1 N 0≤n
(number of occurences of w) 421

CHAPTER 4

C. Heuberger and H.Prodinger considered a symmetric system with 3 q[181] have q4 , and for which an even base q, and digits a j ∈ − , . . . , 2 2 n= ajq j j≥0

Such a system is a priori redundant because of the existence of both ±q/2 but can q be made unique by the condition that |a j | = q/2 implies 0 ≤ sign(a j )a j+1 < . 2 This expansion is called symmetric signed digit expansion by P. J. Grabner, C. Heuberger and H. Prodinger [151], who study the subblock counting problem in such cases. If a block b = (bs , . . . , b0 ) is given, we denote its value by v(b) = bk q l . We l

use the Iverson notation [P], i.e. [P] = 1 if property P is true, and = 0, otherwise. Then the number of subblock occurrences of b will be [(ak+r −1 (n), . . . , ak (n)) = b] sb (n) = k≥0

A block b is called admissible, it if represents the number v(b) in the symmetric sb (n). Then the following theorem holds signed digit expansion. Let Sb (N ) = n
true: Let q ≥ 2 be an even integer and r ≥ 1. For an admissible block b = q (br −1 , . . . , b0 ) with |br −1 | < and b = (0, . . . , 0), the number of occurrences of 2 the block b in the symmetric signed digit expansions of the positive integers less than N satisfies Sb (N ) =

Q(b0 ) N logq N + h 0 (b)N + N Hb (logq N ) + o(N ) + 1)

q r (q

(207)

where Q(η) = q + 2 for η = 0; q if η = ±q/2; 1, otherwise; h k (b)e2kπi x , Hb (x) = k∈Z\{0}

with h k (b) =

2kπi log q Rmin (b0 ) ζ − , [v(b) < 0] + q −r v(b) + r 2kπi(log q + 2kπi) log q q (q + 1)

2kπi Rmax (b0 ) −r −ζ , [v(b) < 0] + q v(b) + r for k = 0, log q q (q + 1) 422

SPECIAL ARITHMETIC FUNCTIONS

Rmin (b0 ) − h 0 (b) = logq [v(b) < 0] + q v(b) + r q (q + 1)

Rmax (b0 ) −r − logq [v(b) < 0] + q v(b) + r − q (q + 1)

Q(b0 ) 1 1 1 1 − r r+ + + r −1 − q (q + 1) 2 log q q +1 q (q + 1)

−r

q $ q% , − (η − 1)mod q ≥ 2 2 q% q $ Rmax (η) = + ηmod q < 2 2 The function Hb (x) is a periodic continuous function of period 1, and mean 0. (ζ (s, x) denotes the Hurwitz zeta function.) The case r = 1 was discussed in [181], and that of q = 2, r = 1 (via Dirichlet series and Mellin-Perron summation formula) in [340]. The following central limit theorem is valid, too: 2 1 Q(b0 ) lim car d n < N : sb (n) < r logq N + x Vb logq N = N →∞ N q (q + 1)

and where

Rmin (η) = −

1 =√ 2π

x

e−t

2 /2

(208)

dt

−∞

where r −1 Q(b0 ) 1 1+2 [bi = b0 , . . . , br −1 = br −1−i ] + Vb = r q (q + 1) qi i=1

Q(b0 ) Q(b0 ) q 1− − 2(r − 1) r +2 r q (q + 1) q +1 q (q + 1)

15

Infinite sums and products involving the sum-of-digits function

There exist many interesting infinite sums or products involving the sum of digit or related functions. The sum s(n) of decimal digits of n has the generating function ∞ n=0

x s(n) t n =

∞

n

n

n

(1 + xt 10 + x 2 t 2·10 + · · · + x 9 t 9·10 )

n=0

423

(209)

CHAPTER 4

for |t| < 1, from which we can deduce that ∞ n=1

s(n) 10 = log 10 n(n + 1) 9

(210)

For a proof of (210) see e.g. J. M. Borwein and P. B. Borwein [32]. More generally, ∞ sq (n) q = log q (211) n(n + 1) q −1 n=1 For q = 2 there exist sums as (see J. P. Allouche [4] and J. P. Allouche and J. Shallit [10]): ∞ 2n + 1 π2 s2 (n) 2 = , (212) n (n + 1)2 9 n=1 ∞ 8n 3 + 4n 2 + n − 1 17 (s2 (n))2 = + log 2 2 2 4n(n − 1)(4n − 1) 24 n=1

For

∞

1 1 s2 (n) k − n (n + 1)k n=1

=

2k (1 − 21−k )ζ (k) k 2 −1

(213)

(214)

see [32]. Let D(n) = ∗ i, where the star means that the product is taken over those i where the ith binary digit of n is non-zero. Then ∞ n=0

π/2 e − e−π/2 (−1)s2 (n) =2 16s2 (n) (D(n))4 π2

(215)

∗ Let t (n) = i, where the sum is taken over the non-zero digits of n base 2 (e.g. t (10112 ) = 4 + 0 + 2 + 1 = 7). Then ∞ (−1)s2 (n) n=0

t (n) + 1

=

∞

2 · (−1)n 2 n=−∞ 3n + n + 2

(216)

The infinite product

s2 (n) ∞ 2n + 1 (−1) n=0

2n + 2

= 2−1/2

(217)

is due to D. R. Woods and D. Robbins [355]. J. O Shallit [306] generalized it to sq (n). 424

SPECIAL ARITHMETIC FUNCTIONS

Let q ≥ 2, 1 ≤ j ≤ q − 1 and cn , dn be the unique integers satisfying qn ≤ cn , dn < q(n + 1) and sq (cn ) ≡ j − 1 (mod q), sq (dn ) ≡ j (mod q). Then ∞ 1 + cn = q −1/q 1 + d n n=0

(217 )

For q = 2 one gets (217). Another generalization is the following: Let q ≥ 2 even and let an , bn be such that 2n ≤ an , bn < 2(n + 1), and sq (an ) ≡ 0 (mod 2), sq (bn ) ≡ 1 (mod 2). Then √ ∞ q 1 + an = (217 ) 1 + b q n n=0 Other general infinite products are treated in Allouche et al. [7]. For example, s2 (n) s2 (n)

∞ ∞ 2n + 1 s2 (n)·(−1) 2n + 1 −(3/2) 1/4 =2 , = 2−2/5 2n + 2 2n + 2 n=0 n=0 Let qw (n) count the number of (possibly overlapping) occurrences of the block w in the binary expansion of n. Then √

(−1)a0 (n) ∞ 2n 2 = , 2n + 1 2 n=1

a1010 (n) ∞ (4n + 2)(8n + 7)(8n + 3)(16n + 10) (−1) n=0

(4n + 3)(8n + 6)(8n + 2)(16n + 11)

√ 2 = , 2

see J.-P. Allouche and J. O. Shallit [11]. More generally, there exists an effectively computable rational function bw (n) such that 1 log2 (bw (n))X aw (n) = X −1 n≥0 for all X = 1, with |X | ≤ 1. See also J.-P. Allouche, P. Hajnal and J. O. Shallit [8]. Let a(n) = number of odd digits in odd places in the decimal expansion of n, b(n) = number of odd digits in n (e.g. a(901) = 2, a(811) = 1, b(901) = 2, b(811) = 2). Then ∞ a(2n ) n=1 ∞

2n

=

1 , 99

∞ b(2n ) n=1

1 1 b(n) 2 − n (n + 1)2 n=1 425

2n

1 = , 9 (218)

25π 2 = 297

CHAPTER 4

Let u(n) be the ”reflection” of n through the decimal point (e.g. e(123) = 0.321, e(90140) = 0.04109). Then ∞ n=1

10 u(n) = log 10 n(n + 1) 99

(219)

For the sums (215), (216), (218), (219), see [32]. Let g(n) be the number of digits ≥ 5 in n. Then ∞ g(2n ) n=0

2n

2 = , 9

(220)

see E. Levine [242] and D. Bowman and T. White [34]. Let e(n) denote the number of digit blocks of 11 in the binary expansion of n. Then ∞ e(n) 3 1 = log 2 − π (221) n(n + 1) 2 4 n=1 see Allouche [4]. Let v(n) be the number of terminating nines when the first n integers are written [n · 10−i ], and the set of values taken by the function v in base 10. Then v(n) = i≥1

is exactly the set of integers k written as k=

K ((10i − 1)/9)n i , i=1

where 0 ≤ n i < 10 for all i ∈ {1, . . . , K }. R. K. Kennedy et al. [216] proved that lim

M→∞

9 1 car d{v(n) : 1 ≤ n ≤ M} = M 10

By using Roth’s theorem of Diophantine approximation theory, it can be shown that (see [32])

∞ ∞ a(n) . 10 e(2n ) . 3166 and are transcendental (222) = = 10n 9 2n 3069 n=1 n=1 We note that by considering formal power series, J. P. Allouche [5] proved that if R is a polynomial in Q[X ] with R(N) ⊂ N, then the formal power series ∞

s p (R(n))X n

n=0

426

SPECIAL ARITHMETIC FUNCTIONS

is algebraic over F p (X ) (where F p is the field of p elements ( p=prime), and F p (X ) is the field of rational fractions with coefficients in F p ) if and only if the degree of R is less than or equal to 1. (223) If s (k) denotes the kth iterate of s , then the formal power series p p ∞

n s (k) p (n)X

n=0

is algebraic over F p (X ) if and only if k = 0 or k = 1.

16

(224)

Miscellaneous results on digital expansions

We now study some miscellaneous facts related to the digital expansions of numbers. First we consider certain iteration problems. Let f : N → N be an arithmetic function, and consider the f -iterative sequence n 0 = n, n 1 = f (n 0 ), . . . , n i = f (n i−1 ) = f (i) (n 0 ), . . . This sequence is called periodic, when there are integers i 0 ≥ 0, and l ≥ 1 such that, for all i ≥ i 0 , n i+l = n i . The set {n i0 , . . . , n i0 +l−1 } is called a cycle of f and l the length of the cycle. The function f is called periodic, if f has finitely many cycles. R. E. Burkard [45] gave a characterization of periodicity: f is periodic if and only if there exists an N ∈ N and for all n > N a k0 = k0 (n) such that for all n > N , (225) f k0 (n) < n The function f is called uniformly periodic if for all n > N there exists k0 ≥ 1, independent of n, such that f k0 (n) < n. m Let q ≥ 2 be a positive integer, and let n = a j q j (0 ≤ a j ≤ q − 1, j = j=0

0, m − 1), 1 ≤ am ≤ q − 1 be the q-adic expansion of n. Let P : {0, 1, . . . , q − 1} → m P(a j ). (226) N be an arbitrary function, and define f (n) = j=0

In 1960, B. M. Stewart [317] showed that there exists an integer N = N (P) such that f (N ) ≥ N and f (n) < n for all n > N

(227)

Therefore this function is uniformly periodic. Stewart gave also al algorithm for the evaluation of N in (226), which is of crucial importance for the determination of all cycles of f . 427

CHAPTER 4

We note that for special choices of P, result (226) has been rediscovered by many writers after Stewart. For a history of these facts, see also J. S´andor [300]. For P(a) = a t (t ∈ N, fixed), in the case of q = 10, t = 2 all cycles were computed by A. Porges, by K. Is´eki for t = 3, . . . , by K. Is´eki and I. Takada for t = 9. N. Yoshigahara obtained all cycles for q = 100 and t = 2. For general base q, H. Hasse and G. Prichett investigated the cycles in the case t = 2. For References, and related results, see H. J. J. te Riele [286]. For P(a) = a!, q = 10, G. D. Poole found all cycles of length 1, i.e. {1}, {2}, {145}, {40585}. P. Kiss

found all other cycles (i.e. two of length 2 and one of length 3). q (binomial coefficient), and q = 10, P. Kiss found all cycles For P(q) = a (one of length 1 and

of length 2). one q , and q = 10, P. Kiss found also all cycles (one each of length For P(a) = a! a 2, 4, 26 and 39). See [286]. A similar function is m f (n) = P(a j ), (228) j=0

where P : {0, 1, . . . , q − 1} → {0} ∪ N. B. M. Stewart [317] showed that there exists an integer N ≥ 1 for which f (N ) ≥ N and f (n) < n for all n > N , only in the case that 1 ≤ M ≤ M < B and P(a) ≥ a for at least one a ∈ {0, . . . , q − 1} Here M =

max

1≤a ≤q−1

(229)

P(a ) and M = max{P(0), M }.

Moreover, he showed that in all other cases either N = 0 or N doesn’t exist. For the particular case of P(a) = a + t (t ≥ 0 fixed), q = 10, S. S. Wagstaff, Jr. made a through computational study of the ten cases 0 ≤ t ≤ 9 (for t ≥ 10 we have f (n) > n for all n). He showed that for t = 1 and n > 18, either f (n) < n or f ( f (n)) < n, hence in this case f is uniformly periodic. For 2 ≤ t ≤ 6 he gave numerical evidence and a heuristic argument that every sequence remains bounded (so is periodic), whereas for t ≥ 7 virtually every sequence tends to infinity. Let now t m f (n) = a j = (sq (n))t (230) j=0

By Stewart’s method, it can be shown that for any fixed q and t, this function is uniformly periodic. S. P. Mohanty and H. Kumar in 1979 found all cycles for q = 10, 2 ≤ t ≤ 10. 428

SPECIAL ARITHMETIC FUNCTIONS

The function f (n) = n − n

(231)

where n is the integer formed by arranging the base q digits of n in descending order, and n by arranging them in ascending order, was first studied by D. R. Kaprekar [203], [204]. For q = 10 he discovered that when we start with a four-digit-number, not all digits being identical, then within 8 steps the cycle {6174} is reached. We note that the following convention is generally adopted: if n is a k-digit number and f (n) has k − 1 digits, then a leading zero is added to f (n). (E.g. (q = 10) n = 1121, f (n) = 2111 − 1112 = 0999, f (2) (n) = 9990 − 0999 = 8991). When we neglect numbers all whose digits are equal, this means that iterating f on k-digit numbers leads always to k-digit numbers. J. H. Jordan studied cycles of length 1 for any base q; for b = 2, 3, . . . , 12, C. W. Trigg listed all cycles for two-digit and five-digit numbers. H. Hasse presented a detailed study of the problem for two-digit numbers, for general q. G. Prichett gave a complete characterization of all cycles in the case of five q-adic integers. A. L. Ludington in 1979 showed that for any fixed base q there exists an r0 ∈ N such that for all r -digit numbers with r > r0 there is no cycle of length 1, while G. D. Prichett, A. L. Ludington and J. F. Lapenta [279] showed that for q = 10 the only cycles of length 1 are 495 (for 3-digit numbers) and 6174 (for 4-digit numbers). For a variation of the two-digit Kaprekar routine, see A. L. Young [358]. Finally, let f (n) = (n + t)∗ (t ∈ N) (232) where t is fixed and m ∗ is the number obtained by reversing the digits of m. This function is not periodic (put t = 10), since the number of cycles is infinite, as was shown by Gorzkowski (see [286]) by the followimg example: Let n 0 = 102k+3 + 10k+1 + 1, and consider the iteration of f at n 0 . Then this f iterative sequence has a period length 36 · 10k (for any fixed k). W. Sierpinski [311] gave results, by showing that f is periodic for t = 1, 2, 3, 4, 7, 8, 9, 11. For t = 6 the question is open (q = 10). The iteration of s(n) has been studied also by K. T. Atanassov [13] (this is a particular case of (sq (n))t studied above). For an application to ”staggered sums”, see A. G. Shannon and A. F. Horadam [307]. U. Balakrishnan and B. Sury [14] considered f (n) = s(d) (233) d|n

and proved that for all n > 1 there is an N = N (n) such that f (N ) (n) = 15 429

(234)

CHAPTER 4

For the iteration of f (n) = σ (n), σ (n) − n see Chapter 1, while for the iteration of f (n) = ϕ(n) and related functions, see Chapter 3. Two properties of the digits of Fibonacci numbers are the following. Let Am (n) be the number of Fibonacci numbers Fn with n ≤ N and whose first digit in base q is m (where 1 ≤ m ≤ q − 1). Then L. C. Washington [348] proved that the Benford law holds true:

m+1 1 Am (N ) = logq (235) lim N →∞ N m Let now x(k) be the number of indices n such that Fn has k decimal digits. Then x(k) ∈ {4, 5} for k ≥ 2. Let A(N ) be the number of k with 2 ≤ k ≤ N such that x(k) = 5. Then A. Guthmann [165] proves that A(N ) = α N + o(1) as N → ∞, (236) √ 1+ 5 − 4 = 0.78497 . . . The proof is based on Baker’s where α = log 10/ log 2 method on linear forms in logarithms. There exist result also for base q palindromes. If n = a0 + a1 q + · · · + am q m with ai ∈ {0, . . . , q − 1}, am = 0, then n is called a base q palindrome (not confusing with the notion of a palindromic sequence x0 , . . . , xn for which xk = xn−k for 0 ≤ k ≤ n) if ak = am−k for all k = 0, 1, . . . , m (thus the sequence of digits is a palindromic sequence). For some properties of base q palindromes, see e.g. I. Korec [233], M. Harminic and R. Sot´ak [178]. In 1998 F. Luca [244] has considered the palindromes in a row of the Pascal triangle. He posed the following questions: 1. Given a positive integer q > 1, find the values of n for which, if one writes each binomial coefficient belonging to the n + 1th row of the Pascal triangle in base q, the sequence resulting by concatenating all these blocks of digits (in the natural order), is a palindrome. By abuse of terminology, we will refer to the n + 1th row of the Pascal triangle for such n as a base q-palindrome. In fact the question is equivalent to the following one: 2. What are the values of n for which

n (237) k is a base q-palindrome for all k = 0, 1, . . . , n? In [244] it is shown that if q = 10, then n ≤ 4. 430

(238)

SPECIAL ARITHMETIC FUNCTIONS

Further, in [245] Luca showed that for any q > 1 there exist only finitely many n’s satisfying the requirement of question 2. More precisely: (239) 1) There exists a constant N (q) such that, if n has the property that all binomial coefficients of list (217) are base q palindromes, then n ≤ N (q). 2) N (2) = 3, N (4) = 5, N (6) = 7. Moreover, if q ∈ {2, 4, 6}, then N (q) < q. For palindromes, as sequences, see C. S. Queen [280]. In 1959 R. L. Goodstein [148] introduced automorphic numbers. A positive integer a is said to be automorphic in a scale q > 1, with index k, if a2 ≡ a

(mod q k )

(240)

For example, 376 is automorphic in the scale 10 with index 3, since 3762 ≡ 141376. He proves the following results: There are automorphic numbers in the scale q, with all indices, if and only if n is expressible as a product of two relatively prime factors. (241) Let q be the product of relatively prime factors u, v, and let k ≥ N be chosen so that u k ≡ 1 (mod v N ). Then the remainder when u k is divided by q N is an automorphic number in the scale q with index N . (242) Finally, Goodstein proved that if q is a product of relatively prime factors u, v and k is chosen so that u k ≡ 1 (mod v), N −1

then the remainder when u kv is divided by q N is an automorphic number in the scale q with index N . (243) In fact, theorem (243) remains true, if one replaces u by any w divisible by u and prime to v. A. R. Pargeter [269] had identified four automorphic numbers in scale 10. In 1993 T. Gagen [136] defined n-automorphic numbers (in scale 10), with index k by an ≡ a

(mod 10k ),

(244)

and examined the case n = 5 in detail. For k = 2a · 5b + 1 he obtained some more general results. In 1991 E. B. Knisch [239] introduced modular idempotent numbers a, having the property a 2 ≡ a (mod 10l(n) ), (245) where l(n) denotes the number of decimal digits of n. For example, 52 = 25, 762 = 5776 so both 5 and 76 are modular idempotents, but 57762 = 3362176 shows that 5776 is not of this type. 431

CHAPTER 4

Let n =

a j · 10 j−1 (0 ≤ a j ≤ 9), and put τ (n, k) =

5k j=1

a j · 10 j−1 (k ≥ 1),

j≥1

and γ (n) = min{τ (n 2 , k) : τ (n 2 , k) > n, k ≥ 1}. Let P1 = 5, Q 1 = 6, and define for n ≥ 2 Pn , Q n recursively by Pn+1 = γ (Pn ), where γ (Q n ) =

k

Q n+1 = γ (Q n ) + (10 − 2sk ) · 10k−1 ,

s j · 10 j−1 , sk = 0.

j=1

Then for all n ≥ 1, Pn and Q n are modular idempotents. In fact, m ≥ 2 is modular idempotent if and only if m = Pn or m = Q n for some n ≥ 1. (246) Finally, we mention that there exists an extensive literature on other expansions (which we do not consider here) and various digit functions and various property (e.g. ergodic) related e.g. to continued powers and roots (see e.g. D. J. Jones [197], [198], G. P´olya and G. Szeg¨o [276]), continued fractions (see e.g. W. B. Jones and W. J. Thron [199], I. J. Good [147], D. Hensley [179], O. Jenkinson and M. Pollicott [194], beta-expansions (see e.g. A. R´enyi [285] and W. Parry [270], D. W. Boyd [36], F. Blanchard [30], V. A. Rohlin [287], J. B. Pesin [273]), f -expansions and BolyaiR´enyi expansions (see e.g. A. R´enyi [285], O. Jenkinson and M. Pollicott [195]), etc. For various representations of real numbers, see J. Galambos [139].

432

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457

Chapter 5

STIRLING, BELL, BERNOULLI, EULER AND EULERIAN NUMBERS

5.1

Introduction

This Chapter is divided into two major parts: Stirling and Bell numbers, and Bernoulli, Euler and Eulerian numbers. These classical topics occur in practically every field of mathematics, in particular in combinatorial theory, finite difference calculus, numerical analysis, number theory, and probability theory. Our aim is to study the many aspects of these numbers, and to point out important connections or applications in number theory and related fields. In fact this Chapter underlines the many connections and aspects of certain parts of Discrete mathematics to almost the entire mathematics.

5.2 1

Stirling and Bell numbers Stirling numbers of both kinds, Lah numbers

Let S(n, k) denote the number of partitions of a set with n elements into k nonempty blocks. Then S(n, k) > 0 for 1 ≤ k ≤ n, and S(n, k) = 0 for 1 ≤ n < k. Put S(0, 0) = 1 and S(0, k) = 0 for k ≥ 1, S(n, 0) = 0 for n ≥ 1. These numbers were introduced by J. Stirling in his book ”Methodus Differentialis” [424] from 1730 (see also O. Schl¨omilch [394], J. F. Steffensen [423], 459

CHAPTER 5

N. Nielsen [333], G. Boole [56], Ch. Jordan [237] for early references, and various other notations), and are called now as the Stirling numbers of the second kind. These numbers are treated also e.g. in J. Riordan [365], [367], L. Comtet [116], P. A. Mac Mahon [295], N. E. N¨orlund [338], L. M. Milne-Thomson [310], M. Aigner [15], P. J. Davis [131], M. Abramovicz and I. A. Stegun [1], A. O. Gelfond [180], F. W. J. Olver [343], I. Tomescu [448], etc. See also D. S. Mitrinovi´c and R. S. Mitrinovi´c [311], where various tables of Stirling numbers can be found, too. The Stirling numbers of the second kind S(n, k) satisfy the recurrence relation S(n, k) = S(n − 1, k − 1) + k S(n − 1, k)

(n, k ≥ 1)

(1)

and, if we denote by (x)n = x(x − 1) . . . (x − n + 1) the falling factorial power, then n xn = S(n, k) · (x)k (2) k=0

where (x)0 = 1 by definition. This gives a ”horizontal” generating function of S(n, k). The ”vertical” generating function is given by

S(n, k)

n≥0

tn 1 = (et − 1)k n! k!

(k ≥ 0)

(3)

(Here one can take n ≥ k in place of n ≥ 0, since S(n, k) = 0 for n < k). Similarly, the ”rational” generating function is 1 S(n, k)u n−k = (k ≥ 1) (4) (1 − u)(1 − 2u) . . . (1 − ku) n≥0 In fact, S(n, k) can be explicitly calculated with the aid of binomial coefficients: k 1 i k (k − i)n (−1) (5) S(n, k) = k! i=0 i and have the following vertical, resp. horizontal recurrences (see e.g. [116]) n−1 n−1 S(l, k − 1), S(n, k) = l l=k−1 S(n, k) =

n

S(l − 1, k − 1)k n−l ,

(6) (7)

l=k

S(n, k) =

n−k (−1)i k + 1i S(n + 1, k + i + 1), i=0

460

(8)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

where xn = x(x + 1) . . . (x + n − 1) (with x0 = 1) denotes the rising factorial power. The number of all partitions of a set with n elements is B(n) =

n

S(n, k),

(9)

k=1

called also a Bell number (or exponential number). These numbers verify the recurrence n n Bk (k ≥ 0), B(n + 1) = k k=0 where B(0) = 1 by definition. The generating function of B(n) is given by tn B(n) = exp(et − 1) n! n≥0

(10)

(11)

The numbers B(n) can be represented also as the sum of a convergent series (Dobinski’s formula) 1 kn 1 1n 2n 3n B(n) = = + + + ... , (12) e k≥0 k! e 1! 2! 3! see also G. P´olya and G. Szeg¨o [352] for these basic properties. The notation B(n) for Bell numbers should not be confused with Bn , the nth Bernoulli number, which can be obtained (formally) from (1 + B)n − Bn = 0,

(13)

where B0 = 1. This gives 1 + 2B1 = 0, 1 + 3B1 + 3B2 = 0, 1 + 4B1 + 6B2 + 4B3 = 1 1 1 1 0, . . . , implying that B1 = − , B2 = , B4 = − , B6 = , . . . and B2n+1 = 0 2 6 30 42 for n ≥ 1. The generating function of (Bn ) is tn t (14) Bn = t n! e −1 n≥0 The Bernoulli numbers can be expressed by the Stirling numbers of the second kind as follows (see e.g. [237]): Bn =

n+1 (−1)k k! k=1

k+1 461

S(n, k)

(15)

CHAPTER 5

For Bernoulli numbers, very important in many parts of Mathematics, see e.g. N. Nielsen [334], L. Saalsch¨utz [373], K. Dilcher, et al. [144]. (For an electronic, updated version, see K. Dilcher [142]). The Bernoulli and Stirling numbers are important in some inversion formulas. For example, if n n F(n) = f (k) − λ f (n) (n = 0, 1, . . . ) k k=0 then

n+1 1 n+1 f (n) = F(k)Bn+1−k , if λ = 1, n + 1 k=0 k

and f (n) = −

n n m=0

m

F(m)

n−m k=0

k!S(n − m, k) , if λ = 1; (λ − 1)k+1

and reciprocally, see Z. Sun [431]. Let s ∗ (n, k) denote the number of permutations of an n-element set that have k disjoint cycles. Then s ∗ (n, n) = s ∗ (n, 1) = 1, s ∗ (n, k) = 0 for n < k, and s ∗ (n, k) = s ∗ (n − 1, k − 1) + (n − 1)s ∗ (n − 1, k)

(16)

The numbers s ∗ (n, k) are called the unsigned Stirling numbers of the first kind. These numbers satisfy a counterpart of (2), namely: xn =

n

s ∗ (n, k)x k ,

(17)

k=0

where xn is the rising factorial power, and s ∗ (n, 0) = 1 if n ≥ 1, s ∗ (0, 0) = 0. The numbers s ∗ (n, k) are nonenegative in contrast with the (normal) Stirling numbers of the first kind s(n, k) given by s ∗ (n, k) = |s(n, k)| = (−1)k+n s(n, k)

(18)

Then in place of (17) one can write (x)n =

n

s(n, k)x k

(19)

k=0

One has the recurrence formulae: s(n, k) = s(n − 1, k − 1) − (n − 1)s(n − 1, k) (n, k ≥ 1) 462

(20)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

with s(n, 0) = s(0, k) = 0, s(0, 0) = 1 = s(n, n); n−1

ks(n, k) =

(−1)n−l−1

l=k−1

s(n + 1, k + 1) =

n! s(l, k − 1), l!(n − l)!

n (−1)n−l (l + 1)(l + 2) . . . (n)s(l, k),

(21)

(22)

l=k

(n − k)s(n, k) =

n

(−1)

l=k+1

s(n, k) =

n

l−k

l s(n, l), k−1

(23)

s(n + 1, l + 1)n l−k

(24)

tn tn k x , x = (1 − t)−x = n! n! n≥0

(25)

tn tn k (x)n , x = (1 + t)x = n! n! n≥0

(26)

l=k

One can write also

s ∗ (n, k)

n,k≥0

s(n, k)

n,k≥0 n

s ∗ (n, k)u n−k = (1 + u)(1 + 2u) . . . (1 + (n − 1)u),

(27)

k=1 n

s(n, k)u n−k = (1 − u)(1 − 2u) . . . (1 − (n − 1)u)

(28)

k=1

From (26) one can deduce n≥k

s(n, k)

logk (1 + t) tn = , n! t!

(29)

which is similar to (3). From (27) one obtains n

s ∗ (n + 1, k + 1)x k = (x + 1)(x + 2) . . . (x + n),

(30)

s ∗ (n + 1, n + 1 − l)u l = (1 + u)(1 + 2u) . . . (1 + nu).

(31)

k=0 n l=0

463

CHAPTER 5

The numbers s(n, k) may be expressed in terms of the Stirling numbers of the second kind (due to O. Schl¨omilch [394], see also [237]): s(n, k) =

n−k

(−1)

m=0

m

n−1+m n−k+m

2n − k S(n − k + m, m) n−k−m

(32)

Let L ∗ (n, k) denote the number of ways to partition an n-element set into k nonempty (linearly ordered) queues. Then L ∗ (n, n) = L ∗ (n, 1) = 1, L ∗ (n, k) = 0 for n < k, and L ∗ (n, k) = L ∗ (n − 1, k − 1) + (n + k − 1)L ∗ (n − 1, k)

(33)

These numbers are called the unsigned Lah numbers (or Stirling numbers of the third kind, see [457]). In 1954 I. Lah [269] proved that xn =

n

L ∗ (n, k)(x)k

(34)

k=0

where L ∗ (n, 0) = 1 if n = 0, L ∗ (0, 0) = 0. The (ordinary) Lah numbers L(n, k) are defined by L(n, k) = (−1)n L ∗ (n, k)

(35)

Then one has xn =

n (−1)k L(n, k)(x)k ;

(x)n =

k=0

n

(−1)k L(n, k)xk ;

k=0

see also I. Lah [270] and M. Petkovˇsek and T. Pisanski [347] (where the history of Lah numbers is also considered). For connections to Physics, see M. Schork [396].

2

Identities for Stirling numbers We mention several other identities for the Stirling numbers of both kinds: n+1

s(n, i)S(i, m) =

i=m n+1

S(n, i)s(i, m) =

i=m

464

0, n = m 1, n = m

(36)

0, n = m 1, n = m

(37)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

((36) and (37) are called also the orthogonality properties of the Stirling numbers of both kind); n+1 i s(n, i) = s(n − 1, m) + s(n − 1, m − 1), (38) m i=m and by inversion s(n, k) =

n

(−1)

m+k+1

m=k−1

m s(n − 1, m) k−1

(39)

These lead also to other formulae (see [237], pp. 187-189): n+1

s(i + 1, m + 1)S(n, i) = (−1)

m+n

i=m

n , m

n s(i, m)S(n + 1, i + 1) = , m i=m

n+1

n+1 n

i

i=m

S(i, m) = S(n + 1, m + 1),

(41) (42)

n , m−1

(43)

n+1 n n+i S(i, m) = m S(n, m), (−1) i −1 i=m

(44)

i s(n, i) = n · s(n, m) m−1

(45)

n+1

(40)

is(i, m)S(n, i) = (−1)

n+m

i=m

n+1

(−1)

m+i

i=m n

s(n, i + 1)S(i, m) = (−1)n+m+1

i=m

(n − 1)! m!

(46)

For the unsigned Stirling numbers s ∗ (n, k) of the first kind, one has (see Ch. Jordan [237], pp. 153, 165, 194, 339): n s ∗ (i, m) i=m

i!

=

s ∗ (n, m + 1) , (n − 1)!

∞ s ∗ (n, k) = 1, (k + 1)! k=0

465

(47)

(48)

CHAPTER 5

∞ s ∗ (n, k) k=1

k · k!

= ζ (n + 1) (where ζ is the Riemann zeta function), ∞ 1 k=1

k

·

s ∗ (n, k) = ζ (n + 1) − 1. (k + 1)!

(49) (50)

By using the metric theory of Pierce expansions, J. O. Shallit [398] has proved that ∞ m−1 n s ∗ (n, k) 1 s ∗ (n, k) =1− = s ∗ (i, m) for n ≥ 1, m ≥ 1; (k + 1)! (k + 1)! m! k=m k=0 i=1 ∞ k=1

H (k)

s ∗ (n, k) = ζ (2) + ζ (3) + · · · + ζ (n + 1), (k + 1)! ∞

H (k + 1)

k=1

s ∗ (n, k) = n + 1, (k + 1)!

(51) (52)

1 1 + · · · + is the kth harmonic number. 2 k As an application of (51), a formula for ζ (3), due to W. E. Briggs et al. [61] can be obtained: ∞ 1 H (k) ζ (3) = (53) 2 k=1 k 2

where H (k) = 1 +

Another consequence of (51) is ζ (3) =

∞ H (k)(H (k − 1) − 1) k(k + 1) k=1

(54)

For harmonic sums and the zeta function, see also [181]. For the arithmetic properties of H (n), see e.g. D. W. Boyd [60], A. Gertsch [182]. For the computation of Euler’s constant via H (n), see E. A. Karatsuba [243]. See also J. S´andor [383]. By using certain basic Stirling polynomials, H. Niederhausen [327] obtains i + j + qn s(i + j + ln, i + j + (l − 1)n) = i + j + (l − 1)n n j i + j + ln i + j + (l − 1)n −1 i + qk = · · i + (l − 1)k j + (l − 1)(n − k) i + lk i + (l − 1)k k=0 ·s(i + lk, i + (l − 1)k)s( j + l(n − k), j + (l − 1)(n − k)) 466

(55)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

where i, j, q, l, n are nonnegative integers so that all expressions in (55) are well defined. For l = 1, q = 0 and m = i + j + n, (55) implies the well known result: m− j m i+j s(k, i)s(m − k, j), s(m, i + j) = k i k=i

(56)

see G. C. Rota et al. [371]. For l = q = 0, m = i + j − n, the identity −1 i j i + j −1 i + j m i s(i + j, m) = s(i, k) s( j, m − k) m k m−k i k k=m− j

(57)

follows, where i < m, j < m, i + j ≥ m. By applying Nielsen polynomials, one obtains m− j m k − i m −i − j i + j s(m, i + j) = s(k, i)s(m − k, j), i i+j i k k=i+1

(58)

for i + j ≤ m, which is analogues to (56). There can be obtained identities for S(n, k), too; as well as mixed identities. We quote only the following: i + j − qn S(i + j − (l − 1)n, i + j − ln) = i + j − ln n i + j − (l − 1)n i + j − ln −1 i − qk j = · · · i − lk j − l(n − k) i − lk i − (l − 1)k k=0 ·S(i − (l − 1)k, i − lk)S( j − (l − 1)(n − k), j − l(n − k))

(59)

For l = q = 0, m = i + j − n one obtains an identity dual to (56), while for l = 1, q = 0, m = i + j + n we get an identity dual to (57). For similar identities, see H. W. Gould [196], too. Identities involving the Bernoulli and Stirling numbers of the first kind have been proved by S. Shirai and K. Sato [399]. Let m ≥ 0, 1 ≤ k ≤ m + 1 be integers. Then one has Bk s(m, k − 1) = k = m!

m+1

(−1)t

t=1

p1 +···+ pt =m+1 q1 +···+qt =k (qi ≥0) pi ≥1

s( p1 + 1, q1 + 1) . . . s( pt + 1, qt + 1) ( p1 + 1)! . . . ( pt + 1)! 467

(60)

CHAPTER 5

Let f (x, a) = a/(1 − (1 − x)a ) for a = 0. By comparing the coefficients of two distinct expansions of f (x, a) about x = 0, Shirai and Sato prove also that m+1 k=1

m+1 Bk (−1)t s(m, k − 1) = (−1)n m! k t=1

p1 +···+ pt =m+1 ( pi ≥1)

1 ( p1 + 1) . . . ( pt + 1)

(61)

In [338] the Bernoulli numbers of order k, Bn(k) are defined by (x − 1)(x − 2) . . . (x − k) =

k k r =0

r

(k+1) r Bk−r x

(62)

On the other hand, by (19), (x − 1)(x − 2) . . . (x − k) =

k

s(k + 1, r + 1)x r ,

r =0

which by (62) implies k − 1 (k) Bn s(k, k − n) = n

(k ≥ n + 1)

(63)

The Bernoulli numbers of higher order satisfy the recurrence (−1)

n

Bn(k)

n−1 n + 1 (k) 1 i Bi = (−1) (k − i) i n(n + 1) i=0

(64)

For certain special values of Bn(k) , see [311]. In 1976 L. Carlitz [77] introduced the numbers b(n, k) and B(n, k) by s(n, n − k) =

k

b(k, j)

j=1

S(n, n − k) =

k j=1

n+ j −1 , 2k

n+ j −1 B(k, j) 2k

(65)

Then one has b(k, j) = (2k − j)b(k − 1, j − 1) + jb(k − 1, j) for 1 ≤ j ≤ k, b(1, 1) = 1, b(1, j) = 0 for j > 1; 468

(66)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

and B(k, j) = (k − j + 1)B(k − 1, j − 1) + (k + j − 1)B(k − 1, j) for 1 ≤ j ≤ k, B(1, 1) = 1, B(1, j) = 0 for j > 1. (67) In fact, b(k, j) = B(k, k − j + 1) for 1 ≤ j ≤ k. (68) The following explicit formulae are true: B(k, k − j + 1) =

j 2k + 1 S(t + k, t); (−1) j−t j −t t=0

j j−t 2k + 1 b(k, k − j + 1) = s(t + k, t) (−1) j −t t=0

(69)

G. Mastroianni [296] used the Stirling numbers in order to study certain linear operators from Approximation theory. He then proved certain identities for generalized Stirling numbers. J. L. Littlejohn and R. Wellman [288] give new applications of S(n, k) to operatorial theory and differential equations. For identities concerning determinants of matrices whose entries are s ∗ (n, k) and S(n, k), see B. S. Chandramouli [101]. Various identities are proved for the generalized Stirling numbers (see the next section).

3

Generalized Stirling numbers

We now study certain generalized Stirling numbers. A. A. Broder [63] defined regular Stirling numbers (or r -Stirling numbers for short) Sr (n, k) for 0 ≤ r ≤ k ≤ n, as the number of partitions of the set {1, 2, . . . , n} into k nonempty disjoint subsets, such that the numbers 1, 2, . . . , r are in distinct subsets. It is easy to see that S0 (n, k) = S1 (n, k) = S(n, k) = Stirling number of the second kind. Broder proved that R(n, k, r ) = Sr (n + r, k + r ), (70) n n where R(n, k, x) = S(m, k)x n−m (x ∈ R) are the Stirling polynomials of m m=0 the second kind introduced by L. Carlitz [85]. R(n, k, x) has an expression by means of divided differences (see [63], [85]) R(n, k, x) = [x, x + 1, . . . , x + k]t n , 469

(71)

CHAPTER 5

implying Sr (n + r, k + r ) = [r, r + 1, . . . , r + k]t n

(72)

For divided differences and combinatorial identities, see L. Verde-Star [464]. The recurrence formula R(n, k, x) = (x + k)R(n − 1, k, x) + R(n − 1, k − 1, x) (0 < k ≤ n)

(73)

was given by Carlitz [85]. More relations were given by E. Newman [320]: R(n, k, x + 1) = (k + 1)R(n, k + 1, x) + R(n, k, x) (0 ≤ k ≤ n); (74) n−k−1 n (k − 1)k R(n, k, x) = n(k − 1)R(n − 1, k − 1, x) + [(x + k − 1)· j +k j=0 ·( j + k)R( j + k − 1, k − 1, x) − ( j + 1)R( j + k, k − 1, x)] (1 ≤ k ≤ n) (75) Neuman proves also inequalities for R(n, k, x). For example: (x + k)n − k(x + k − 1)n (x + k)n ≤ R(n, k, x) ≤ , for 0 ≤ k ≤ n, k! k!

(76)

with equality only for k = 0; R(n + 1, k, x)R(n − 1, k, x) ≤ (R(n, k, x))2 ≤ n n+1−k · R(n + 1, k, x)R(n − 1, k, x), (77) n−k n+1 where 0 ≤ k < n. The first inequality tells us that the sequence (R(n, k, x))n is logarithmically concave. If 1 ≤ k ≤ n, then ≤

k R(n, k, x) ≥ n R(n − 1, k − 1, x);

(78)

and if 1 < k < n, then x R(n − 1, k, x) <

n−k R(n, k, x) < (x + k)R(n − 1, k, x) < R(n, k, x). n

For any k ≥ 0, 1

1

R(k + 1, k, x) ≥ R(k + 2, k, x) 2 ≥ R(k + 3, k, x) 3 ≥ . . .

(79)

This give inequalities for the r -Stirling numbers Sr (n, k). For example, (76) implies (r + k)n (r + k)n − k(r + k − 1)n ≤ Sr (n + r, k + r ) ≤ for k ≤ n, k! n! which for r = 0 reduces to a result by H. Wegner [480]. 470

(80)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

By defining the r th complete symmetric functions h r (r ≥ 0) by h r = h r (x0 , . . . , xn ) = x0i0 . . . xnin

(81)

i 0 +···+i n =r

n +r terms and (where i 0 , . . . , i n ∈ {0, 1, . . . , r } and the sum involves r (x0 , . . . , xn ) ∈ Rn+1 + ), in [322] E. Neuman obtains the identity Sr (n + r, k + r ) = h n−k (r, r + 1, . . . , r + k)

(82)

Here certain new identities and inequalities for Sr , as well as for the q-binomial coefficient of Gauss are deduced. The generalized nth symmetric mean is n +r Hr (x0 , . . . , xn ) = x0i0 . . . xnin (83) n i +···+on =r 0

K. V. Menon [303] proved the logarithmic convexity of the means (83) for r = 1, 2, 3. In 1979 D. W. DeTempe and J. M. Robertson [443] have proved that Hr2 (x0 , x1 ) ≤ Hr −1 (x0 , x1 )Hr +1 (x0 , x1 )

(84)

The logarithmic convexity of Hr (x0 , . . . , xn ) without any restriction to r and n has been proved by E. Neuman in [323]. (85) He prove also the strong logarithmic concavity of (S(n, k))n , i.e. S 2 (n, k) > S(n − 1, k)S(n + 1, k) (2 ≤ k ≤ n)

(86)

We note that the strong logarithmic concavity of the sequence (S(n, k))k has been established by L. H. Harper [213] and E. H. Lieb [287]: S 2 (n, k) > S(n, k − 1)S(n, k + 1) (2 ≤ k ≤ n)

(87)

In the proofs of (85) and (86) the B-splines by H. B. Curry and I. J. Schoenberg [125] are applied. An identity connecting Sr (n, k) to the ”hyperharmonic numbers” Hnr is Hnr = Sr (n + r, n + 1)/n! where Hn0 =

n 1 Hir −1 for r, n ≥ 1. , Hnr = n i=1

471

(82 )

CHAPTER 5

For (82 ) and other identities, e.g. m n! m Sr (n + r, k) = Sr −t (n + r − m, k − t) t (n + t − m)! t=0 for 0 ≤ m ≤ r , see A. T. Benjamin et al. [42]. The numbers Hnr have been introduced by H. J. Conway and R. K. Guy [118], and in fact, n +r −1 r (Hn+r −1 − Hr −1 ), Hn = r −1 where Hn are the ordinary harmonic numbers. For another class of Stirling numbers, namely p-Stirling numbers, see R. Merris [306]. B. The Stirling numbers of fractional order have been introduced and studied by P. L. Butzer, M. Hauss and M. Schmidt [72]. Let be the Euler gamma function, and define for a ∈ R the binomial coefficient functions (x + 1) x = (88) (x ∈ R, x ∈ Z− ) a (a + 1)(x − a + 1) where Z− = {−1, −2, −3, . . . }. The falling and rising factorial functions will be (x)a =

(x + 1) (x ∈ Z− ), (x − a + 1)

xa =

(x + a) (x + a ∈ Z− 0) (x)

(89)

− where Z− 0 = Z ∪ {0}. The Stirling functions s(a, k) (k = 0, 1, . . . ) of first kind will be defined via,

(x)a =

∞

s(a, k)x k

(|x| < 1)

(90)

k=0

or equivalently by 1 s(a, k) = k!

d dx

k

(x)a

x=0

(a ∈ R, k = 0, 1, . . . )

(91)

We note that the series in (90) is well-defined for all a ∈ R on account of the properties of the gamma function. The series is absolutely and uniformly convergent for |x| < 1. For all a ∈ R, k ≥ 0 there hold the relations (see [72]) ∞ s( j, k) sin π(a − j) s(a, k) = · , (a + 1) j! π(a − j) j=k

472

(92)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

and s(a, k) =

∞ a s( j, k)s(a − j, 0) j j=k

(a ∈ Z− )

(93)

both series being uniformly convergent in a. The Stirling functions of the first kind s(a, k) satisfy the triangular recurrence relation s(a, k) = s(a − 1, k − 1) − (a − 1)s(a − 1, k), 1 (94) s(a, 0) = (1 − a)s(a − 1, 0) = (1 − a) Further, for a > n ≥ 0 they satisfy s(a + 1, k) =

n

(−1) j (a) j s(a − j, k − 1) + (−1)n+1 (a)n+1 s(a − n, k)

(95)

j=0

There holds the complex integral representation (w + 1) 1 1 dw, s(a, k) = · 2πi K (w − a + 1) w k+1

(96)

where K is a Jordan-curve in C for which 0 ∈ int K , and K ⊂ {z ∈ C : |z| < 1}. The Stirling functions of the second kind S(a, k) will be defined in analogy with (91) by 1 (a ≥ 0, k = 0, 1, . . . ) (97) S(a, k) = k x a x=0 k! where is the forward difference operator, namely f (x) = f (x + 1) − f (x), j f (x) = ( j−1 f )(x), j ∈ N. The following formulae hold true: k 1 k+ j k S(a, k) = ja, (−1) k! j=0 j 1 S(a, k) = 2πi

C

(a ≥ 0, k ≥ 0),

(w − 1)a dw (w − 1) . . . (w − k − 1)

(a > 0),

(98)

(99)

where C is an arbitrary rectifiable Jordan-curve in the semi-plane Re z > 1 containing all n ∈ {2, 3, . . . , k + 1}. For a > 0 there holds ∞ za = S(a, k)(z)k , (100) k=0

473

CHAPTER 5

1 . The series converges abthe abscissa of convergence being λ ≤ max 0, a − 2 solutely for |z| ≥ 1 + λ and uniformly on all compact subsets of the half plane Re z ≥ λ + ε for arbitrary ε > 0. For a > 1, k ∈ N they satisfy the triangular recurrence relation S(a, k) = k S(a − 1, k) + S(a − 1, k − 1), S(a, 0) = S(0, k) = 0, S(0, 0) = 1.

(101)

We note also the generating functions ∞

S(a + bn, k)

n=0 ∞

k xn 1 b (−1)k+ j j a e j x (a, b ≥ 0) = n! k! j=0

S(a, k)x k = e−x

k=0

∞ ka k=0

k!

(102)

x k (a ≥ 0, x ∈ R)

One has also lim

a→∞

S(a, k) 1 = , a k k!

lim

a→∞

S(a + 1, k) = k. S(a, k)

(103)

In [70] Butzer and Hauss prove several results for S(α, k), α ≥ 0, k ∈ N, viewed as a function of α. Among the results obtained are proofs of the continuity and differentiability of S, real integral representations, a representation in terms of the Weyl derivative of fractional order α, and connections with Bernoulli, Stirling and Bernstein polynomials, and with Bernoulli numbers of fractional order. C. Starting from a given sequence t = (tk )k≥0 , in 1972 L. Comtet [117] has given the following generalization of classical Stirling numbers. The Stirling numbers of the first kind st (n, k) associated to the sequence t are defined by n−1

(x − tk ) =

k=0

n

st (n, k)x k ,

(104)

k=0

and the Stirling numbers of the second kind, St (n, k), associated to the sequence t are given by n St (n, k)(x − t0 ) . . . (x − tk−1 ) (105) xn = k=0

Particularly, for tk = 1 one gets the binomial coefficients, while for tk = k the usual Stirling numbers. 474

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

I. Aldanondo [17] has given a generalization of finite differences by Dt0 xm = xm ,

Dtn+1 xm = Dtn xm+1 − tn Dtn xm

where t = (tk )k≥0 is a given sequence. Defining the shift, resp. identity operators E and I by E xm = xm+1 , I xm = xm , one can write: Dtn xm =

n−1 (E − tk I )xm k=0

Therefore Dtn xm

=

n

st (n, k)xm+k ,

xm+n =

k=0

n

St (n, k)Dtk xm

(106)

k=0

S. Toader and Gh. Toader [446] have introduced the Stirling numbers associated to a matrix T = (tn,m )n,m≥0 . Define the finite differences DT0 xm = xm , DTn+1 xm = DTn xm+1 − tn,m DTn xm (n, m ≥ 0), and define sT (n, m, k) and ST (n, m, k) by DTn xm =

n

sT (n, m, k)xm+k ,

xm+n =

k=0

n

ST (n, m, k)DTk xm

(107)

k=0

From the definition of DTn and by (107) one can deduce that sT (n, m, n) = 1, sT (n, m, n − 1) = −

n

tn−k,m+k−1 , . . . ,

k=1

sT (n, m, 0) = (−1)n tn−1,m . . . t0,m , and the triangular recurrences sT (n + 1, m, k) = sT (n, m + 1, k − 1) − tn,m sT (n, m, k),

(108)

where sT (n, m, k) = 0 for k < 0 or k > n. Similarly, for the numbers ST (n, m, k) we have ST (n + 1, m, k) = ST (n, m + 1, k − 1) + tk,m ST (n, m + 1, k)

(109)

The following vertical recurrences are valid: sT (n + 1, m, k) =

n

(−1)n− j sT ( j, m + 1, k − 1)

j=k−1

ST (n + 1, m, k) =

n

n

ti,m

i= j+1

ST ( j, m + n − j + 1, k − 1)

m+n− j−1 i=m

j=k−1

475

(110) tk,i

CHAPTER 5

If one considers the matrices sT (m) = (sT (n, m, k))n,k≥0 ,

sT (m) = (ST (n, m, k))n,k≥0 ,

then sT (m)ST (m) = S)T (n)sT (n) = I

(111)

These are generalizations of the classical results. For generalizations of orthogonality properties like (111), see K. S. Miller [307], H. W. Gould [197], S. Tauber [437] (”quasi-orthogonality”), [438] (”bilinearly recurrent orthogonality”). For Stirling numbers generalization via methods of generating functions, and recurrent sequences, see e.g. L. Toscano [449], J. M. Sixdeniers, K. A. Penson, and A. I. Solomon [403]. The matrix approach is adopted also by T. Bickel [48], where one can find symmetries and dualities e.g. between Lah and Stirling numbers, between matrices constant on rows and those constant on columns, between matrices and their inverses; etc. D. In 1979 L. Carlitz [86], and in 1985 F. T. Howard [221] have introduced and studied the degenerate weighted Stirling numbers. Let 0 ≤ k ≤ n be integers, λ, θ = 0 real numbers, |x| < 1. Then define the numbers s(n, k, λ|θ ) and S(n, k, λ|θ ) by θ−λ

(1 − x)

1 − (1 − x)θ θ

k = k!

∞

s(n, k, λ|θ )

n=k

and (1 + θ x)λ/θ ((1 + θ x)1/θ − 1)k = k!

∞

xn n!

S(n, k, λ/θ)

n=k

xn n!

(112)

(113)

There are the Howard degenerate weighted Stirling numbers [221], which could be defined also by basis-transformations. E.g. by (x + λ|θ )n =

n

S(n, k, λ/θ )(x)k ,

(114)

k=0

where (t|θ )n =

n−1

(t − kθ ) (n ≥ 1).

k=0

In fact the equivalence between (113) and (114) may be verifed by means of the recurrence relations S(n + 1, k, λ|θ ) = S(n, k − 1, λ|θ ) + (k − nθ + λ)S(n, k, λ|θ )

(115)

for n ≥ k ≥ 1, with S(0, 0, λ|θ ) = S(n, n, λ|θ ) = 1 and S(n, 0, λ|θ ) = (λ|θ )n . 476

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

The Carlitz degenerate Stirling numbers [86] s(n, k|θ ) and S(n, k|θ ) are special cases when λ = 0. For λ = 0, θ → 0 one reobtains the classical Stirling numbers s(n, k) and S(n, k). See also [87]. Another related class of generalized Stirling numbers, called Dickson-Stirling numbers have been defined by (see L. C. Hsu, G. L. Mullen and P. J. S. Shiue [226]) Dn (x, α) =

n

S(n, k, α)(x − α)k

(n = 1, 2, . . . )

(116)

k=0

where Dn (x, α) =

[n/2] i=0

n−i n (−α)i x n−2i , with D0 (x, α) = 2. n−i i

In [227] the numbers S(n, k, λ|θ ) and S(n, k, α) have been applied in order to introduce two kinds of generalized Eulerian polynomials and numbers, which are associated with the construction of certain summation formulas for extended arithmeticgeometric series. For the numbers S(n, k, α) we quote (see [227]): ∞

Dn (k, α)x k = x α

k=α

n j=0

j!

S(n, j, α)x j , (1 − x) j+1

(117)

where α ≥ 0 is a given integer and |x| < 1. Another class of degenerate Stirling numbers, introduced via certain exponential Bell polynomials appears also in [222]. Hsu and Shiue [228] have defined a class of generalized Stirling numbers S(n, k; α, β, γ ) as follows: (x|α)n =

n

S(n, k; α, β, γ )(x − γ |β)n

(118)

k=0

where (x, α)n is defined as in (114). For some combinatorial and statistical applications of these numbers, see R. B. Corcino, L. C. Hsu and E. L. Tan [120]. For combinatorial applications of some weighted Stirling and related numbers, see also C. A. Charalambides [102]. In [229] L. C. Hsu and H. Q. Yu defined a ”Stirlingtype pair” {S 1 (n, k), S 2 (n, k)} = {S(n, k; α, β, r ), S(n, k; β, α, −r )}. For α = 1 the S 1 and S 2 are essentially Howard’s degenerated weighted Stirling numbers. In [229] quasi-Schl¨omilch formulas, while in [120] orthogonality relations, generating functions, congruence properties, etc., are worked out. 477

CHAPTER 5

E. A common generalization of the Stirling numbers, and Lah’s numbers are the Sα,β (r, m) numbers studied by E. G. Tsylova [457]. These are defined by the identity m

Sα,β (r, m)

r =0

m−1

(x + βh) =

h=0

m−1

(x + αk)

(119)

k=0

It is easy to see that from S−1,0 (k, m), S0,−1 (k, m) and S1,−1 (k, m) one can reobtain the numbers s(m, k), S(n, k) and L(n, k), respectively. Starting with the generating function

r ∞ 1 1 − (1 − αx)β/α xm = Sα,β (r, m) (120) r! β m! m=r and applying the Cauchy residue theorem, Tsylova proves also that if α = β, and r, m → ∞ so that 0 < m − r = O(m 1/2 ), then m−r m 1 (α − β)r Sα,β (r, m) = · (1 + o(1)) (121) 2 r For generalized Stirling and Lah numbers, see also C. G. Wagner [470]. F. General factorial numbers were introduced by L. Verde-Star [465], or (n) H. Niederhausen [328]. The general factorial powers xa,b and factorial numbers f a,b (n, k), F a,b (n, k) (where n, k are nonnegative integers, and a = b real numbers) are defined by x − nb x (n) −1 xa,b = a−b a−b n−1 and

(n) xa,b =

f (a,b) (n, k)x k

(122)

(k) F (a,b) (n, k)xa,b

(122 )

k≥0

xn =

k≥0

appear in [328]. Here (·)n−1 denotes a falling factorial. Since f 1,0 (n, k) = s(n, k) and F 1,0 (n, k) = S(n, k), the factorial numbers a,b f (n, k) and F a,b (n, k) are generalizations of the Stirling numbers. Some properties of factorial numbers are f a,b (n, 0) = F a,b (n, 0) = δn,0 and F a,b (n, k) f a,b (k, m) = δn,m (123) k≥0

(i.e. orthogonality), where δn,m is the Kronecker delta; f a,b (n, n) = (a − b)−n ; F a,b (n, n) = (a − n)n ; ck f ac,bc (n, k) = f a,b (n, k); cn F ac,bc (n, k) = F a,b (n, k); f a,b (n, k) = (−1)n f b,a (n, k); F a,b (n, k) = (−1)k F b,a (n, k). 478

(124)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

The numbers f a,b (n, k) can be expressed in terms of Stirling numbers of the first kind: n j −1 a,b s(n − 1, j − 1) (b(1 − n) − a) j−k (a − b)− j , f (n, k) = k − 1 j=k j −1 a,b (−bn) j−k (a − n)− j , s(n, j) f (n, k) = k − 1 j=k

n

(125)

for all n ≥ k ≥ 1. Closed form representations of F a,b (n, k) and f a,b (n, k) do exist, but are slightly complicated for the latter (see [329]) F a,b (n, k) =

k ( ja + (k − j)b)n (−1)k− j , j!(k − j)! j=0

(126)

f a,b (n, k) = (n − k)(a − b)k−n ·

j j γ =0

γ

γ

(−1)

k n+k 2k 1 · n − k j=0 k − j j!(n + j)

b γ+ j a−b

j+k

1 for n ≥ k ≥ 0. For a = = −b one obtains the so called central factorial numbers, 2 studied by P. L. Butzer, et al. [73], where (126) appears in this particular case. But the non-central Stirling numbers by M. Koutras [260] are covered, too. Indeed, (a − b)−n F a,b (n, k) is a non-central Stirling number, with non-centrality parameter kb/(b − a), in the sense of [260]. Similarly, (a − b)k f a,b (n, k) = sc (n, k) + (n − 1 − c)sc (n − 1, k), where sc (n, k) is the non-central Stirling number of the first kind with nb non-centrality parameter c = − 1. b−a For f 1+c,c (n, m) with n ≥ m ≥ 0 there exist identities, as: n i(1 + c) ki 1 (−1)n−m n! ∗ 1+c,c (n, m) = , (127) f i (1 + c)m i ki k i ! i=1 where the sum is taken for k1 + · · · + kn = m, 1 · k1 + · · · + n · kn = n and ki ≥ 0. For c = 0 this is a well-known statistic on permutations, where ki is the number of cycles of length i. Alternatively, f 1+c,c (n, m) = (−1)n−m

m n! (li (1 + c) − 1)(li −1) /li ! m! l +···+lm =n i=1 1

479

(128)

CHAPTER 5

where all li ≥ 1. Finally, there is a third sum of products: n−1 n(c + 1) − l − 1 1+c,c n−m (n − 1)! (n, m) = (−1) f n −1−l (m − 1)! l=m−1 l

m−1

1 li 1 +···+lm−1 =l i=1 (129) M. Kontras [260] derived the above sum for noncentral Stirling numbers of the first kind. For proofs of (127)-(129), see H. Niederhausen [330]. For F a,b (n, k) there exists the generating function formula (a, b, d ∈ R): tn F a,b (n, k)x k = exp{x(eat − ebt )}, (130) n! n≥0 k≥0 implying that

n≥k

F a,b (n, k)

tn 1 = (eat − bat )k , n! k!

n− j i+j n a,b F (n, i + j) = F a,b (m, i)F a,b (n − m, j), i m m=i n n F a,b (n, k) = (dk)n− j F a−d,b−d ( j, k), j j=k n a,b F (n, k) = F a,d (i, j)F d,b (n − i, k − j) i i, j

(131) (132) (133) (134)

Here formula (133) is equivalent to an identity of Verde-Star [465]. For (130)(132), see [328], and for (133)-(134), see [330]. The motivation for defining generalized factorial powers in the form from (122) comes from the recurrence (n) (n) (n−1) − (x + b)a,b = nxa,b (x + a)a,b (n) for all n ≥ 1. Together with the initial values (0)a,b = δ0,n , this recurrence shows (n) that {(x)a,b : n ≥ 0} is a sequence of binomial type. For combinatorial and statistical applications of f a,b (n, k) and F a,b (n, k), see [328], [331]. In [331] it is described a new bijection from n-permutations with η-colored head and τ -colored tail records to positive integer n-sequences where each member stays between certain bounds. This bijection extends a construction given by R. P. Stanley [416] for Stirling numbers. F. The theory of q-analogs has a long history, going back to Gauss who studied the q-binomial coefficients n n (q n − 1)(q n−1 − 1) . . . (q n−k+1 − 1) = = , for q > 1; k q k (q − 1)(q 2 − 1) . . . (q k − 1)

480

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

n = for q = 1, where q is a positive integer. Recently there has been considerable k interest in q-Stirling numbers. The q-Stirling numbers of the first kind, s[n, k], were introduced in H. W. Gould’s paper [198]. He studied also the q-Stirling numbers of the second kind S[n, k], but these numbers were earlier introduced by L. Carlitz [90], [91]. The numbers S[n, k] were studied also by S. C. Milne [308]. I. M. Gessel [185] gave an interpretation for s[n, k] as the generating function for an inversion statistic. For the numbers S[n, k], S. C. Milne [309] showed that these numbers could be viewed in terms of inversions on partitions, and B. E. Sagan [375] showed that there was also a version of the major index. S. C. Milne [309] and M. Wachs and D. White ([376]) proved that the q-Stirling numbers S[n, k] count restricted growth functions using various statistics. Milne gave also a vector space interpretation as follows. Let q be a prime power, and let V be a vector space over the Galois field Fq . Given any set of vectors S ⊂ V , let S denote the linear span of S. Now any sequences of one dimensional subspaces of V (i.e. lines) l1 , l2 , . . . , ln determines a flag V0 ⊂ V1 ⊂ · · · ⊂ Vk , obtained by deleting the repeated subspaces in 0 ⊂ l1 ⊂ · · · ⊂ l1 , l2 , . . . , ln . Now fix the above flag of dimension k, and let {v1 , . . . , vk } be a basis for Vk such that {v1 , . . . , vi } is a basis for Vi for all i. Then the group of upper unitriangular matrices in this basis acts on the sequences of n lines generating the given flag of dimension k. Milne [309] showed that S[n, k] is the number of orbits under this action, regardless the initial choice of a flag. The q-analogue of a positive integer n is defined by [n] = 1 + q + · · · + q n−1 The numbers s[n, k] can be defined by the recurrence s[n, k] = s[n − 1, k − 1] − [n − 1]s[n − 1, k] for n ≥ 1

(135)

and s[0, k] = δ0,k . The signless q-Stirling numbers of the first kind s ∗ [n, k] satisfy s ∗ [n, k] = s ∗ [n − 1, k − 1] + [n − 1]s ∗ [n − 1, k] for n ≥ 1 s ∗ [0, k] = δ0,k

(135 )

Finally, the numbers S[n, k] are defined by S[n, k] = S[n − 1, k − 1] + [k]S[n − 1, k] for n ≥ 1 (136) S[0, k] = δ0,k n For combinatorial interpretations involving inversions of , s ∗ [n, k], S[n, k], k see B. E. Sagan [376]. For partition lattice q-analogs related to q-Stirling numbers, 481

CHAPTER 5

see T. A. Dowling [146] and C. Bennett, K. J. Dempsey and B. E. Sagan [43]. For certain other statistics on the set of partitions of {1, 2, . . . , n} (e.g. the mak or mak’ functions of Steingrimsson) involving S[n, k], see G. Ksavrelof and J. Zeng [263]. There exist many identities involving the q-Stirling numbers. For example (see [198])

k j +n k− j k + n −k S[n, k] = (q − 1) (−1) , k− j j j=0

k n k− j n − j −k s[n, k] = (q − 1) q j ( j+1)/2 (−1) k − j j j=0

(137)

The following two identities are due to Carlitz [90], [91]: n k

=

n n j=k

[m]n =

n

j

(q − 1) j−k S[ j, k],

q k(k−1)/2

k=0

m k

(137 ) [k]!S[n, k]

Applying various M¨obius inversions to the first formula of (137 ), we can deduce other equivalent identities: n n j n−l ) ( 2 = (q − 1)n− j s ∗ [n, j], q l l j=l

(q − 1)

n−l

n j n− j n , S[n, l] = (−1) j l j=l

(137 )

n n− j n j−l j ( ) 2 q s [n, l] = (−1) j l j=l

n−l ∗

(q − 1)

For combinatorial proofs of these identities (including (137 )), see A. de M´edicis and P. Leroux [302]. The q-analogue of a Frobenius identity is due to A. M. Garsia and J. B. Remmel [175]: n k S[n, k][k]!x k [m]n x m = q (2) (1 − x)(1 − xq) . . . (q − xq k ) m≥0 k=0 482

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

The double generating functions are due to I. Gessel [185] and to Garsia and Remmel [176], respectively:

s ∗ [n, k]u k

n≥0 k≥0 n n≥0 k=0

∞ xn 1 − qk x = ; [n]! k=0 1 − (1 + (1 − q)n)q k x

q (2) S[n, k] k

uk n x = n!

(−1)m

m≥0

um [m]!

e[m]x

m≥0

um [m]!

(137 )

Denoting Sq (n, k) = S[n, k]; sq (n, k) = s[n, k], and extending the definitions of Sq and sq for all n ∈ Z (given by (137)) Gould proved also that sq −1 (−n − 1, k) = q k Sq (n, k) Sq −1 (−n − 1, k) = q k sq (n, k), n ∈ Z

(138)

He also discussed the relationship of these numbers to the q-Bernoulli numbers of higher order as defined by L. Carlitz [91], [93]. For certain other q-binomial coefficients, connected to the ordinary Stirling numbers, see [199]. k Let S[n, k] = q −(2) S[n, k], and the q-exponential polynomial given by en [z] =

n

S[n, k]z k .

k=0

In 1999 J. Cigler [109] has studied the determinant of a Hankel matrix, proving that

n+1 n+1 det( ei+ j [z])0≤i, j≤n = q ( 3 ) [0]![1]! . . . [n]!z ( 2 )

(139)

Recently R. Ehrenborg [147] has considered other determinants involving qStirling numbers, or ordinary Stirling numbers. For example, he proved that det(S[k + i + j, k + j])0≤i, j≤n = q (

)−(k3) [k]0 [k + 1]1 . . . [k + n]n ,

k+n+1 3

(140)

where S[k + i + j, k + j] are q-Stirling numbers of the second kind. His methods permits also to show that det

S(k + i + j, k + j) (k + i + j)!

= 0≤i, j≤n

2k(n+1) , (2k)!!(2k + 2)!! . . . (2k + 2n)!!

(141)

where S(k + i + j, k + j) are ordinary Stirling numbers of the first kind, and m!! = m(m − 2) . . . 2 is the double-factorial. 483

CHAPTER 5

For Hankel determinants of Bell polynomials, see P. Delsarte [132], who showed that (142)

N det(Bi+ j (x))0≤i, j≤N = k! x N (N +1)/2 k=0

See also A. Junod [238] for more results. For non-central q-factorial coefficients and q-Stirling numbers, see Ch. A. Charalambides [103]. G. The p, q-Stirling numbers have been introduced by M. Wachs and D. White [469]. The p, q-analogue of nonnegative integers are defined by [0] p,q = 0, [n] p,q =

pn − q n = p n−1 + p n−2 q + · · · + q n−1 for n ≥ 1. p−q

The signless p, q-Stirling numbers of the first kind s ∗p,q [n, k] and the p, qStirling numbers of the second kind S p,q [n, k] are defined by s ∗p,q [n, k] = s ∗p,q [n − 1, k − 1] + [n − 1] p,q s ∗p,q [n − 1, k] (n ≥ 1)

(143)

with boundary conditions s ∗p,q [0, 0] = 1, s ∗p,q [n, k] = 0 for k < 0 or k > n. Similarly, S p,q [n, k] are defined by S p,q [n, k] = S p,q [n − 1, k − 1] + [k] p,q S p,q [n − 1, k] (n ≥ 1),

(144)

with the analog boundary conditions as above. We note that there exist variations to these coefficients, for example S p,q [n, k], S p,q [n, k] defined by (see [375]); (145) S p,q [n, k] = q k−1 S p,q [n − 1, k − 1] + [k] p,q S p,q [n − 1, k], S p,q [n, k] = q k−1 S p,q [n − 1, k − 1], S p,q [n − 1, k − 1] + ( p[k − 1] p,q + q k−1 ) where [n] p,q = 1 + pq + ( pq)2 + · · · + ( pq)k−1 . For a unified combinatorial approach for the p, q-Stirling numbers s ∗p,q and S p,q , see A. de M´edicis and P. Leroux [302], where one can find also the following generating functions formula and recurrence relations: s ∗p,q [n, r ]y n−r x r = x(x +[1] p,q y)(x +[2] p,q y) . . . (x +[n −1] p,q y), (for n ≥ 1), r ≥0

(146) r ≥0

S p,q [k + r, k]x r =

1 1 1 · ... ; 1 − [1] p,q x 1 − [2] p,q x 1 − [k] p,q x

484

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

s ∗p,q [n + 1, k + 1] =

n

[n] p,q [n − 1] p,q . . . [ j + 1] p,q s ∗p,q [ j, k];

(147)

j=k

S p,q [n + 1, k + 1] =

n j [k + 1]n− p,q S p,q [ j, k]; j=k

(n − k)S p,q [n, k] =

n−k−1

j+1 j+1 S p,q [n − 1 − j, k]([1] p,q + · · · + [k] p,q );

(148)

j=0 n

(−1)n−k s ∗p,q [n, k]S p,q [k, m]

=

k=m n

1, n = m 0, n = m

S p,q [n, k](−1)k−m s ∗p,q [k, m]

k=m

s ∗p,q [n, k]

=

(orthogonality)

1, n = m 0, n = m

n j−k ∗ = (−1) j−k [n] p,q s p,q [n + 1, j + 1], j=k

S p,q [n, k] =

n−k (−1) j [k +1] p,q [k +2] p,q . . . [k + j] p,q S p,q [n +1, k + j +1], (149) j=0

n (1 − [2] p,q )(1 − [3] p,q ) . . . (1 − [k − 1] p,q )S p,q [n, k] = n − 1

(n ≥ 2).

k=2

The last identity is similar to an identity for Sq [n, k] due to A. Mercier [305]: n (−1)k q k−2 [k − 2]!Sq [n, k] = n − 1 for n ≥ 2 k=2

For far-reaching generalizations and analogs, see A. de M´edicis and P. Leroux [301]. Let U = (A, w), where A = (ai )i≥0 denotes a strictly increasing sequence of nonnegative integers, and w : N → K denotes a function from N to a ring K . The UStirling numbers of the first kind (without sign) and of the second kind, are defined by (150) sU∗ (n, k) = sU∗ (n − 1, k − 1) + wn−1 sU∗ (n − 1, k), SU (n, k) = SU (n − 1, k − 1) + wk SU (n − 1, k) with initial boundary conditions sU∗ (0, k) = δ0,k = SU (0, k), sU∗ (n, 0) = w0 w1 . . . wn−1 , SU (n, 0) = w0n . For a general study of U-Stirling numbers, see [301]

485

CHAPTER 5

(where another class of Stirling numbers, namely the partial p, q-Stirling numbers, are studied, too). For q-analogue triangular numbers, which arises in the context of distance geometry, see K. B. Stolarsky [426]. In [427] Stolarsky remarks that ([n]/n)

1 n−1

=

qr + qs 2

1 r +s

(151)

1 1 qn − 1 is an identity for n = −2, −1, − , and 2, where [n] = is extended to 2 2 q −1 all real numbers n, and r = r (n), s = s(n) are the roots (in some order) of the polynomial Pn (z), where 2Pn (z) = 50(1 + n)2 z 2 − 30(1 + n)(1 + n 2 )z + (n − 2)(n − 1)(1 + n 2 ) 2

In fact√r (n) → ρ, √ s(n) → σ as n → 0, where ρ = φ 2 /5, σ = φ /5, with 1− 5 1+ 5 φ= ,φ = (φ being the golden number). 2 2 Computer algebra shows that ([n]/n)

1 n−1

−

qr + qs 2

1 r +s

= (n − 2)(1 + n)(2 + n)(−1 + 2n)(1 + 2n)·

(q − 1)7 991 + 10n − 9n 2 (q − 1)6 8 9 − + 10 5 4 (q − 1) + O(q − 1) · 6 4 3 2 · 3 · 5 · 7 27 · 34 · 52 · 7 2 ·3 ·5 ·7 (152) Relation (151) is an identity for only five values of n listed, but for no other value of n. We note that the cases n = 0 and n = 1 the left side must be understood as a limit. This limits are in fact famous means, as the logarithmic and identric means. For these means see e.g. H. Alzer [19], H. Alzer and S. L. Qiu [23], B. C. Carlson [99], E. B. Leach and M. C. Sholander [274], A. P. Pittenger [351], J. S´andor [384], [385], E. Neuman and J. S´andor [324], [325], K. B. Stolarsky [428], M. K. Vamanamurthy and M. Vuorinen [461], C. E. M. Pearce and J. Peˇcari´c [344], J. S´andor and Gh. Toader [387], L. Losonczi and Z. P´ales [290], J. S´andor and T. Trif [388], etc. For an extension of the Gauss q-binomial coefficient to sequences, see W. A. Kimball and W. A. Webb [253]. For an application of q-Stirling numbers in the construction of a q-analogue of power series, see J. Satoh [391]. A q-analogue of Bell numbers has been introduced by S. C. Milne [308], [309]. m Let Sq satisfy the recurrence (in fact Sq [n, m] = q ( 2 ) Sq [n, m]):

Sq [n, m − 1] + [m]q Sq [n, m], Sq [n + 1, m] = q m−1 486

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

and put Bq (n) =

n

Sq [n, m].

m=0

Then Bq (n) satisfies the recurrence relation n n j q Bq ( j) Bq (n + 1) = j j=0 Recently, C. G. Wagner [471] has introduced another q-analogue by Bq∗ (n) = q n Bq (n) and studied it particularly for q = −1 (B−1 (n) arises in fermionic oscillators, see M. Schork [396]). One has B−1 (n) = (−1)n , if n ≡ 0 (mod 3); = (−1)n+1 , if n ≡ 1 ∗ (mod 3), and = 0, if n ≡ 2 (mod 3). Similarly, B−1 (n) = +1, if n ≡ 0 (mod 3); = −1, if n ≡ 1 (mod 3); = 0, if n ≡ 2 (mod 4). The generating function of B−1 (n) is given by ∞ xn ω −ω2 x 1 B−1 (n) = , e−ωx − e n! 1−ω 1−ω n=0 √ −1 + i 3 where ω = . 2 For a common generalization of binomial coefficients, Stirling numbers, and Gaussian coefficients, see B. Voigt [467]. H. Finally, we mention certain other generalizations of Stirling numbers. B. Richmond and D. Merlini [364] and P. Flajolet and H. Prodinger [161] have defined and studied Stirling numbers of complex argument. Therefore they have given a solution to the problem of R. L. Graham, D. E. Knuth and P. Patashnik [200] which asks for a good generalization of the Stirling numbers of the second kind S(n, k) for complex numbers n and k. See also G. Kemkes et al. [247]. For Stirling numbers of the first kind, see B. Gyires [206]. The number of permutations of n objects having ai cycles of length i is n!/(1a1 a1 !2a2 a2 ! . . . n an an !) and summing up such expressions in the right way gives n the refined Stirling numbers of the first kind, having m i cym 0 m 1 . . . m k−1 cles of length i (mod k) for some given positive integer k. These numbers have been introduced by J. Katriel [244], who considered also refined Stirling numbers n of the second kind, as the number of partitions of n objects into m 0 m 1 . . . m k−1 m i non-empty blocks of cardinality i (mod k). With the help of generating functions, recurrence relations can be derived. 487

CHAPTER 5

By studying a self-adjoint Legendre differential operator, W. N. Everitt, L. L. Littlejohn and R. Wellman [158] have introduced the Legendre-Stirling numbers P S(n, k), which is the coefficient of x n−k in the Taylor expansion of the rational k (1 − m(m + 1)x)−1 . function m=1

L. A. Sz´ekely [435] has defined the so-called Holiday numbers of the first and second kind, which provide results analogous to similar results for Stirling numbers of the first and second kind. H. Niederhausen [332] has studied Stirling numbers in the algebra of formal Laurent series.

4

Congruences for Stirling and Bell numbers

We now study congruences for Stirling numbers. A. Let p be a prime number. Since s( p, 1) = ( p − 1)!, by Wilson’s theorem one can write s( p, 1) ≡ −1

(mod p) (153) p On the other hand, by s( p, p − 1) = − ≡ 0 (mod p), by induction, and 2 relation (23) applied to n = p one gets that s( p, k) ≡ 0 (mod p) for 2 ≤ k ≤ p − 2,

(154)

where p is a prime, and s( p, k) are Stirling numbers of the first kind. See e.g. [116] pp. 54-55 for more details. Another proof of (154) follows by Lagrange’s theorem (x p ) = x(x − 1) . . . (x − p p + 1) ≡ x p − x (mod p), By (19), (x p ) = s( p, k)x k ≡ x p − x (mod p) k=0

which by s( p, p) = 1 and (153) gives (154). For this idea of using Lagrange and Wilson theorems for generalized Stirling numbers, see E. T. Bell [37]. Since s ∗ (n, k) = (−1)k+n s(n, k) (see (18)), so (154) holds true also for s ∗ ( p, k). For an extension of Fermat’s little theorem, with application to Stirling numbers - congruences, see K. Dilcher [143]. We note that (154) has an important application in the arithmetic properties of generalized harmonic numbers H (n, r ) = 1/(n 0 n 1 . . . n r ) (where r ≥ 0, n ≥ 1 are integers), since H (n, r ) =

n 0 +n 1 +···+nr ≤n

(r + 1)! ∗ s (n + 1, r + 2), n! 488

(155)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

see A. Gertsch [182]. Other congruence properties from [182] with applications, are e.g. s ∗ ( p + 1, r + 2) ≡ 0

(mod p 2 ) if 0 < r < p − 3 is odd ( p = prime);

s ∗ (2 p, r + 2) ≡ 0 (mod p 2 ) if 0 < r < p − 1 is odd

(156) (157)

By letting n = p in (16), by (154) one can deduce immediately s ∗ ( p − 1, k) ≡ 1

(mod p) (1 ≤ k ≤ p − 1)

Selecting n = p − 1, by the above result, and induction on k we get (see F. T. Howard [220]): s ∗ ( p − 2, k) ≡ 2 p−k−1 − 1

(mod p) (0 ≤ k ≤ p − 2)

More generally, if p is a prime, h > 0, 0 ≤ m < p, then h h ∗ (−1)h−i s ∗ (m, k − h − i( p − 1)) s (hp + m, k) = i i=0

(mod p)

A. Nijenhaus and H. S. Wilf [336] have proved that if k + n is odd, then s ∗ (n, k) is divisible by the odd part of n − 1, and Howard [220] has improved this assertion: if k + n is odd, then n(n − 1) ). s ∗ (n, k) ≡ 0 (mod 2 A common generalization of Wilson’s theorem, and other congruence properties is s ∗ (n + p, k) ≡ ( p − 1)s ∗ (n, k − 1) + s ∗ (n, k − p) (mod p), (158) see B. E. Sagan [377]. Taking n = 0, k = 1 in (158) and recalling that s ∗ (0, k) = δ0,k , we can deduce Wilson’s theorem ( p − 1)! ≡ p − 1 (mod p) ≡ −1 (mod p). Using the fact that s ∗ (n + 1, 1) = n! ≡ 0 (mod p) for n ≥ p, by induction on k, from (158) one gets s ∗ (n, k) ≡ 0 (mod p), if n > kp (159) Let n = n 1 p + n 0 (0 ≤ n 0 < p), k = k1 ( p − 1) + k0 (0 ≤ k0 < p − 1) in (158). Then via induction on n one can deduce ∗ k−1 n 1 s (n, n − k) ≡ (−1) s ∗ (n 0 , n 0 − k0 ) (mod p) (160) k1 This implies a result of H. Gupta [205]: p) if p − 1 np − k, and s ∗ (np, k) ≡ 0 (mod n s ∗ (np, k) ≡ (−1)m (mod p) if m( p − 1) = np − k m 489

(161)

CHAPTER 5

B. The method of proof of (158) is by using M¨obius inversion over the lattice of subgroups of a given Abelian group acting on a set. Such an idea originates to J. Peterson [346] who in 1872 used the action of the cyclic group of order p to prove the congruences of Fermat and Wilson. This method has been rediscovered many times (see L. E. Dickson [139]), but a systematic development was still lacking until the papers by G. C. Rota and B. E. Sagan [372] (who investigated congruences from groups acting on functions), J. H. Smith [414] (who has considered wreath products), or I. M. Gessel [186] (congruences for groups acting on graphs). The method can be summarized as follows. Let G be a finite group with identity element e, and let S be a finite set on which G acts. Given s ∈ S, the stabilizer of s is G s = {g ∈ G : gs = s} and the orbit of s is Os = {t ∈ S : t = gs for some g ∈ G}. It is well known that car d Os = car d G/car d G s . An element s ∈ S is said to be aperiodic whenever G s = e. By the above result, s is aperiodic iff car d Os = car d G, so we immediately have the following result: The number of aperiodic elements of S is divisible by car d G. (162) Theorem (162) is the base of proving many congruences. The only thing needed is the determination of the number of aperiodic elements. Let L(G) denote the lattice of all subgroups H ≤ G ordered by inclusion. For H ∈ L(G) define α(H ) = car d{s ∈ S : G s = H } Then (162) can be restated as α(e) ≡ 0

(mod |G|),

(163)

where |G| = car d G. Calculating α directly is difficult, but there is another function β which is easier to work with. Let β(H ) = car d{s ∈ S : G s ⊃ H }. Then clearly β(H ) =

α(K )

(164)

K ≥H,K ∈L(G)

By using M¨obius inversion (see Chapter 2), one obtains µ(H )β(H ) ≡ 0 (mod |G|),

(165)

H ∈L(G)

where µ is the M¨obius function of L(G). In order to obtain congruences from (165) we need only to substitute various groups G and sets S on which they can act. 490

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

For example, let G = C p the cyclic group of order p ( p prime). Then µ(e) = 1, µ(C p ) = −1, so (165) becomes β(e) ≡ β(C p ) (mod p)

(166)

Let S be the set of all permutations σ of {1, 2, . . . , n + p} with k cycles, with C p acting by conjugation. Then car d S = s ∗ (n + p, k), and (158) follows from (166). r By considering G = C rp (when µ(G) = (−1)r p (2) ), one can deduce an extension of (159): s ∗ (n, k) ≡ 0 (mod pr ) if n > kr p (167) The method of (165) applies also to the Stirling numbers of the second kind. Let S be the set of all partitions π of {1, 2, . . . , n + p} into k subsets, which is counted by S(n + p, k). The action on C p on subsets extends naturally to such partitions. Then β(e) = car d S = S(n + p, k), To compute β(C p ) there are two cases. If {1} is a singleton block of π then so are {2}, {3}, . . . , { p}. This means that there are S(n, k − p) ways to choose the rest of the blocks of π . If 1 is in a block of size at least two, then 2, 3, . . . , p are also in the same block. Thus the set {1, 2, . . . , p} is acting like a single element, which we shall denote by P, so we are just partitioning the set {P, p + 1, . . . , p + n} into k blocks. This gives a final contribution of S(n + 1, k) to β(C p ), and hence (166) implies S(n + p, k) ≡ S(n + 1, k) + S(n, k − p) (mod p)

(168)

This can be proved also by induction on n. From (168), via an inductive argument on h, the following can be proved (see [220]): If p is prime and h > 0, 0 ≤ m < p, then h h S(hp + m, k) ≡ S(m + h − i, k − i p) (mod p) (168 ) i i=0 Nijenhuis and Wilf [336] showed that if k + n is odd, the S(n, k) is divisible by the odd part of k, and Howard [220] improved this to the following: if k + n is odd, then k(k + 1) S(n, k) ≡ 0 (mod ) 2 This can be further refined: if n + k is odd, then S(n, k) ≡ 0

(mod

k 2 (k + 1) ), if k|n, 2

S(n, k) ≡ 0

(mod

k 3 (k + 1) ), if k 2 |n 2

and

491

CHAPTER 5

For n = 0 (168) gives S( p, 1) ≡ 1 (mod p); S( p, k) ≡ 0 (mod p) for 2 ≤ k ≤ p − 1,

(169)

in analogy with relation (154) for s( p, k). p By (169) and x p = S( p, k)(x)k one can reobtain (with a new proof) the k=0

Lagrange congruence (x) p ≡ x p − x (mod p). By repeated application of (168), and an inductive argument, one can obtain a result of H. E. Becker and J. Riordan [35]: If j is defined by p j ≤ k < p j+1 , then S(n + p j ( p − 1), k) ≡ S(n, k) (mod p) In the case of G = C rp one can deduce a more complicated recurrence (see also [377]): j r r i j − 1 + δi, j · (−1)i S(i, j)[− p( p − 1)]i− j l i j=0 i=0 l=0 ·S(n + (r − i) p + l, k − ( j − l) p) ≡ 0

(mod pr )

(170)

The periods of the Stirling numbers S modulo pr can be derived from the recursion above. One has (L. Carlitz [94]): If p j ≤ k < p j+1 , then S(n + pr + j ( p − 1), k) ≡ S(n, k) (mod pr )

(171)

See also H. Tsumura [451]. The minimum period of (S(n, k))n≥0 (mod p) is computed by A. Nijenhuis and H. S. Wilf [336]. They proved that this period is ( p − 1) p j if k ≥ p, and l.c.m.[o(1), . . . , o(k)] for k < p, where j is the greatest integer such that p j < k, and o(i) denotes the multiplicative order of i in the integers mod p (lcm denotes least common multiple). (171 ) A generalization of (169) appears in [372]: If a = [k/ p], then S( p j , k) ≡ S( p j−r , k) (mod p j−a−r +1 )

(172)

In particular, if 1 < k < p, then S( p j , k) ≡ 0

(mod p) ( j ≥ 1) 492

(173)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

C. The method can be applied also to the Bell numbers B(n), defined by (9). By letting G = C p , and S = the set of all partitions of {1, 2, . . . , n + p}, one can deduce a result of J. Touchard [450] from 1933: B(n + p) ≡ B(n + 1) + B(n) (mod p)

(174)

By iteration this implies a result of M. Hall [210]: B(n + ( p p − 1)/( p − 1)) ≡ B(n) (mod p)

(175)

We note that for n = 0, relation (174) implies B( p) ≡ 2 (mod p) for any prime p

(176)

For generalizations, see Williams [485] or Becker and Riordan [35], who considered the iterated Stirling numbers introduced by E. T. Bell [38], [39]. For s ≥ 0 define the numbers S(n, k, s) by S(n, k, 1) = S(n, k);

S(n, k, s) =

n

S(n, i, 1)S(i, k, s − 1)

(177)

i=0

When s = 0, the equation (177) forces S(n, k, 0) = δn,k . The corresponding Bell numbers are n S(n, k, s) (178) B(n, s) = k=1

Becker and Riordan prove that if p m−1 ≤ s < p m = q, then B(n + pq − 1, s) ≡ B(n, s) (mod p),

(179)

i.e. B(n, s) has the period pq − 1 modulo p. For the period of S(n, k, s) one has: Let j−1 j qm = pq , where q = p m . If qm−1 < s ≤ qm , and qm p s ≤ k < qm p s , then S(n + qmj (qm − 1), k, s) ≡ S(n, k, s) (mod p)

(180)

For extensions of these results, see L. Carlitz [94]. An extension of Hall’s result (175) is proved by W. F. Lunnon, P. A. B. Pleasants and N. M. Stephens [293]:

B n+p

r

pp − 1 p−1

493

≡ B(n)

(mod pr )

(181)

CHAPTER 5

pp − 1 is the minimal period of Bn p−1 (mod p), and showed that if the period of the residues is equal to T for a given prime p, then there exists a number c, depending on p, such that C. Radoux [360] conjectured that T =

B(c + q) ≡ 0

(mod p) for 1 ≤ q < p.

See also [357], [358] on arithmetic properties of B(n). Radoux’s conjecture has been verified for all primes p ≤ 17 (see N. Kahale [239]). In [239], for a given prime p, there is constructed an integer c such that the Bell numbers B(c + 1), B(c + 2), . . . , B(c + p − 1) (182) are divisible by p. For maximum zero strings of Bell numbers (mod p), see J. W. Layman [271], and J. W. Layman and C. L. Prather [272]. For generalizations of Carlitz’s and Radoux’s or Layman’s, Layman and Prather’s results, see A. Mazouz [297], [298]. For the p-adic method, see also a paper from 1976 by D. Barsky [33]. For congruences for the so-called Barrucand numbers, having the generating function exp(xe x ), see A. Mazouz [299]. The congruence B(mp) ≡ B(n + 1) (mod p) (183) is due to L. Comtet [116]. A. Gertsch and A. M. Robert [184] have proved that if p k n, then B(np) ≡ B(n + 1) (mod p k+1 ) (184) This was a conjecture of M. Zuber (see [184]). H. Tsumura [451] proved that B( p k ) ≡ (B( p) − 1)k + 1

(mod p),

for all k ≥ 1. L. Carlitz proved that B(m + nω p ) ≡ B(m) (mod p k+1 ), with ω p = 1 + p + · · · + p p−1 . A. Junod has recently shown that B(m + np ) ≡ s

n n j=0

B(m + np) ≡

j

s n− j B(m + j)

n n j=0

j

(mod p k+1 ),

B(m + j) (mod p k+1 )

494

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

where p k n and p is odd. When p = 2, one has modulo 2nZ2 : n ∗ n n n, if n is odd, and m ≡ 2 + (mod 3) B(m + j) ≡ B(m + 2n) − 4 j 0, otherwise j=0 For all these results, see [238]. The Bell polynomials Bn (x) are defined by n n Bk (x) (n ≥ 0), Bn+1 (x) = x k k=0

B0 (x) = 1.

Then B(n) = Bn (1). In 1976 C. Radoux [359] proved the following congruence: For k ≥ 1 and p prime, k

Bn+ pk (x) ≡ Bn+1 (x) + (x p + · · · + x p )Bn (x) (mod p A[x]),

(185)

where A is either the ring Z of integers or the ring Z p of p-adic integers (and f (x) ≡ g(x) (mod p A[x]) means that f (x) = g(x) + ph(x), where h(x) ∈ A[x] - the ring of polynomials with coefficients in A). For a new proof, see [184]. For x = 1 and k = 1, (185) reduces to Touchard’s congruence (174). I. E. Shparlinkij [401] has proved that the congruence B(n) ≡ λ (mod p) (186) 2p − 1 . has for any integer λ, a solution n with 0 ≤ n ≤ p−1 A. Gertsch [183] has applied the theory of congruences of Bell numbers in the study of Kurepa’s left-factorial function ! p = k p = 0! + 1! + · · · + ( p − 1)! (see [312], ch. XII, and [266]). Gertsch proved that k p ≡ B( p − 1) − 1

(mod p)

for an odd prime p. A. Junod [238] has recently shown that Kurepa’s conjecture (i.e. k p ≡ 0 (mod p)) is equivalent to the following: If p n, then p (B( p n − 1) − 1) for all odd primes p, and integers n ≥ 1. For a connection of Kurepa’s conjecture with Bernoulli numbers, see V. S. Vladimirov [466]. Congruences for Bell polynomials associated to a partition π of monomials have been studied by I. M. Gessel [186]. Associate with each partition π of {1, 2, . . . , n} 495

CHAPTER 5

a monomial π(x) = x1i1 x2i2 . . . xnin where i h is the number of blocks of size j in π . Define the polynomial Bn (x) by π(x), π a partition of {1, . . . , n}. Bn (x) = Then Bn+ p (x) ≡

p x1 Bn (x)

+

n i

i

xi+ p Bn−i (x) (mod p)

(187)

For congruences modulo m (m arbitrary positive integer) we quote the following results: n µ(d)a!S , a ≡ 0 (mod n) (188) d d|n In particular, if a < p, and p is the smallest prime divisor of n, then n , a ≡ 0 (mod n), µ(d)S d d|n

(189)

for any positive integer a. Similarly,

ϕ(d)a!S

n d

d|n

, a ≡ 0 (mod n),

(190)

for any positive integer a, where µ is the classical M¨obius function, while ϕ is the classical Euler totient function. For (189) and (190), see [372]. The following results can be found in [377]:

µ(d)

i n/d n/d

d|n

µ(d)

d|n

d|n

µ(d)

i

i=0

n/d i=0

n/d i

i

i

≡0

(mod n),

n S(i, j)d i− j B m + − i ≡ 0 (mod n), d j=0

n/d n/d i=0

n S(i, j)d i− j S m + − i, k − d j ≡ d j=0

(d − 1)n/d−i

(191) (192)

n s ∗ (i, j)d i− j s ∗ m, k − d j − + i ≡ d j=0

i

≡0

(mod n)

(193)

D. Let P(n, k) = k!S(n, k) denote the number of ”preferential arrangements” of {1, 2, . . . , n} with k blocks (a preferential arrangement order on the blocks of π ).

496

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

Thus P(n, k) is the number of surjections {1, 2, . . . , n} → {1, 2, . . . , k}. Such arrangements were first studied by Touchard [450], and rediscovered by O. A. Gross [202] and I. J. Good [195]. P(n, k) has period p − 1 (mod p) for all k, and for k ≥ p we have P(n, k) ≡ 0

(mod p)

(194)

For n ≥ pr −1 , P(n, k) has period pr −1 ( p − 1) and is congruent to 0 for sufficiently large k. In fact, Touchard [450] proved that P(m + ϕ(n), k) ≡ P(m, k) for any m ≥

max{ piai −1

: i = 1, r }, where n =

r

(mod n),

(195)

piai is the prime factorization of

i=1

n (and ϕ is Euler’s totient). n P(n, k). Then (see [450]): Let P(n) = k=1

P(m + ϕ(n), k) ≡ P(m, k)

(mod n),

(196)

for any m ≥ max{ piai −1 : i = 1, r }. In 1977 A. T. Lundell [292] has studied other divisibility properties of P(n, k) and of n,m = g.c.d.{P(n, k) : m ≤ k ≤ n} (197) He proved the following congruence: Let p be a prime, n = Ad + r where 0 ≤ r < d = p α ( p − 1), α ≥ 0 if p is odd, α ≥ 1 if p is even. If 1 ≤ t ≤ A, then t−1 A−1−i i A k!S(id + r, k) ≡ (−1) P(n, k) + (−1) i t −1−i i=0 t

≡ (−1)

k+t

(−1)

q≥0

·

t−1 i=0

(−1)i

pq

k ( pq)r · pq

A A−1−i ( pq)id t −1−i i

where ε = 1 if p is an odd prime, ε = 2 if p = 2.

497

(mod p t (α + ε))

(198)

CHAPTER 5

The following results give the exact conditions for p t to divide n,m for p an odd prime and t < p: Let p be an odd prime, n = A( p − 1) + r , where 0 ≤ r < p − 1, and let A = a0 + ai pi be the p-adic expansion of A with aγ = 0 if A = a0 (see i≥γ

Chapter 4). If A = a0 , let µ = max{a0 , t − γ − 1}. Then for 1 ≤ t < p, (1) If 1 ≤ t ≤ a0 and t ≤ r, then p t |n,m iff m > (t − 1)( p − 1) + r ; (2) If 1 ≤ t ≤ a0 and t > r, then p t |n,m iff m > t ( p − 1) + r ; (3) If A = a0 , and 0 ≤ a0 < t, then p t n,m ; (4) If A = a0 , 0 < a0 ≤ µ < t, and t − γ ≤ r, then p t |n,m iff m > µ( p − 1) + r ; (5) If A = a0 , 0 < a0 ≤ µ < t, and r < t − γ , then p t |n,m iff m < max{a0 , t − γ }( p − 1) + r ; (6) If A = a0 , 0 = a0 ≤ µ < t and t ≤ r, then p t |n,m iff m > µ( p − 1) + r ; (7) If A = a0 , 0 = a0 ≤ µ < t, and r < t, then p t |n,m iff m > t ( p − 1) + r.

(199)

For the divisibility of n,m by powers of 2, a table is provided in [292]. Let γ p (m) be the p-adic order of m (i.e. the highest power of the prime p which divides m). In 1994 T. Lengyel [278] showed that γ2 (P(2n , k)) = k − 1 if n ≥ l(k),

(200)

where l(k) is a function depending on k. For k ≥ 5, l(k) can be chosen so that l(k) ≤ k −2. In [279] Lengyel has shown that the 5-adic analysis of the Fibonacci and Lucas numbers Fn , L n plays a major role in determining γ5 (P(n, k)) for n = a · 5q , k = 2b · 5z , where a, b, q are positive integers such that (a, 5) = (b, 5) = 1, and 4|a, while z is a nonnegative integer. For instance, when q is sufficiently large and z > 0, one has the congruences b

P(n, k) ≡ −2 · 5 2 ·5 −1 L b·5z P(n, k) ≡ 2 · 5

b·5z −1 2

z

Fb·5z

(mod 5q+1 ), if b is even;

(mod 5q+1 ), if b is odd

(201)

Notice that for n = a·5q , 4|a, (a, 5) = 1, and q sufficiently large, γ5 (P(n, k)) can depend on n only if k is odd. More general results can be found in I. M. Gessel and T. Lengyel [187]. Let n = a( p − 1) pq , where p is a prime, (a, p) = 1, q sufficiently 498

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

large, and k/ p is not an odd integer. Then

k−1 + τ p (k), γ p (P(n, k)) = p−1

(1 ≤ k ≤ n),

(202)

where τ p (k) ≥ 0 (τ p (k) = 0 if k = multiple of ( p − 1)). They prove also that for any odd prime p, if k/ p is an odd integer, then γ p (P(a( p − 1) pq , k)) > q

(203)

k−1 + If q is sufficiently large, and k ≡ 5 (mod 10), then γ5 (P(n, k)) = 4 τ5 (k), where τ5 (k) = γ5 (k + 1) if k ≡ 9 (mod 20); γ5 (k) if k ≡ 10 (mod 20); γ5 (k + 2) if k ≡ 18 (mod 20); 0, otherwise. (204) For τ p (k) given by (202), Gessel and Lengyel conjecture that: 1) if k is divisible by 2 p, but not by p( p − 1), then

τ p (k) = γ p (k); 2) if k + 1 is divisible by 2 p, but not by p( p − 1), then τ p (k) = γ p (k + 1). F. W. Clarke [111] conjectures that if p is an odd prime and n > k, then p divides P(n, k) less than n times. For γ2 (S(n, k)) see D. M. Davis [129]. These exponents are important in algebraic topology (see also M. Crabb and K. Knapp [123] and D. M. Davis [130]). Let s p (m) denote the sum of digits in the p-ary representation of m (see Chapter 4 for this function). In [277] T. Lengyel proves that γ p (S(n, n − k)) ≥

s p (n − k) − s p (n) − k( p − 2) +1 p−1

(205)

for every prime p ≥ 3 an integers 1 ≤ k ≤ n. He proves also that for all 1 ≤ k ≤ 2n , γ2 (S(2n , k)) = s2 (k) − 1,

(206)

and that for all odd a ≥ 3 and all integers 1 ≤ k ≤ n, γa (P(n, n − k)) ≥ γa (n!) − k + 1

(207)

For all a ≥ 3 with γ2 (a) ≥ 2, the inequality (207) holds for 1 ≤ k ≤ n. For k = 1 we have γa (P(n, n − 1)) = γa (n!(n − 1)/2) ≥ γa (n!) − 1. 499

CHAPTER 5

For modified Stirling numbers divisibility, see J. Gonz´alez [194]. E. In 1990 F. T. Howard [224] has obtained certain congruences for Stirling numbers, in terms of the 2r th Bernoulli numbers B2r . Let p be an odd prime, and suppose p t n. Assume that 0 < 2r < 2 p − 2 and 1 < 2r + 1 < 2 p − 2. Then we have n n−1 B2r (mod p 2t ), s ∗ (n, n − 2r ) ≡ − 2r 2r (208) 2 n − 1 (2r + 1) n B2r (mod p 3t ), s ∗ (n, n − 2r − 1) ≡ − 2r + 1 4r n n + 2r B2r (mod p 2t ), S(n + 2r, n) ≡ 2r 2r (209) n 2 (2r + 1) n + 2r + 1 B2r (mod p 3t ). S(n + 2r + 1, n) ≡ 2r + 1 4r If n = p and 0 < 2r < p − 1, 1 < 2r + 1 < p − 1, congruences (208) reduces to results of J. W. L. Glaisher, while (209) reduce to results of N. Nielsen. In [224] there are proved also congruences in terms of Bernoulli numbers of higher order. As we have seen in relations (62), (63), the Stirling numbers of the first kind s(n, k) are connected to the Bernoulli numbers of order k and degree by s(k, k − n k + n (−k) k − 1 (k) Bn . n) = Bn . One can deduce similarly that S(k + n, k) = n n By studying congruence properties of Bn(k) , A. Adelberg [4], by the application of the above identities, has proved the following congruences for s(l, l − p), resp. S(l + p, l): If p is an odd prime, and l is a positive integer divisible by p, then l −1 2 l /2 (mod pl 2 ) s(l, l − p) ≡ (210) p In particular, γ p (s(l, l − p)) ≥ 2γ p (l), and γ p (s(l, − p)) = 2γ p (l) if γ p (l) > 1.

(211)

Similarly, if p is an odd prime, and p|l, then S(l + p, p) ≡ ((l/ p) + 1)l 2 /2 so

(mod pl 2 ),

γ p (S(l + p, p)) ≥ 2γ p (l), and γ p (S(l + p, p)) = 2γ p (l) if γ p (l) > 1 500

(212) (213)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

R. S´anchez-Peregrino [382] has studied congruences for ”poly-Bernoulli numbers” defined by n (−1)m m! S(n, m). Bnk = (−1)n (m + 1)k m=0 He proved that if k ≥ 2 and ( p − 1)|n, k + 2 ≤ p ≤ n + 1, then p k Bnk ≡ −1 (mod p) for primes p. For the numbers Bnk , see M. Kaneko [240]. F. In 1963 L. Carlitz [88] has considered congruences for S(n, k) for k ≡ r (mod m). For example n−m−1 S(m, 2m) ≡ (mod 2); m−1 (214) n−m−1 S(n, 2m + 1) ≡ (mod 2), m

l S(n, 3m) ≡ m−1

(mod 3) (m = n − 2l − 2),

l S(n, 3m + 1) ≡ m

(mod 3) (m = 2n − 2l − 1 or n − 2l − 2),

l S(n, 3m + 2) ≡ m

(mod 3) (m = n − 2l − 2)

(215)

Let θ0 (n) denote the number of coefficients S(n, 2m) that are odd, and θ1 (n) denote the number of coefficients S(n, 2m + 1) that are odd. Then by (214) one can write θ1 (n) = θ0 (n + 1) and θ0 (2n) = θ0 (n) + θ0 (n − 1), θ0 (2n + 1) = θ0 (n + 1). Let ω0 (n) = θ0 (n + 2) = number of coefficients S(n + 2, 2m) that are odd. Then we have ω0 (2n) = ω0 (n) + ω0 (n − 1), ω0 (2n + 1) = ω0 (n) (216) Among the properties of ω0 we mention the following: ω0 (2r ) = r + 1,

ω0 (2r s − 1) = ω0 (s − 1)

(r ≥ 0, s ≥ 1).

If n = 2r0 + 2r0 +r1 + · · · + 2r0 +r1 +···+rk (r0 ≥ 1, . . . , rk ≥ 1) and n j = 2r j + 2r j +r j+1 + · · · + 2r j +···+rk 501

(0 ≤ j ≤ k)

(i.e. n 0 = n),

CHAPTER 5

then ω0 (n j ) = (1 + r j )ω0 (n j+1 ) − ω0 (n j+2 )

(0 ≤ j < k)

(217)

where n k+1 = 0. When r0 = r1 = · · · = rk = r we get ω0 (n) =

εk+2 − ε−k−2 ε − ε−1

(r = 1),

(218)

1 where ε = {1 + r + (1 + r )2 − 4} and n = 2r (1 + 2r + 22r + · · · + 2kr ). 2 For r = 1, we have ω0 (2(2k − 1)) = k + 1. For upper bounds on ω0 (n) we quote ω0 (n) ≤ (1 + r0 )(1 + r1 ) . . . (1 + rk ) and log2 n , ω0 (n) < k+1

(219)

where the last one is optimal for r0 = r1 = · · · = rk = r . Put S(x) = ω0 (n). 0≤n≤x

Then S(2n + 1) = 2S(n) + S(n − 1),

S(2n) = S(n) + 2S(n − 1)

(220)

1 Since S(2r − 1) = (3r + 1) and S(3 · 2r ) = 3r +1 + 2r + 2, by putting λ = 2 log 3/ log 2, it follows that lim

r →∞

so lim inf n→∞

1 S(2r − 1) = , r λ (2 − 1) 2

lim

r →∞

S(3 · 2r ) = 31−λ , (3 · 2r )λ

(221)

S(n) S(n) < lim sup λ , proved by other methods by P. T. Bateman, too (see λ n n n→∞

[88]). For the distribution of Stirling (and Euler) numbers mod p in arithmetical triangles, see B. A. Bondarenko et al. [55]. For a given prime p, and positive integer N , they examine the number of elements of Stirling number having residues mod p equal to k for k ≤ p − 1 in both row N and the triangle with base N in the table of these coefficients. G. The associated Stirling numbers of the first kind d(n, k), and of the second kind b(n, k) (see [116]) are defined by k n ∗ , s (n, n − k) = d(2k − j, k − j) 2k − j j=0 502

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

and S(n, n − k) =

k

b(2k − j, k − j)

j=0

n 2k − j

respectively. These numbers can be defined also by means of the generating functions (− log(1 − x) − x)k = k!

∞

d(n, k)x n /n!,

n=2k

(e x − x − 1)k = k!

∞

b(n, k)x n /n!

n=2k

These imply at once that d(n, k) = (n − 1)d(n − 1, k) + (n − 1)d(n − 2, k − 1), with d(0, 0) = 1, d(n, 0) = 0 (n ≥ 1); d(n, 1) = (n − 1)! (n > 1); d(n, k) = 0 (n < 2k or k < 0), and b(n, k) = kb(n − 1, k) + (n − 1)b(n − 2, k − 1) with b(0, 0) = 1; b(n, 0) = 0 (n ≥ 1); b(n, 1) = 1 (n > 1); b(n, k) = 0 (n < 2k or k < 0). The numbers d(n, k) and b(n, k) are connected with s ∗ (n, k) and S(n, k) by the following formulae: k j n s ∗ (n − j, k − j), (−1) d(n, k) = j j=0 b(n, k) =

k n S(n − j, k − j) (−1) j j j=0

The above relations imply at once that d(n, k) ≡ 0 d( p, k) ≡ 0 b( p, k) ≡ 0 (mod p),

(mod (n − 1))

(n ≥ 2)

(mod p) (k ≥ 2), p prime; b( p + 1, k) ≡ 0

(mod p)

(k ≥ 2)

More generally, one has (see Howard [220]): If h > 0, 0 ≤ m < p, then d( ph + m, k) ≡ (−1)h d(m, k − h) (mod p), 503

CHAPTER 5

and b( ph + m, k) ≡

h h b(m + h − r, k − r ) (mod p) r r =0

Thus, b( ph + m, k) ≡ 0

(mod p) if m + h < k or m + h = k and m > 0, b(kp, k) ≡ 1

(mod p).

On the other hand, if n − k = ( p − 1)w, then w−1 b(n, k) ≡ (mod p); k−1 and if n − k = ( p − 1)w + v (1 ≤ v ≤ p − 2), then p−1−v w b(n, k) ≡ b(m + v, m) (mod p) k−m m=0 p−1 If v ≤ , the upper limit of summation can be replaced by v. 2 H. We now study congruences for q-Stirling numbers and related q-coefficients (see (135)-(136) for definitions). For the q-Stirling numbers of the second kind, B. E. Sagan [376] proved that if k ≡ k0 (mod d), then S[n, k] ≡ S[n − 1, k − 1] + [k0 ]S[n − 1, k] (mod [d]),

(222)

where as usual, [d] = 1 + q + q 2 + · · · + q d−1 . More generally, S[n, k] =

n

S[m − 1, d − 1]S[n − m, k − d] (mod [d])

(223)

m=d

for all n ≥ d. The generating function of S[n, k] is given by n≥0

S[n, k]x n =

xk , (1 − x)(1 − [2]x) . . . (1 − [k]x)

and one has the following congruence: If k = k1 d + k0 , where 0 ≤ k0 < d, then n≥0

S[n, k]x n ≡

xk {(1 − x) . . . (1 − [d − 1]x)}k1 (1 − x) . . . (1 − [k0 ]x)

(mod [d]) (224)

504

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

For the case d = 2 one can write: ∗ n − k2 − 1 (mod [2]), (225) S[n, k] ≡ n−k ∗ k k where is the integer part of . 2 2 Related to the periods of S[n, k], one has the following analogue of the Nigenhuis-Wilf result for S(n, k) (see (171 )). Let p be a prime. Then the sequence (S[n, k])n≥0

(mod [ p]) (or

(mod p))

has minimum period 1 if k = 0, 1; ( p − 1) p j if k ≥ p; p · lcm[0(1), . . . , 0(k)], if 1 < k < p, where j is the least integer such that p j ≥ k. (226) The sequence (S[n, k])n≥0 (mod [d]) is not periodic unless k = 0, 1 or d = 2, 3 and k ≤ d. In these cases the minimum period is 1 if k = 0, 1 or k = d = 2; 6, if k = 2, 3 and d = 3. (227) For the q-Stirling numbers s ∗ [n, k] the analogs of relations (222), (225) are the following: If n − 1 ≡ m (mod d), then

and

s ∗ [n, k] ≡ s ∗ [n − 1, k − 1] + [m]s ∗ [n − 1, k] (mod [d]),

(228)

s ∗ [n, k] ≡ s ∗ [n − 2, k − 2] + s ∗ [n − 2, k − 1] (mod [2])

(229)

The generating function of s ∗ [n, k] is given by s ∗ [n, k]x k = x(x + [1])(x + [2]) . . . (x + [n − 1]), k≥0

and reducing this equation modulo [2] and taking the coefficients of x k , we get: n ∗ ∗ 2 (mod [2]), (230) s [n, k] ≡ n−k n ∗ n where is the integer part of . As (225), this relation can be proved by combi2 2 natorial arguments, too. The coefficients s ∗ [n, k] are periodic in a trivial sort of way: For any d ≥ 1, s ∗ [n, k] ≡ 0 for n ≥ kd + 1. 505

(mod [d])

(231)

CHAPTER 5

This follows by induction on k and (135 ) applied to n = kd. For proofs of (224)-(231), see Sagan [376]. For the q-binomial coefficients of Gauss, one has ([376]): n n − d n − d

(232) ≡ + (mod φd (q)) k−d k k for all n ≥ d, where φd is the dth cyclotomic polynomial (see Chapter 3). The q-analogue of the famous Lucas congruence theorem for binomial coefficients (see Chapter 2) was first published in a paper by G. Olive [342] (and perhaps was known to Gauss) and rediscovered by J. D´esarm´enien [133], A. Blass [50], V.Strehl [430]. A different q-analogue is given in R. D. Fray [166]. The proofs from [342]-[50], [166] were all algebraic, the one from [430] uses combinatorial arguments. The result can be stated as follows: Let n = n 1 d + n 0 , k = k1 d + k0 , with 0 ≤ n 0 , k0 < d. Then n n n

1 0 ≡ (mod φd (q)) (233) k k1 k0 For a Lucas-type congruence for S(n, k), see R. S´anchez-Peregrino [382]. The q-multinomial coefficients can be defined as follows. Let k1 , . . . , kl be nonnegative integer, and put n = k1 + k2 + · · · + kl . Then

n n − k1 n − k 1 − k 2 − · · · − kl n = ... k2 kl k1 k 1 , k 2 , . . . , kl

n For n = k1 + · · · + kl , define = 0. Then (232) and (233) have the k 1 , . . . , kl following extensions to these coefficients (see [376]):

n k 1 , . . . , kl

≡

l i=1

n−d k1 , . . . , ki − d, . . . , kl

(mod φd (q)),

(234)

for all n ≥ d, and if n = n 2 d + n 0 , ki = ki1 + ki0 , with 0 ≤ n 0 , ki0 < d for 1 ≤ i ≤ l. Then

n nl n0 ≡ (mod φd (q)) k 1 , . . . , kl k11 , k21 , . . . , kl1 k10 , k20 , . . . , kl0 nl where is the ordinary multinomial coefficient. k11 , . . . , kl1 506

(235)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

For Lucas’ theorem for generalized binomial coefficients, see D. Wells [482], D. F. Bailey [30]. See also Kimball and Webb [253]. For congruences involving the number of involutions of permutations, see S. Chowla, I. N. Herstein and W. K. Moore [106], S. Chowla, I. N. Herstein and W. R. Scott [107], L. Moser and M. Wyman [316], B. E. Sagan [377]. For derangements and cycle indicators, see J. Riordan [367], I. M. Gessel [186] and B. E. Sagan [377]. For combinatorial properties and applications, see also L. Comtet [116].

5

Diophantine results

In this section we shall study some Diophantine results on the Stirling numbers of the second kind S(n, k). In 1989 A. Pint´er [348] (published in 1992) has shown that if a ≥ 1 is an integer, and if S(n, n − a) is an mth power for some n > a, m ≥ 3, then n < C1 = C1 (a), where C1 is an effectively computable positive constant depending on a. (236) The proof is based among others on results of A. Baker [32] on the solutions of hyperelliptic equations, and A. Schinzel and R. Tijdemans’s [393] results on the equation y m = P(x). Theorem (236) implies a result of R. Tijdeman [444] which says that there are effectively computable upper bounds for the solutions of equation n+a = y m (a > 1, m > 1, n ≥ 1, y > 1 integers) a with a = 2, m ≥ 3 and a = 3, m ≥ 2. See also K. Gy¨ory [207], [208], for equations of this type. Further A. Pint´er [349] has shown that for fixed 1 < a 0 is a constant. B. Brindza and A. Pint´er [62] have studied equations of type S(x, x − a) = by z ,

(238)

and S(x, a) = by z

(239)

For equation (238) they proved that if x > 2b · 16 · a , and y > 1, then z(7.5 + log z)−2 < 11000(log b + 8a log a + 3a). The same bound is valid in the case of s ∗ (n, k) (signless Stirling numbers of the first kind). a

507

8a

CHAPTER 5

For equation (239) they proved that if y > 1 (with a > 1), then z is bounded by an effectively computable constant in terms of a and b. Further, if z ≥ 3 is fixed, or z = 2 and a = 1, 3 then max(x, y) can be effectively bounded in terms of a and b. The main tool in the proofs is the theory of linear forms in logarithms. Recently, M. Klazar and F. Luca [254] have considered more general Diophantine problems. Let d > 1 be an integer, and let P, Q, R ∈ Q[X 1 , . . . , X m ] be nonconstant polynomials such that P does not divide Q and R is not a dth power. Let Si = S(n, ki ). Then, if k1 , . . . , km are any sufficiently large distinct positive integers, then Q(S1 , . . . , Sm ) ∈Z P(S1 , . . . , Sm )

(240)

for only finitely many positive integers n. Further, R(S1 , . . . , Sm ) = x d

(241)

for on;y finitely many pairs (x, n) ∈ N2 . The proofs of those results rely primaly on finiteness results of P. Corvaja and U. Zannier [122], which in turn are based on Schmidt’s subspace theorem, and hence are innefective in terms of bounding the size of solutions.

6

Inequalities and estimates

Inequalities for Stirling numbers have been considered earlier, too (see e.g. relation (80) for the r -Stirling numbers, or (76)-(79) for the Stirling polynomials of the second kind. A. L. H. Harper [213] and E. H. Lieb [287] established the strong log-concavity of the sequence (S(n, k))k (see relation (87)); in fact they proved the more precise estimate k−1 2 S (n, k) S(n, k − 1)S(n, k + 1) ≤ (242) k The strong log-concavity of (S(n, k))n is due to E. Neuman [323], see relation (86). He also showed the log-concavity of (R(n, k, x))n , see relation (77). B. C. Rennie and A. J. Dobson [362] have proved by elementary arguments, that 1 n n−r 1 2 r (243) (r + r + 2)r n−r −1 − 1 ≤ S(n, r ) ≤ 2 2 r D. C. Kurtz [267] defined the generalized Stirling numbers of the second kind (A(n, k)) by A(n, k) = A(n − 1, k − 1) + f (k)A(n − 1, k); A(1, 1) = 1, A(1, k) = 0 for k > 1, A(n, 0) = 0 for n ≥ 1, 508

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

where 1 = f (1) ≤ f (2) ≤ f (3) ≤ . . . is an increasing arithmetic function. He showed that n n−k ( f (k)) ( f (k))n−k (1 ≤ k ≤ n, n ≥ 1) ≤ A(n, k) ≤ (244) k For a more general recurrence, see J. B. Kim and Y. M. Lee [251], where it is shown also that expressions like s ∗ (n, k)r n−k and S(n, k)r n−k are strong log-concave (with respect to k). The sequence (s ∗ (n, k))k is strong log-concave. In fact a more precise result is due to E. Neuman [321]: k ∗ n−k+1 ∗ s (n, k − 1)s ∗ (n, k + 1) ≤ s (n, k), n−k k+1

(245)

for 1 ≤ k ≤ n. The proof is based on the theory of B-splines. Neuman proves also the following relations p p(n − k) + k p n S (n, k) ≤ S( p(n − k) + k, k), (246) k k for p = 0, 1, . . . ; 0 ≤ k ≤ n; where equality holds when p = 0, 1 and for arbitrary value of p in the case n = k. If 0 ≤ k ≤ n, then the following inequalities are also true: (n + 3 − k)(n + 4 − k) (n + 1 − k)(n + 2 − k) 2 S (n + 2, k) ≤ S(n, k)S(n + 4, k) (n + 1)(n + 2) (n + 3)(n + 4) (247) and n+1−k 2 n+2−k S (n + 1, k) ≤ S(n, k)S(n + 2, k) (248) n+1 n+2 J. C. Ahuja [13] defined numbers C(m, k) and D(m, k) by C(m, k) =

2m−k+1

(−1)

j+k

j=m+1

2m − k ∗ s ( j, j − m) j

with a similar expression for D(m, k) in which s ∗ ( j, j − m) are replaced by S( j, j − m). These numbers are suitably defined for k = 0, m ≤ k + 1 and (m, k) = (1, 0). It is shown that these numbers are strong log-concave with respect to k. (249) B. In [291] L. Lov´asz and K. Vesztergombi have studied the sum Sk =

k

(m!S(k, m))2

m=1

509

CHAPTER 5

and proved that for large k Sk ∼ (k!)2 /(2π(1 − log 2))(log 2)2k ≈ 0.72019(k!)2 · (2.081)k

(250)

The numbers m!S(k, m) (denoted by P(k, m) in (194)) for m = 1, 2, . . . , k, are distributed asymptotically normal as k → ∞. (251) n n The numbers P(n) = n!S(k, n) = P(k, n) (studied first by Touchard, k=1

k=1

see Part D of 5.24) occur also in the study of the following question: Let k ≥ 1 be an integer, and put S0,n = 1k + 2k + · · · + n k , S1,n = S0,1 + S0,2 + · · · + S0,n , . . . , Si+1,n = Si,1 + Si,2 + · · · + Si,n . Then √ P(k) + 1 2n + 1 Sn,n = √ ; (251 ) lim n+1 n→∞ 4 2π see [294]. T. Popoviciu conjectured the existence of an odd integer λ, with the limit λ being equal to √ . It is easy to see that λ = P(k) + 1 is odd. 2π The sequence (s ∗ (n, k))k is unimodal in k. Let el = el (n) denote the sum of the products of the first n positive integers (i.e. the lth elementary symmetric function). Then el (n) = s ∗ (n + 1, n + 1 − l) (see e.g. [116]). By the unimodality, it follows that there exists a value k = f (n) which maximizes ek (n); i.e. e1 (n) < e2 (n) < · · · < e f (n)−1 (n) ≤ e f (n) (n) > e f (n)+1 (n) > · · · > en (n) = n! In 1951 J. M. Hammersley [211] proved that e f (n)−1 (n) < e f (n) (n) for 1 ≤ n ≤ 188, and that 

f (n) =

(252)

∗

 h  , + 2 3 3  log(n + 1) + γ − log(n + 1) + γ − 2 2 (253) where [x]∗ is the integer part of x, γ is Euler’s constant, ζ is the Riemann zeta function, and −1.1 < h < 1.5. Thus, by a simple computation, if n > e5 , then

1 ∗ ≤ n − f (n) ≤ [log n]∗ (253 ) log n − 2

  = n − log(n + 1) + γ − 1 + 

ζ (2) − ζ (3)

510

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

Hammersley conjectured that (252) holds true for all n ≥ 1, i.e. the value of k for which ek (n) is maximal, for a given n is unique. This was proved in 1953 by P. Erd¨os [153]. The unimodality of (S(n, k))k was proved by L. H. Harper [213] who proved that there are at most two consecutive values of K n with S(n + k) ≤ S(n + K n ) for all k. He showed also that n as n → ∞ (254) Kn ∼ log n In 1978 E. R. Canfield [74] proved that if n is sufficiently large, and k0 = k0 (n) is the solution of the equation k0 (k0 + 2) ln(k0 + 2)/(k0 + 1) = n, the either [k0 ]∗ or [k0 +1]∗ , and possible both, yield(s) K n with S(n +k) ≤ S(n + K n ) for all k. The technique used involves the relation S(n, k) = p(n, k)(exp(r ) − 1)k n!/(r n k!),

(255)

in which p(n, k) is the probability that X 1 + · · · + X k = n, where X 1 , . . . , X k are independent random variables, each taking the value i with probability (exp(r ) − 1)−1r i /i! for i ≥ 1, and the parameter r is chosen to be ln(k0 + 2). By means of asymptotic analysis involving the characteristic function ϕ(ζ ) = E{exp(iζ (X −µ))}, an accurate approximation for p(n, k) can be obtained, valid for m − 1 ≤ k ≤ m + 2, where m = [k0 ]∗ . Using this approximation it follows S(n, m) > S(n, m − 1) and S(n, m + 2) < S(n, m + 1). √ n Kn n For an asymptotic evaluation of , when √ → ∞, − log n → ∞, we Kn Kn n quote V. V. Menon [304]. E. R. Canfield and C. Pomerance [75] proved that the number of n ≤ x such that S(n, k) has a repeated maximum is O(x c ) for some c < 1

(256)

C. A general theorem of log-convexity and concavity is due to E. A. Bender and E. R. Canfield [41]: If 1, Z 1 , Z 2 , . . . is a log-convex sequence of nonnegative integers, and the sequence (a(n)) is defined by

∞ ∞ Zj j a(n) n x = exp x , n! j n=0 j=0 then the sequence

a(n) n!

is log-concave, and (a(n)) is log-convex. 511

(257)

CHAPTER 5

Particularly, the Bell sequence (B(n))n≥0 is log-convex, while

B(n) n!

is log-

concave. (258) The weaker result that (B(n)) is convex can be found e.g. in [116]. More generally, N. Asai et al. [28] have defined the Bell numbers of order k, by bk (n) , expk (0)

B (k) (n) =

(n ≥ 0),

where expk (x) = exp(exp(. . . (exp(x)))), ! k times

and bk (n) are given by expk (x) =

∞ bk (n) n=0

n!

xn.

Since exp2 (0) = e, one gets ee

x −1

=

∞ B (2) (n) n x , n! n=0

i.e. B (2) (n) ≡ B(n). By applying Asai et al. [28] proved that the sequence (B (k) (n))n≥0 is log (k) (257), B (n) convex, is log-concave. (259) n! n≥0 This is an extension of (258) to k ≥ 2. Moreover, the following inequalities hold true: n + m (k) B (n)B (k) (m) (m, n ≥ 0) (260) B (k) (n)B (k) (m) ≤ B (k) (n + m) ≤ n In 1994 K. Engel [152] conjectured that the sequence (xn )n≥0 given by B(n + 1) − 1, B(n)

xn =

which is the average number of blocks in a partition of an n-set, is concave. E. R. Canfield [76] proved the conjecture for all sufficiently large n. (261) In the proof, the asymptotic of the sequence (B(n)) is used. Namely, 1

B(n) ∼ (R + 1)− 2 exp{n(R + R −1 − 1) − 1}, where R · e R = n → ∞. This is due to L. Moser and M. Wyman [317]. 512

(262)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

Other asymptotic relations for B(n) can be found in A. M. Odlyzko [339]: B(n) ∼ n!(2π R 2 exp(R))−1/2 exp(exp(R) − 1)R −n ;

(262 )

or if n = w log(w + 1), then B(n) ∼ (log w)1/2 w n−w w w as n → ∞ n n +O . More precise estimate is Here w = log n (log n)2 n log log n n(log log n)2 n log log n n + w= + +O . log n (log n)2 (log n)3 (log n)3

(262 )

The procedure can be iterated indefinitely to obtain expressions for w with error terms O(n(log n)−α ) for as large a value of α as desired. B(n) occurs also in the asymptotic expansion of the number an of Boolean sublattices of the Boolean lattice of subsets of {1, 2, . . . , n} (having the generating function zn an = exp(2z + exp(z) − 1)). One has n! n≥0 an ∼

n log n

2 B(n) as n → ∞,

see [339], and the References therein. M. A. Khan and Y. H. H. Kwong [248] have determined the number f (N , r ) of pairwise disjoint finite sequences of rolling an N -faced dice in which the ith sequence consists of ri trials. This is given by f (N , r ) =

m ∗ N ! r j S(ri , t j ), tj t j ! i=1,i= j

(263)

where r = (r1 , . . . , rm ), and the sum is taken over t1 , . . . , tm ≥ 1 with t1 + · · · + tm = N . For m = 2, r1 = p, r2 = q, it is shown that f (N , p, q) attains its minimal value over all p, q such that p + q = C and | p − q| ≤ 1. (264) D. The convexity or unimodality properties of sequences appear in many fields of mathematics, see e.g. the survey article of R. P. Stanley [419]. The q-analog of log-concavity has been defined by Butler and Stanley (see B. E. Sagan [378]). The qanalog of the relation ≤ will be denoted by ≤q . Given two polynomials f (q), g(q) ∈ R[q], we write f (q) ≤q g(q) if and only if g(q) − f (q) ∈ R+ [q]. 513

CHAPTER 5

Equivalently, for each k ≥ 0, the coefficient of q k in f (q) is less than or equal to the corresponding coefficient in g(q). (This is a partial order on R[q]). A sequence ( f k (q))k≥0 is said to be q-log-concave if f k−1 (q) f k+1 (q) ≤q ( f k (q))2 for all k > 0, and it is strongly q-log-concave if f k−1 (q) f k+1 (q) ≤q f k (q) fl (q) for all 0 < k ≤ l. For example, if [n] = [n]q = 1 + q + · · · + q n−1 , the the sequence ([n])n≥0 is strongly q-log-concave. L. M. Butler [67] and C. Krattenthaler [262] proved that the q-binomial coefficients

n n n+ j , , k k≥0 k+ j k n≥0 j≥0 are strongly q-log-concave. (265) P. Leroux [284] and B. E. Sagan [379], [378] proved that the q-Stirling sequences (s ∗ [n, k])k≥0 , (S[n, k])k≥0 , (S[n, k])k≥0 , (s ∗ [n + j, k + j]) j≥0 , (S[n + j, k + j]) j≥0 are all strongly q-log-concave. (266) In [67], [262] the above results were proved by an inductive argument. All the theorems are proved via combinatoric methods in [378]. This method was used also in [284]. The notions of q-log-concavity can be extended to sequences of polynomials f k (q) in N[q], when k ∈ Z. The sequence ( f k (q))k∈Z = . . . , f −2 (q), f −1 (q), f 0 (q), f 1 (q), f 2 (q), . . . is q-log-concave if f k−1 (q) f k+1 (q) ≤q ( f k (q))2 and strongly q-log-concave if f k−1 (q) fl+1 (q) ≤q f k (q) fl (q) for l ≥ k. The sequence q 2 , q + q 2 , 1 + 2q + q 2 , 4 + 2q + q 2 is q-log-concave, but not strongly q-log-concave, In [379] it is shown e.g. that for fixed n ≥ 0, the sequences n , (S[n, k])k∈Z , (s ∗ (n, k))k∈Z k k∈Z are strongly q-log-concave.

(267) 514

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

As a corollary of a more general result, in [379] appears also: For fixed k ≥ 0, the sequences

n , (s ∗ [n, n − k])n≥0 , (S[n, n − k])n≥0 n − k n≥0 are strongly q-log-concave. (268) ∗ As we can remark, the sequence (s [n, k])n≥0 is missing from the q-log-concavity results. It is absent because it is not q-log-concave even in the case q = 1 (take e.g. k = 1). P. Leroux [284] conjectured that for this sequence an ”almost” log-concavity holds true, namely s ∗ [n − 1, k]s ∗ [n + 1, k] ≤ [2](s ∗ (n, k))2

(269)

This has been proved in the case q = 1 by A. de M´edicis [300], and by L. Habsieger [209] in the general case (even in a more general setting of sequences). For a notion and results on p, q-log-concavity, see [284], [379]: The p, q-Stirling numbers S p,q [n, k] (see (150 )) are p, q-log-concave in n, i.e. for n ≥ 1 and 2 ≤ k ≤ n − 1, S p,q [n, k − 1]S p,q [n, k + 1] ≤ p,q (S p,q [n, k])2 ,

(269 )

where f ≤ p,q g is to be interpreted as ”g − f is polynomial in the variables p and q with non-negative coefficients”. More generally, for n ≥ 1, 2 ≤ k ≤ l ≤ n − 1 we have (269 ) S p,q [n, k − 1]S p,q [n, l + 1] ≤ p,q S p,q [n, k]S p,q [n, l] Relation (269 ) holds true also when S p,q is replaced with s ∗p,q . The p, q-unimodality in k if false even for p = 1, since the q-Stirling numbers ∗ s [n, k] and S[n, k] fail to be q-unimodal in k. E. The asymptotic development of the Stirling numbers of the first kind s(n, m) originates from Ch. Jordan [237] and L. Moser and M. Wyman [318]. Jordan proved that for m fixed one has (1 ≤ m ≤ n): s ∗ (n, m) ∼ (n − 1)!(log n + γ )n−1 /(m − 1)!, as n → ∞,

(270)

where γ is Euler’s constant. Moser and Wyman proved that (270) is valid also for m = o(log n). If m, n → ∞ such that m ≤ n − O(n α ), α fixed, 0 < α < 1, then √ s(n, m) ∼ (−1)n+m (n + R)/R m (R) 2π H ,

(271)

515

CHAPTER 5

where R is the unique solution of n−1 k=0

R = m, R+k

and

n−1

H =m−

k=0

√

R2 (R + k)2

Finally, if n − o( n) ≤ m ≤ n, then s(n, m) ∼ (−1)

n+m

n−m 1 n m m 2

(272)

In the proof, the saddle point method is used. Another procedure, which is probabilistic, was developed by E. A. Bender [40] to obtain the asymptotic behavior of general numbers a(n, k), with special applications to asymptotic behavior of the Stirling numbers. Previously L. H. Harper [213] had shown that a certain normalized sum of the S(n, k) tends to a Gaussian integral as n → ∞. Ch. Knessl and J. B. Keller [255] have used asymptotic methods of applied mathematics such as the ray method and the method of matched asymptotic expansions, to obtain asymptotic expansions for n 1. In this case, (−1)n−m s(n, m) ∼

n2 2

(n − 1)! (log n)m−1 if n = O(1); (m − 1)!

n−m

1 if n − m = O(1); (273) (n − m)!

n m − n mn −1/2 + 2 exp (n − m)(log β − 1) + n log 1 + ∼ A(β) β β β m < 1, if 0 < n ∼

where β is the unique solution of em/β = 1 + n/β, and the function A(β) has the limiting forms A(β) ∼ 1 for β 1;

A(β) ∼ (2πβ)−1/2 for β 1

√ D. Gardy [174] has proved by the saddle point method that e.g. if n = m+o( m), then n m n−m ∗ s (n, m) = (1 + o(1)) (274) 2 m 516

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

and if n − m → ∞ with n − m = o(m 2/3 ), then n m n−m (n−m)2 /6m ∗ s (n, m) = e (1 + o(1)) 2 m

(275)

In 1993 N. M. Temme [439] proved that n , s (n, m) ∼ e g(t1 ) m ∗

B

(276)

holds uniformly for 0 < m ≤ n as n → ∞. Here B = ψ(u 1 ) − n log(1 + t1 ) + m log t1 , with t1 = by

g(t1 ) =

1 [m(n − m)/nψ (u 1 )]1/2 u1

m and u 1 the unique positive solution of ψ (u) = 0, where ψ is given n−m ψ(u) = log[(u + 1)(u + 2) . . . (u + n)] − m log u.

H. S. Wilf [484] provided a considerably more explicit formula when m = O(1), m ≥ 1. He proved that for any fixed m we have (log n)m−2 (log n)m−1 (log n)m−2 1 ∗ , s (n, m) = γ1 + γ2 + · · · + γm + O (n − 1)! (m − 1)! (m − 2)! n (277) where the γ j are the coefficients in the expansion ∞ 1 (6γ 2 − π 2 )z 3 2γ 3 − γ π 2 + 4ζ (3) 4 γ j z j = z + γ z2 + = + z + ... (z) 12 12 j=1

(γ is Euler’s constant). In fact the coefficients (γn ) satisfy the recurrence " n−2 1 n− j−1 γ · γn + (n ≥ 1, γ1 = 1). (−1) ζ (n − j)γ j+1 γn+1 = n j=0 The asymptotic sign fluctuations of the sequence (γn ) have been studied by W. K. Hayman [215]. See also M. A. Evgrafov [159] (see pp. 92-94). Wilf finds for fixed m also the complete asymptotic series of the Stirling numbers of the first kind, by showing that this is obtained by equating the coefficients of x m on both sides of the relation " ∞ x log n 1 ∗ e (278) s (n + 1, m + 1)x m ≈ ψ j (x)/n j , exp n! m xr (x) j=1 517

CHAPTER 5

where the polynomials ψ j (x) are given by j (−x) j+1 j − 1 B j+1−s (−x)s + ψ j (x) = j ( j + 1) s=1 s − 1 s( j + 1 − s)

( j = 1, 2, . . . )

with Bk being Bernoulli numbers. This provides improvements of relation (277). For example, with error O(n −3 ) one finds that k 1 ∗ (log n)r s (n + 1, m + 1) = m! r! r =0

+

γk+1−r +

1 (γk−r + γk−r −1 )+ 2n

1 (−2γ − 3γ + 2γ + 3γ ) + O(n −3+ε ) k−r k−r −1 k−r −2 k−r −3 24n 2

(279)

H.-K. Hwang [230] has proved that formula (277) holds for m 1≤ in the mrange (log n) . More m ≤ θ log n for any θ > 0, if the error term is replaced by O m!n generally, he showed that for any θ > 0 and k ∈ N, one has asymptotically k s ∗ (n, m) m,r (log n) = +O (n − 1)! nr r =0

(log n)m m!n k+2

,

(n → ∞)

(280)

uniformly for 1 ≤ m ≤ θ log n, where m,r (X ) are certain polynomials in X of degree m − 1. One may roughly interpret theorem (280) as saying that the Stirling numbers ∗ s (n, m) are asymptotically Poisson distributed of parameter (log n). This certainly implies the asymptotic normality of s ∗ (n, m)/n! in the sense of convergence in distribution. A more convenient uniform asymptotic expansion is the following (see [230]): s∗ (n, m) satisfy, uniformly with respect to m, 2 ≤ m ≤ θ log n (θ > 0); m−1 , r= log n

µ−1 s ∗ (n, m) 1 g (k) (r ) (log n)m−1 pk (m) + Z µ (m, n) , (281) = + n! n(m − 1)! (1 + r ) k=2 (log n)k where pk (m) is a polynomial in m of degree [k/2]∗ (k = 0, 1, 2, . . . , m − 1), g(w) = 1 , µ is any fixed positive integer such that 2 ≤ µ ≤ m, and (1 + w) Kµ (log n)m µ/2 Z µ (m, n) = O (m − 1) + (log n)µ m!n 518

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

with

|g (µ) (z)| ≤ K µ for |z| ≤ θ.

In particular, p2 (m) = − p5 (m) = −

m−1 , 2

p3 (m) =

(m − 1)(5m − 11) , 30

m−1 , 3

p4 (m) =

p6 (m) = −

(m − 1)(m − 3) , 8

(m − 1)(3m 2 − 32m + 53) 144

In general, pk (m) =

∗ [k/2]

(−1)k− j (m − 1) j

∗∗

j=1

1c2 2c3 . . . (k − 1)ck , c2 !c3 ! . . . ck !2!c2 3!c3 . . . k!ck

where ∗∗ indicates that 2c2 + 3c3 + · · · + kck = k, c2 + c3 + · · · + ck = j, and c2 , c3 , . . . , ck ≥ 0. The following recurrence holds true: (k + 1) pk+1 (m) = −kpk (m) + (1 − m) pk−1 (m) p0 (m) = 1,

(k ≥ 1),

p1 (m) = 0

Result (281) is derived via applications of the singularity analysis of P. Flajolet and A. M. Odlyzko [160], and A. Selberg’s method [397]. According to a remark in [397] related to the distribution of the arithmetic function ω(n) (number of distinct prime factors of n, see D. S. Mitrinovi´c and J. S´andor [312], Chapter 5), we could replace the principal term of (281) by (log n)m /(m!n), but then the error term would become worse when m = θ (log n). R. Chelluri, L. B. Richmond and N. M. Temme [104] have defined s ∗ (x, y) for positive real x and y by (u + x + 1) du 1 · y+1 , (282) s ∗ (x, y) = 2πi C (u + 1) u where C is any Jordan curve which circles the origin in the counterclockwise sense. (u + x + 1) is a polynomial in u for integral x and has no singularities Note that (u + 1) for positive nonintegral x for |u| < x. Let ψ(u) = log[(u + 1)(u + 2) . . . (u + [x])] − y log u; u 1 be the unique positive solution of ψ (u) = 0 (see [439] for a proof that y u 1 is unique), and let t1 = and B = ψ(u 1 ) − x log(1 + t1 ) + y log t1 . Finally, x−y [x]−1 1 put g(t1 ) = [y(x − y)/xψ (u 1 )]1/2 and let H = y − u 21 /(u 1 + h)2 . Then u1 h=0 x uniformly for 0 < y ≤ x as x → ∞, when x and y are s ∗ (x, y) ∼ e B g(t1 ) y 519

CHAPTER 5

integers (this is exactly (276)). Moreover if 0 < y ≤ (log x)1/2 then for integral x and y x!(log x + γ ) y−1 (283) s ∗ (x, y) = {1 + O[(log x)−1/2 ]} y! If (log x)1/2 < y ≤ x − x 1/3 then for real x and y s ∗ (x, y) =

(x + 1 + u 1 ) y+1

(u 1 )(2π H )1/2 u 1

[1 + O(1/y)]

(284)

where the O-constant is independent of y. If x − x 1/3 ≤ y ≤ x, then independently of y for integral x and y, x + 1 y x−y ∗ s (x, y) = [1 + O(x 1/3 )] (285) y+1 2 The proofs are based on the Moser-Wyman [318] method, as well as that of Temme [439]. For Stirling numbers of fractional order s(a, k) (a ∈ R, k ∈ N), see (90), (92) or (97), (102)-(103). F. Asymptotic expansions of Stirling numbers of the second kind S(n, m) can be found in several papers, too. The asymptotic behavior of S(n, m) for n and m large seems to have been obtained first by Laplace (see e.g. L. Moser and M. Wyman [319], F. N. David and E. E. Barton [128]). In recent times, Ch. Jordan [237] (p. 173) has given the formulae S(n, m) ∼

mn , m!

S(n, n − m) ∼

n 2m m!2n

(n → ∞)

(286)

These formulae, however are applicable only for small values of m. In 1948 L. C. Hsu [225] obtained the complete asymptotic expansion

f 2 (k) f 1 (k) n 2k f t (k) −t−1 + 2 + · · · + k + O(n 1+ S(n + k, n) = k ) (t < k), 2 k! n n n (287) where f 1 , f 2 , . . . are polynomials in k whose coefficients can be found; e.g. 1 f 1 (k) = (2k 2 + k), 3

f 2 (k) =

1 (4k 4 − k 2 − 3k), 18

1 (40k 6 − 60k 5 − 2k 4 − 63k 3 + 133k 2 − 48k). 810 This is a generalization of (286), but it is shown to be a complete asymptotic expansion for fixed k (i.e. when m is fixed in (286)). f 3 (k) =

520

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

Moser and Wyman [319] have given the exact formula n −m m−1 k S(n, n − m) = q Am kq m k=0

(288)

dm 2 w , Am = (P (w)e ) , with Pk (w) polynomials in w obk w=0 n−m k dwm ∞ tained by the MacLaurin expansion of F(q, w) = pk (w)q k about q = 0 for k=0

∞ F(q, w) = exp (−1)k+1 2ζ (2k)w 2k q 2k−1 /k(2π )2k . where q =

k=1

For example, P0 (w) = 1, P1 (w) = w 2 /12, P2 (w) = w4 /288, P3 (w) = 6 w /10368 − w4 /1440. One finds that

n + k n k (k)6 (k)2 (k)4 (k)4 1 1+ S(n + k, n) = + + ... , + − 2 6n 72n 2 1296 180 n 3 n (289) which has a similar form to (287). However, this is an exact formula. Moser and Wyman [319] proved that (289) behaves as an asymptotic formula for large n when √ k = o( n). For the range 0 < m < n, lim (n − m) = ∞, the following is proved in [319]: n→∞

S(n, m) ∼ Q

∞ k=0

∞

−∞

exp(−t )b2k dt /(m R) 2

k

,

(290)

where R is given as the solution of R(1 − e−R )−1 = n/m, Q = A/(m R H )1/2 , with H = e R (e R −1− R)/2(e R −1)2 , A = n!(exp(R)−1)m /2π R n m!, b0 = 1, and b2k are polynomials in φ containing only even powers of φ, with φ = θ (m R H )1/2 , |θ | < π . For example, if we compute the first two terms in (290), we obtain the following expression:

15(C3 (R))2 3C4 (R) n!(exp(R) − 1)m 1 S(n, m) ∼ 1− (291) − 2R n m!(π m R H )1/2 mR 16R 2 H 3 4R H 2 where 1 C3 (R) = [R +(R −3R 2 + R 3 )(e R −1)−1 +(3R 3 −3R 2 )(e R −2)−2 +2R 3 (e R −1)−3 ], 6 C4 (R) =

1 [R + (R − 7R 2 + 6R 3 − R 4 )(e R − 1)−1 + 24

+(18R 3 − 7R 2 − 7R 4 )(e R − 1)−2 + (12R 3 − 12R 4 )(e R − 1)−3 − 6R 4 (e R − 1)−4 ]. 521

CHAPTER 5

W. E. Bleick and P. C. C. Wang [51] by using the method of Moser and Wyman have obtained a complete asymptotic expansion of S(n, m) in powers of (n + 1)−1 . Convergence is demonstrated for m < (n + 1)2/3 /[π + (n + 1)−1/3 ]. The expansion when divergent is still useful when used as an aymptotic series. For example, the first two terms give: 2 + 18G − 20G 2 (e4 + 1) n!(eu − 1) K 1 − + S(n, m) ∼ (2π(n + 1))1/2 m!u n (1 − G)1/2 24(n + 1)(1 − G)3 3G 3 (e2u + 4eu + 1) + 2G 4 (e2u − eu + 1) + 24(n + 1)(1 − G)3 where u=t−

∞

m m−1 (te−t )m /m!,

m=1

K = (n + 1)(1 − e−u )/u,

G=

t=

(292)

n+1 , m

∞ u 1 B2k u 2k /(2k)! = 1 − u + eu − 1 2 k=1

The bracket expression in (292), augmented by an additional inverse power of (n + 1), is approximated by 1−

1 1 + for small u, 2 6u(n + 1) 72u (n + 1)2

and by 1−

1 1 for large u. + 12(n + 1) 288(n + 1)2

By using the ray method, Ch. Knessl and J. B. Keller [255] have obtained the following asymptotic expansions for n 1: S(n, m) ∼

n2 2

n−m

1 if n − m = O(1); (n − m)!

(293)

−1/2 1 n · exp[m − n + n log(mb) − m log(bm − m)], ∼ 2π m − 1 − b

if n/m ∈ (1, ∞) where b is defined by b = (b − 1) exp

n , bk

522

b ∈ (1, ∞)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

√ D. Gardy [174] has shown that if e.g. n − m = o( m) and n − m → ∞, then n m m−k (1 + o(1)); (294) S(n, m) = 2 m and if n − m → ∞, with n − m = o(m 2/3 ), then n m m−k −(n−m)2 /6m e (1 + o(1)) S(n, m) = 2 m

(295)

As we can see, (294) is similar to (274) for s ∗ (n, m), but (295) differs from the similar formula (275) for s ∗ (n, m). N. M. Temme [439] proved the relation n A n−m (296) f (t0 ) S(n, m) ∼ e m m n−m uniformly as n → ∞ for δ < m ≤ n, δ being a positive constant. Here t0 = , m 1/2 t0 f (t0 ) = , x0 is the unique positive solution of the equation (1 + t0 )(x0 − t0 ) m x0 = 1 − e−x0 and A = φ(x0 ) − mt0 + (n − m) log t0 , n where φ(u) = −n log u + m log(eu − 1). In 1999 P. Flajolet and H. Prodinger [161] have defined the Stirling numbers of complex arguments x, y by x! 1 · S(x, y) = z −x−1 (e z − 1) y dz, (297) y! 2πi C where C is a Hankel contour which starts at −∞, circles the origin and goes back to −∞ subject to |Im z| < 2π . Here x! = (x + 1), where is Euler’s gamma function. R. Chelluri, L. B. Richmond and N. M. Temme [104] have generalized relation (296) to real numbers x and y, by using the definition (297). Let u 0 be the unique y solution of u 0 = 1 − e−u 0 . Let x x−y t0 = , φ(u) = −x log u + y log(eu − 1), y f (t0 ) =

A = φ(u 0 ) − yt0 + (x − y) log t0 , 1/2 t0 , 2H0 (u) = eu (eu − 1)−1 − ueu (eu − 1)−2 . (1 + t0 )(u 0 − t0 ) 523

CHAPTER 5

Then S(x, y) ∼ e y

A x−y

x f (t0 ) y

(298)

uniformly as x → ∞ for δ < y ≤ x (δ > 0 is fixed). Furthermore if δ < y ≤ x−x 1/3 , then x! (eu 0 − 1) y 1 S(x, y) = · [1 + O(x −1 )], (299) · y! 2u 0x [π u 0 y H0 (u 0 )]1/2 where the O-constant depends only on δ. Finally, if x − x 1/3 ≤ y ≤ x, then S(x, y) =

1 2x−y

·

y 2(x−y) [1 + O(x −1/3 )], (x − y)!

(300)

where the O-constant is independent of y. They further showed that the function S(x, y) is a log-concave function of y for δ < y ≤ x and x large enough. (301) This extends the classical result on log-concavity of (S(n, k))k ; see (87) and (242). Relation (301) shows that the generalized Stirling numbers S(x, y) have a unique maximum for δ < y ≤ x for large x. In the proof of (301), a method of B. Richmod and D. Merlini [364] is applied. Asymptotic properties of S(n, m) and s ∗ (n, m), when n, m → ∞, 1 < a ≤ n ≤ b < ∞, where a, b are constants, are given in A. N. Timashev [445]. Here m linear recursion relations for the coefficients of m −k in the asymptotic expansions of S(n, m) and s ∗ (n, m) are obtained. Let n Pn (z) = S(n, k)z k k=1

be the horizontal generating function of S(n, k). This admits the representation z(et −1) e n! Pn (z) = dt, 2πi γ0 t n+1 where γ0 is a simply closed curve of positive orientation encircling 0. It is known that Pn has real, simple zeros not greater than 0. In 2001 C. Elbert [150] established an oscillatory asymptotic of Pn (nz), when z ∈ (−e, 0); a uniform asymptotic of Pn (nz) on every compact subset of C \ [−e, 0]; and an Airy asymptotic of Pn (nz) in a neighborhoud of −e. It follows that for large enough n, most of the zeros of Q n , given by Q n (z) = Pn (nz), lie in [−e, 0]. The position of the zeros of Q n are investigated in [151]. For 1 n ≥ 1, define the distribution function Fn of Q n by Fn (η) = Nn (η), η ≤ 0, where n Nn (η) = car d{µ ∈ {1, . . . , n} : xn,µ ≤ η}. 524

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

Elbert proves that Fn converges weakly to a distribution function F, and using Stieltjes transforms determines certain representations for F and F . Previous results for Pn (1) are well known, see F. E. Binet and G. Szekeres [49] and N. de Bruijn [65]. For the asymptotic approximations of r -Stirling numbers, see R. B. Corcino, L. C. Hsu and E. L. Tan [121]. An asymptotic formula for the Bell numbers was given by Moser and Wyman (see relation (262)). For an asymptotic relation for a generalization of Bell numbers, see C. B. Corcino [119]. For asymptotics on Stirling numbers for complex argument, see G. Kemkes, C. F. Lee, D. Merlini and B. Richmond [247]. For another kind of generalization of s ∗ (n, k) to s ∗ (z, k) where z ∈ C, k ∈ N, involving hypergeometric functions, see V. Adamchik [2].

5.3 1

Bernoulli and Euler numbers Definitions, basic properties of Bernoulli numbers and polynomials

As we have seen in 5.2., the Bernoulli numbers (see (14) of 5.2.) are strongly related to Stirling numbers, too (see e.g. relation (15) of 5.2.). For general references, see [334], [373], [144], [142]. For Bernoulli numbers of higher order, connected to Stirling numbers, see (63), (210) of 5.2. A. Recall that the coefficient Bm in the expansion ∞ t Bn n = t t e −1 n! n=0

(1)

is called the nth Bernoulli number. We note that there is another (older) notation for Bernoulli numbers (see e.g. T. J. I’A. Bromwich [64], p. 297) when the Bernoulli numbers Bn are defined by ∞ t (−1)n−1 Bn 2n t = 1 − + t et − 1 2 n=1 (2n)!

(1 )

It is easy to see that the coefficients (Bn ) are connected with (Bn ) given by (1) via

Bn = |B2n | = (−1)n+1 B2n

(n ≥ 1)

Ramanujan (see Berndt [46], I.) defined the signless Bernoulli numbers of arbitrary index s by writing Bs∗ = 2(s + 1)(ζ (s)(2π )−s 525

CHAPTER 5

∗ By (7) and (8), clearly B2m = (−1)m+1 B2m > 0 (m ≥ 1). The Bernoulli numbers first appeared in Jacob Bernoulli’s (1654-1705) posthumous work ”Ars Conjectandi” (1713) in connection with sums of powers of the form

f k (n) = 1k + 2k + · · · + n k

(k ≥ 0 integer)

The numbers f k (n) can be calculated via the recurrence relation k+1 k+1 k+1 f 1 (n) + · · · + f k (n), (n + 1) − 1 = f 0 (n) + 1 k where f 0 (n) = n. But they can be explicitly obtained by the formula k+1 k+1 1 Bk+1− j n j f k (n − 1) = j k + 1 j=1

(2)

(3)

For identities of type ( f 1 (n))2 = f 3 (n), and related divisibility problems, see D. E. Penney and C. Pomerance [345]. From the definition (1) follows at once that n n+1 B0 = 1, Bk = 0 for k ≥ 1; k k=0 and that B2m+1 = 0 for m ≥ 1

(4)

Since Bn x x ex + 1 x x x x n − B1 x = x + = · x = coth , n! e −1 2 2 e −1 2 2 n≥0 we can easily deduce the expansions 1 B2n · 4n x 2n , x n≥0 (2n)! B2n π 4n (4n − 1)x 2n−1 |x| < tanhx = (2n)! 2 n≥1

cothx =

The similar formulae for tg x and cotx are: B2n n n 4 (4 − 1)x 2n−1 , tan x = (−1)n−1 (2n)! n≥1 1 B2n n 2n cotx = (−1)n 4 x x n≥0 (2n)! 526

(5)

(6)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

We note that similar expansions are the following (see [64]) ∞ 2(22n − 1) x =1+ (−1)n−1 B2n x 2n sin x (2n)! n=1

log

∞ sin x 22n−1 = (−1)n B2n x 2n x n(2n)! n=1

log cos x =

∞ 22n−1 (22n − 1) n=1

log

n(2n)!

(|x| < π)

(|x| < π),

(−1)n−1 B2n x 2n

|x| <

∞ cosh x − cos x nπ (−1)n B2n x 2n n+1 · = 2 cos x2 2 2n(2n)! n=1

π 2

π |x| < 2

which follow by integration of known power series expansions (e.g. of (6)). One of the most important properties is given by Euler’s formula B2k = (−1)k+1 where ζ (s) =

∞

2(2k)! ζ (2k) (2π )2k

(7)

n −s (Re s > 1) is the Riemann zeta function. A direct consequence

n=1

of (7) is (−1)k+1 B2k > 0 for k ≥ 1 By defining the tangent numbers T (n) by tan z =

(8) ∞ n=0

T (n)

zn , T (0) = 1, we n!

immediately get from (6) and (8) that T (2n) = 0,

T (2n − 1) =

22n−1 (22n − 1) |B(2n)| n

(9)

As an application of (9) and the von Staudt-Clausen divisibility theorem for B(2n) (which we shall study later), R. Mcintosh [232] proved the following primality condition for a Fermat prime n

F(n) = 22 + 1 (n ≥ 0) : F(n) is prime if and only if F(n) T (F(n) − 2)

(10)

B. The Euler numbers (E n ) are defined by ∞ En n 2 = t t −t e +e n! n=0

(E 0 = 1, E 2 = −1, E 4 = 5, E 6 = −61, . . . ) 527

(11)

CHAPTER 5

In fact, Euler considered the series expansion of the secant function ∞ 1 (−1)n E 2n 2n = sec x = x cos x (2n)! n=0

|x| <

π 2

(12)

these numbers were later called Euler numbers by H. F. Scherk (1825). We note that there is in use another notation for the Euler numbers, namely (see e.g. [64]) ∞ π E n 2n sec x = (12 ) |x| < x (2n)! 2 n=0 Then clearly, these coefficient are connected with (E n ) by E n = |E 2n | (n ≥ 0) There is also a third notation, namely (see e.g. [238]) sec x =

∞ n=0

E n∗

xn n!

(12 )

Then by (12) and (12 ) these coefficient satisfy ∗ E 2n = E n = (−1)n E 2n ,

The expansion of cosecx = cosecx =

∗ E 2n+1 =0

(n ≥ 0)

1 is given by sin x

∞ (−1)n+1 B2n (22n − 2) 2n−1 1 + x x n=1 (2n)!

(|x| < π, x = 0)

(12 )

For the numbers (S2n ) given by the expansion sec xch x =

∞ S2n 2n x (2n)! n=0

see L. Carlitz [80], Gandhi and Singh [173]. The Euler numbers satisfy n 2n E 2k = 0, E 2k+1 = 0 2k k=0

(k = 0, 1, . . . )

(13)

From (4) it follows that the Bernoulli numbers are rational numbers, while (9) and 528

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

(13) imply that the tangent and Euler numbers are always integers. For the computation of tangent, Euler and Bernoulli numbers, see T. Buckholtz and D. Knuth [66]. C. The Bernoulli and Euler polynomials Bn (x), E n (x) are defined by ∞ tn te xt B (x) = (|t| < 2π ); n et − 1 n! n=0

∞ tn 2e xt E (x) = (|t| < π) n et + 1 n! n=0

(14)

Then it is immediate that

1 , Bn (1) = Bn (n = 1), B1 (1) = −B1 Bn = Bn (0) and E n = 2 E n 2 n

(15)

Therefore, E n = E n (0); in fact E n (0) = −E n (1) =

2 (1 − 2n+1 )Bn+1 , n+1

1 1 = − 1 − n−1 Bn Bn 2 2

but

For values at other rational points, we note that 1 1 1 2 n = (−1) Bn =− 1 − n−1 Bn Bn 3 3 2 3

(16)

(17)

(n = even);

2 1 2 1 1 n+1 n = (−1) E n = Bn+1 − 2 Bn+1 = En 3 3 n+1 3 6 1 1 (1 − 2n+1 ) 1 − n Bn+1 (n = odd); n+1 3  1 1     − 2n 1 − 2n−1 Bn (n = even) 1 3 n = (−1) Bn = Bn  4 4    − n E n−1 (n = odd) 4n 1 1 1 5 1 n = (−1) Bn = 1 − n−1 Bn 1 − n−1 Bn (n = even) 6 6 2 2 3

(18)

=

(19)

(20)

In (15)-(20) there are listed all the known values of Bernoulli and Euler polynomials at rational points, expressed in terms of one and only one Bernoulli or Euler number. It is unknown whether there exist corresponding such formulas for values at other rational points. 529

CHAPTER 5

See A. Granville and Z. W. Sun [201], where connections to Fermat quotients are also pointed out. If m ≥ 3, and n ≥ 2 is even, then one has (see [201])

Bn (k/m) =

1≤k<m/2,(k,m)=1

1 (1 − p n−1 )Bn 2m n−1 p/m

(21)

The basic properties of these polynomials are the following: m

m Bn+1 (m + 1) − Bn+1 E n (m + 1) + (−1)m E n (0) , (−1)m−k k n = n+1 2 k=1 k=1 (22) for m, n = 1, 2, . . .

kn =

Bn (1 − x) = (−1)n Bn (x), E n (1 − x) = (−1)n E n (x) (−1)n Bn (−x) = Bn (x) + nx n−1 , (−1)n+1 E n (−x) = E n (x) − 2x n

(23)

for n = 0, 1, 2, . . . Bn (x) = n Bn−1 (x), E n (x) = n E n−1 (x) (n = 1, 2, . . . ) Bn (x + 1) = Bn (x) + nx n−1 , E n (x + 1) = −E n (x) + 2x n (n = 0, 1, . . . )

Bn (t)dt =

Bn+1 (x) − Bn+1 (a) n+1

(n ≥ 1)

E n (t)dt =

E n+1 (x) − E n+1 (a) n+1

(n ≥ 0)

x

a x

a 1

Bn (t)Bm (t)dt = (−1)n−1

0

m!n! Bm+n (m, n ≥ 1) (m + n)!

(24)

(25)

1

m!n! E n (t)E m (t) = (−1)n · 4(2m+n+2 − 1) Bm+n+2 (m, n ≥ 0) (m + n + 2)! 0 n n n n n−k Bk (x)h , E n (x + h) = E k (x)h n−k Bn (x + h) = k k k=0 k=0 Bn (mx) = m

n−1

m−1

Bn

k=0

E n (mx) = m

n

k x+ m

m−1

(−1) E n

k=0

(n ≥ 0, m ≥ 1) (Raabe’s theorem)

k

(26)

k x+ m

530

(n ≥ 0, m ≥ 1 odd)

(27)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

E n (mx) = −

m−1 k 2 mn (n ≥ 0, m ≥ 2 even), (−1)k Bn+1 x + n+1 m k=0 E n (x) =

n n Ek k 2k

k=0

1 x− 2

(28)

n−k (n ≥ 0)

(29)

There are true also the following Fourier expansions:

1 ∞ cos 2π kx − π n n! 2 Bn (x) = −2 n n (2π ) k=1 k (n > 1, 0 ≤ x ≤ 1; n = 1, 0 < x < 1)

1 ∞ sin (2k + 1)π x − π n n! 2 E n (x) = 4 n+1 n+1 π (2k + 1) k=0 (n > 1, 0 ≤ x ≤ 1, n = 0, 0 < x < 1)

∞ cos 2kπ x (−1)n−1 · 2(2n)! 2n (2π ) k 2n k=1 (n ≥ 1, 0 ≤ x ≤ 1)

(30)

B2n (x) =

∞ sin(2k + 1)π x (−1)n · 4(2n)! E 2n (x) = 2n+1 π (2k + 1)2n+1 k=0 (n ≥ 1, 0 ≤ x ≤ 1, n = 0, 0 < x < 1) ∞ sin 2kπ x (−1)n · 2(2n − 1)! (2π )2n−1 k 2n−1 k=1 (n > 1, 0 ≤ x ≤ 1, n = 1, 0 < x < 1)

(31)

B2n−1 (x) =

∞ cos(2k + 1)π x (−1) · 4(2n − 1)! E 2n−1 (x) = 2n π (2k + 1)2n k=0 (n ≥ 1, 0 ≤ x ≤ 1)

(32)

n

There exists the following relation between the two polynomials: x 2 x x +1 2n E n−1 (x) = Bn (x) − 2n Bn (n ≥ 1) (33) Bn − Bn = n 2 2 n 2

531

CHAPTER 5

For these, and other fundamental properties, see [237], [338], [1], where also tables can be found. For polynomials related to Bernoulli numbers of higher order, see L. Berg [44]. For the zeros of Bernoulli, as well as generalized Bernoulli and Euler polynomials, see K. Dilcher [140] and the references therein. D. The Genocchi numbers G n are defined by (see e.g. [116]) ∞ Gn n 2t = t et + 1 n! n=0

(|t| < π)

(34)

It follows that G 0 = 0, G 1 = 1 and G 2n+1 = 0 (n ≥ 1). The numbers G 2n are connected to the Bernoulli numbers by G 2n = 2(1 − 22m )B2n ,

(35)

G 2n = 2n E 2n−1 (0)

(36)

while to E 2n−1 (0) by The Genocchi polynomials are defined by ∞

G n (x)

n=0

2t t x tn = t e n! e +1

(37)

It follows that G n (x + 1) + G n (x) = 2nx n−1 (n ≥ 1);

G n = −G n (1) (n > 1);

G n (x) + (−1)n−1 G n (−x) = 2nx n−1 ; G n (x) + (−1) G n (−x) = 2n n

[N /2]∗ i=1

n G 2i x n−2i , where N = n 2i

(38)

or n − 1, according as n is even or odd; G n (x) + G n (x − 1) = 2n(x − 1)n−1 ; G n (1 − x) = G n (x) if n is odd;

G n (1 − x) = −G n (x), if n is even.

G n (x) is connected with Bn (x) by x

x +1 n − Bn G n (x) = 2 Bn 2 2 532

(39)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

One has the following multiplication theorems G n (mx) = m

n−1

m−1

(−1) G n k

k=0

G n (mx) = −2m

n−1

m−1

k x+ m

(−1) Bn k

k=0

k x+ m

, if m is odd; , if m is even

(40)

For special values, we note that, for n even (n > 1) 1 1 3 1 2 Gn = 0, G n (1) = −G n , G n = Gn , Gn = −G n ; 2 4 4 3 3 while for n odd, 1 3 1 2 = Gn , Gn = Gn G n (1) = G n = 0, G n 4 4 3 3 Generally,

(41)

n 2n m=0

G 2m

2m 1 = 0, G 2m+1 2m 2m + 1 2

22m − 1 = 24m 2m−1 B2m 2 −1

1 4

(42)

Relationships between G n and E n are e.g. E n−1

n n j 1 2 G j, = 2n j=0 j

n−1 2n j n−1 E n−1− j Gn = n (−1) j 2 j=0

(43)

Genocchi polynomials of order k, G n(k) (x) are defined by (see A. F. Horadam [218]) k ∞ 2t tn (k) G n (x) = et x (k = 0, 1, . . . ) (44) t +1 n! e n=0 whence n G n(k) (x) = 0 for n < k, and G (0) n (x) = x ,

having similar properties. The Genocchi numbers of order k are G n(k) = G (k) n (0). For logarithmic polynomials and numbers, see J. M. Gandhi [172], [170]. 533

CHAPTER 5

2

Identities

In what follows we shall select some identities and recurrence relations involving Bernoulli and Euler numbers. 1

A. By computing in two different ways the integrals

1

(x + 1)Bk (x + 1)d x, and

0

x Bk (x)d x, one can deduce (see J. S´andor [386]):

0 k k i=0

i

Bi

1 2k−i+2 − 1 = (Bk+1 + 2k + 1) k −i +2 k+1

By putting k = 2m this implies m 4m + 1 m 2m 22m−2i+2 − 1 = + (22m+1 − 1) (m ≥ 1) B2i m − i + 1 2m + 1 2m + 1 2i i=0

(45)

(45 )

Similar identity is m 2m i=0

2i

B2i

2(m − 1) 1 = m −i +1 2m + 1

(m ≥ 1)

(46)

For the identity m−1 i=0

2m − 1 B2i · 22i = 2m − 1 2i

(m ≥ 2)

see also [386], where a new proof od k−1 (22k − 1)B2k 1 (22i − 1)B2i 2k − 1 = −2 2i 2k − 2i 2 2k i=1

(47)

(48)

is given too. For a proof of (48) by contour-integration arguments, see Gh. Stoica [425]. The following formula has been proved by Gelfand (see [144], [142]): k m Bk+ j k Bm+i m k!m! m−1 k−1 (−1) + (−1) = (49) (k + m + 1)! i −1 m +i j −1 k+ j i=1 j=1 By setting m = k in (49), we have k Bk+i k (−1)k−1 (k!)2 = · i −1 k +i 2 (2k + 1)! i=1 534

(49 )

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

B. The following identities can be found in T. Agoh [7]. k+r k +r m +r +i Bm+i + (−1)k+m+r −1 · r i i=0 ·

m+r m +r k +r + j Bk+ j = 0; r j j=0

k+r k + r m + r + i k+r −1−i (−1) 2 E m+i + i r i=0 m+r m + r k + r + j m+r −1− j k 2 E k+ j = +(−1) r j j=0 r m +r s k +r (−1) = s r −s s=0

(50)

m

(51)

By letting k = m in (50) and then in (51), one can deduce k+r k +r k +r +i Bk+i = 0 i r i=0

(r ≥ 1, odd),

k+r k + r k + r + i k+r −i k+ r2 k + r 2 (r ≥ 0, even) E k+i = (−1) r i r 2 i=0

(50 )

(51 )

If r = 1 in (50 ) and replace k by k − 1, and then letting r = 0 in (51 ) one obtains the following classical formulas which were repeatedly discovered by many authors: k k (k + i)Bk−1+i = 0 (50 ) i i=0 k k k−i 2 E k+i = (−1)k i i=0

(51 )

We quote also the following formulas from [7]: k k i=0

+(−1)

k+m−1

i

(2m+1+i − 1)

Bm+1+i + m +1+i

m Bk+1+ j m (2k+1+ j − 1) = 0, j k+1+ j j=0

535

(52)

CHAPTER 5

k+1 (22k+2−2i − 1)B2k+2−2i = 0, 2i

(53)

k+1 (22k+2−2i − 1)32i B2k+2−2i = (−1)k−1 2k (k + 1) 2i

(54)

∗ [(k+1)/2]

i=0 ∗ [(k+1)/2]

i=0

C. The following famous identity is due to Euler (see B. C. Berndt [46], I): If n ≥ 2, then n k=1

(2n)! B2k B2n−2k = −(2n + 1)B2n (2k)!(2n − 2k)!

(55)

J.-I. Igusa [231] obtained a similar identity by the use of Eisenstein series: For each positive integer n ≥ 4 one has n−2 k=2

B2k B2n−2k B2n (2n + 1)(2n − 6) (2n − 2)! · · = − (56) (2k − 2)!(2n − 2k − 2)! 2k 2n − 2k 2n 6(2n − 2)(2n − 3)

For new proofs of (55) and (56) via the Shintani zeta functions, see M. Eie and K. F. Lai [149]. They prove also the following new Bernoulli identity: For any integer m ≥ 3, | p|=2m

(2m − 4)! B p B p B p B p · 2 p1 −1 · 3 p2 −1 · 5 p3 −1 · 6 p4 −1 = p1 ! p2 ! p3 ! p4 ! 1 2 3 4

257 2m−4 B2m−2 17 1 2m−2 + + ·2 ·3 − 432 48 360 2m − 2 1 1 197 62m−4 B2m−2 + B2m−2 + · − 180 2m − 2 6 3 62m−4 52m−3 1 1 1 1 B2m−3 25B2m−3 − 16B2m−3 − , + · 54 2m − 3 6 3 2m − 3 5 1 B2m =− · − 1080 2m

(57)

where | p| = p1 + p2 + p3 + p4 and p = ( p1 , p2 , p3 , p4 ), with p1 ≥ 0 integers (i = 1, 4). We quote also the following famous Ramanujan identities (see [46]), Part I, p. 122): Let α, β > 0 with αβ = π 2 . If n ≥ 2, then αn

∞ ∞ k 2n−1 k 2n−1 B2n n − (−β) = (α n − (−β)n ) ; 2αk 2βk e −1 e −1 4n k=1 k=1

536

(58)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

n+1 B2n+2−2k B2k · (2k − 1)(2n − 2k + 1)(−β)k α n+1−k = (2k)! (2n + 2 − 2k)! k=0

= (2n − 1)2−1−2n α 1−n

∞ csch2 kα k=1

+(2n − 1)2

−1−2n

(−β)

1−n

k 2n

∞ csch2 kαcothkα k=1

∞ csch2 kβ k=1

+ 2−2n α 2−n

k 2n

−2n

−2

β

2−n

k 2n−1

∞ csch2 kβcothkβ k=1

k 2n−1

+ (59)

D. Identities between Bernoulli numbers are used many times in the proof of some congruences. In [219] F. T. Howard provides an easy proof of the recurrence n−1 m−1 m 1 k Bm = B n j m−k , k n(1 − n m ) k=0 k j=1

(60)

and uses this formula to the proof of von Staudt-Clausen, Frobenius, etc. -type theorems. M. Kaneko [241] puts Bn = (n + 1)Bn , and gives two proofs of the identity n−1 n+1 1 Bn+k , B2n = − n + 1 k=0 k

(61)

one of which uses convergents of continued fraction expansions of f (x) = √ √ ( x/2)coth( x/2), and a very short one (due to D. Zagier) defining a suitable convolution on the set of sequences. For double series we note that in 1970 J. Higgins [217] proved n m k Bn = (−1) k m+1 m=0 k=0 n m

k

(62)

S. Bhargava and C. Agida [47] have shown that m n−1 (−1)m+k m n (m − k)n−1 Bn = k 1 − 2n m=0 k=0 2m+1

(63)

As a generalization of (63), H. Alzer obtained ([22]): m n−1 (−1)m+k m (m − k + 2x)n−1 Bn (2x) = 2 Bn (x) + n m+1 k 2 m=0 k=0 n

537

(64)

CHAPTER 5

For x = 0 this implies (63). In 1989 K. Dilcher [141] extended (64) to n−1 m n k+1 m Aλ (m)(k + λx)n−1 , (−1) Bn (λx) = λ Bn (x) + n k m=0 k=0 where Aλ (m) =

(65)

λ−1 (ei2π j/λ )/(1 − ei2π j/λ )m+1 (λ ≥ 2 is an integer). j=1

Further, Bn (λx) = λn Bn (x)+ ∗ [(n−1)/2] 2m+1 (−1)m+k 2m + 1 Cλ (k, m)(m − k + λx)n−1 , +n k 2m + 1 m=0 k=0

(66)

where Cλ (k, m) = −2−2m−1

(λ−1)/2

{k + (2m + 1 − k) cos 2 jπ/λ}{sin( jπ/λ)2m+2 }

j=1

and λ ≥ 3 is an odd integer. By making use of the Alzer and Dilcher results, P. G. Todorov [447] gave twelve explicit formulas for the Bernoulli and Euler polynomials and numbers, e.g. n m−1 n n k+1 n k n−1 (n ≥ 1) (−1) (67) Bn = (n/2 (2 − 1)) m m=1 k=0 M. Hauss [214] has introduced the ”conjugate” Bernoulli polynomials B n (x) by applying the Hilbert transform to the (1-periodic) Bernoulli polynomials B n (x) = H1 Bn (x), x ∈ [0, 1). The conjugate Bernoulli numbers are defined by B n = B n (0). The following remarkable formula for ζ (2m + 1) (m ≥ 1 integer) is derived: ζ (2m + 1) = (−1)m 22m π 2m+1

B 2m+1 (2m + 1)!

(m ≥ 1)

(68)

E. T. Agoh, K. Dilcher and L. Skula [12] have studied the Euler quotients a ϕ(m) − 1 q(a, m) = , where m ≥ 2 and a are two relatively prime integers. By m Euler’s divisibility theorem (see Chapter 3), q(a, m) is an integer. The Euler quotients (called also as generalized Fermat quotients by M. Lerch [282]) are connected to Bernoulli polynomials and numbers by the identity (see [12]): a−1 a ϕ(m) j − Bϕ(m) aq(a, m) = − Bϕ(m) (69) m Bϕ(m) j=1 a 538

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

3

Congruences for Bernoulli numbers and polynomials. Eulerian numbers and polynomials

The congruences and divisibility properties of Bernoulli and Euler numbers and polynomials are of fundamental importance in many fields of mathematics (for example in the study of the cyclotomic field, see e.g. K. Iwasawa [234], P. Ribenboim [363]). 1 1 1 The first few nonzero terms of (Bn ) are B0 = 1, B1 = − , B2 = , B4 = − , 2 6 30 1 1 5 691 7 3617 B6 = , B8 = − , B10 = , B12 = − , B14 = , B16 = − , 42 30 66 2730 6 510 43867 . B20 is the first one with a composite numerator 174611 = 283 · 617. B18 = 798 Nn with Dn > 0 and (Nn , Dn ) = 1. The denominators are A. Write Bn = Dn completely characterized by the famous von Staudt-Clausen theorem from 1840 (see [422], [115]): If n ≥ 2, even, then Dn = p, (70) p prime,( p−1)|n

and Bn +

1/ p is an integer.

(71)

p prime,( p−1)|n

If Z p denotes the ring of p-adic integers, we see that as a consequence of the above theorem, if ( p − 1) n, the Bn ∈ Z p . Also if ( p − 1)|n, then p Bn ∈ Z p (more precisely, p Bn ≡ −1 (mod p)). (The relation Bn ∈ Z p may be expressed also by saying that Bn is p-integral). More generally, if a, b, c, d, m (m > 1) are integers. and if (b, m) = 1, we a a c a c say that is m-integral. If and are m-integral, we define ≡ (mod m) iff b b d b d m|(ad − bc), i.e. if ad ≡ bc (mod m) with the usual definition of congruence. It is easily seen that the familiar properties of congruence on Z continue to hold. In 1845 von Staudt [421] proved a related result, too: If n ≥ 2 is even, and p is a (72) prime with ( p − 1) n, then if pr |n for some r ≥ 1, then pr |Nn , too. There are many proofs known in the literature for these theorems. For some recent proofs, see Eie and Lai [149] (where (70)-(71) are proved via certain identity for Bernoulli numbers), K. Girstmair [190] (where (72) is proved with the introduction of certain ”cyclotomic” Bernoulli numbers Bm,k (0 ≤ k ≤ n−1), satisfying t/(ζ k et − ∞ Bm,k t m /m!, where ζ is a primitive nth root of unity for n ≥ 2), A. Robert 1) = m=0

[369] (who proved (72) by using a p-adic version of the mean value theorem). 539

CHAPTER 5

There exist many applications of the above theorems. For example, by using the von Staudt-Clausen theorem, R. W. van der Wall [348] (by answering a question ∞ posed by F. Hirzebruch) has shown that if tan x = ar x 2r −1 (|x| < π/2), then r =1

ar−1 ∈ Z if and only if r = 1 or r = 2. (73) P. Erd¨os and S.S. Wagstaff, Jr. [154] have studied the fractional parts of the Bernoulli numbers. By applying among others the von Staudt-Clausen theorem, they proved that the sequence {B2m } is dense in (0, 1) (74) Further, for every k ≥ 1, the set of all m for which {B2m } = {B2k } has positive asymptotic density. (75) 1 (Here {x} = x − [x]∗ denotes the fractional part of x). Let Fx (z) = car d{m ≤ x x : {B2m } < z}. Then the limiting distribution lim Fx (z) = F(z) exists, and is a x→∞ jump function. The convergence is uniform and the sum of the heights of the jumps of F is 1. (76) For applications in the theory of Fermat and Euler quotients, e.g. E. Lehmer [275], D. H. Lehmer [276], Agoh-Dilcher-Skula [12]. For example, in [12] it is proved that for p an odd prime, and n a positive integer the following congruence is true: ( p n −1)/2 1/k ≡ −2q(2, p n ) + p n q(2, p n )2 (mod p n+1 ) (77) k=1, p k

This generalized a result by E. Lehmer [275] (see (69) for q(a, m)). For many other congruences on q(a, m), see [12]. For Wieferich primes (i.e. satisfying q(2, p) ≡ 0 (mod p)) see also R. E. Crandall, K. Dilcher and C. Pomerance [124]. For Wilson quotient, see [11], [8]. By applications of (70), (71) B. J. Powell [354] proved that if p > 3 is a prime such that 2 p − 1 is also prime, then |N2 p−2 | > |N2 p |,

(78)

while in [355] he proposed that for p > 3 prime N2 p ≡ 1 (mod p) p

(79)

and that for each k ≥ 1 there exist infinitely many r, s, t, . . . such that Nk |Nr , Nk |Ns , . . . (80) If p is a prime such that 2 p + 1 is composite (e.g. when p ≡ 1 (mod 3)), then N2 p is divisible by a prime ≡ 3 (mod 4), see Z. I. Borevic and I. R. Safarevic [57]. 540

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

The proofs use also the following famous congruence of Kummer [265]. If n ≥ 1 and p > 3 is a prime such that ( p − 1) 2n, then B2n+ p−1 B2n ≡ 2n + p − 1 2n

(mod p),

(81)

where the congruence relation is in the sense of Z p . More generally, if m ≡ n (mod ( p − 1) p N ) and ( p − 1) m, then Bm Bn (82) ≡ (1 − p n−1 ) (mod p N +1 ) m n In 1910 Frobenius [168] gave a generalization of the Kummer congruence. Vandiver [463] obtained the complementarity congruences, which were extended by Carlitz [89] in many directions. In 1964 J. M. Gandhi [171] obtained congruences for Genocchi numbers (see (34), (35)), similar to the above type congruences. Let Z m (n) = 1m − 2m + 3m − · · · + (−1)n+1 n m . Then, if m ≥ 2, n ≥ 3 odd, F. T. Howard [219] has shown that (1 − p m−1 )

G 2m ≡ −2m Z 2m−1 (n − 1)

(mod n 2 ),

and that if p ≥ 5 prime, and m > 1, then G 2m 2(22m − 1)B2m ≡− ≡ Z 2m−1 ( p − 1) (mod p 2 ) 2m 2m A main ingredient of the proof is the identity m−1 m m (n − n)G m = n k G k Z m−k (n − 1) k k=1 (n, m > 1, n = odd). Eie and Lai [149] have shown that (82) follows from a Bernoulli identity, von Staudt-Clausen’s theorem and Euler’s generalization of Fermat’s little theorem. Kummer’s congruence (82) can be restated in p-adic analysis (see N. Koblitz [259]) by saying that the p-adic integrals are continuous on compact sets. Other useful fact is contained in the following theorem of Adams [3]: (83) If p is a prime, n ≥ 1 and ( p − 1) 2n, p k |2n for some k ≥ 1, then pk |N2n . I. Sh. Slavutskii [405] (see also [411]) attributes both Kummer’s congruence and Adams’s theorem to two obscure pamphlets [421] of von Staudt. Finally, we mention the following congruence of Voronoi (G. Voronoi (1889)): If p is a prime, p a, and ( p − 1) 2n, (a, n positive integers), then

p−1 ja ∗ 2n−1 p−2n B2n (a − a ) j (mod p), (84) ≡ 2n p j=1 541

CHAPTER 5

The following more general version is true (see e.g. J. V. Uspenski and M. A. Heaslet [460] or K. Ireland and M. Rosen [233]): If a, k ≥ 1, (a, k) = 1, then (a 2n − 1)N2n ≡ 2n D2n

k−1

(a j)2n−1

j=1

aj k

∗

(mod k)

(85)

This is essentially due to Voronoi [468]: see also [408] and [353]. B. Various generalizations and extensions, or analogues of the von StaudtClausen, Kummer, Adams, and Voronoi-theorems are known in the literature. In what follows we shall study such relations. Let k be a positive integer and p > 3 a prime. Z. H. Sun [432] (see also [433]) proved that p Bk( p−1) ≡ kp B p−1 − (k − 1)( p − 1) (mod p 2 ), and p Bk( p−1)

(86) k k−1 3 ≡ p B2 p−2 − k(k − 2) p B p−1 + ( p − 1) (mod p ) 2 2

More generally, let k, l, n be positive integers, p a prime, and put h = ϕ( pl ). I. Sh. Slavutskii [411] shows that e.g. p Bkh ≡ p B2h

k(k − 1) (k − 1)(k − 2) − p Bh k(k −2)+( p −1) 2 2

(mod p 2l+1 ) (87)

for p ≥ l + 3. For l = 4 one obtains results of type (86). There are proved also results that generalize theorems due to Vandiver [463], Carlitz [89] and others. We quote also the following congruence from [411]: p Bhk ≡ p − 1 + kpl w p where w p =

(mod pl+1 ) for p ≥ 5,

(88)

( p − 1)! + 1 (integer, by Wilson’s theorem). Relation (88) implies p p Bhk ≡ p − 1 (mod pl ),

(89)

due to L. Carlitz [89] and other theorems due to Beeger, Lerch, Lehmer (see relation (108)). In fact Carlitz proved (89) in the form p Bhk ≡ p − 1 (mod pl ), x ∈ Z p , p ≥ 3, and this is also a particular case of x + −x p m (mod pl ), p Bhk+m (x) ≡ p Bm (x) − p Bm (90) p 542

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

for p ≥ 3, x ∈ Z p , l, k ≥ 1 and m ≥ 0, due to Sun [433]. Here −x p is the least nonnegative residue of −x (mod p), and Bn (x) is a Bernoulli polynomial. Theorem (90) is a consequence of a more general result, namely. Let p, l, m, x be as above. Then l x + −x p l k k( p−1)+m ≡ (−1) p Bk( p−1)+m (x) − p Bk( p−1)+m p k k=0 ≡

if n ∈ S p or ( p − 1) m 0 (mod pl ), pl−1 (mod pl ), if n ∈ S p and ( p − 1)|m,

where

S p = {s : Bs ∈ Z p , m ≥ 1} =

{3, 5, 7, . . . }, if {s| p − 1 s, s ≥ 1}, if

(91)

p = 2, p>2

Another result by Sun says that p Bk( p−1)+m (x) − p

k( p−1)+m

Bk( p−1)+m

x + −x p p

≡

l−1 k l−1−r k − 1 − r · ≡ (−1) r l −1−r r =0 x + −x p r ( p−1)+m · p Br ( p−1)+m (x) − p + Br ( p−1)+m p k l−1 l +(−1) δ(l, m, p) p ≡ l ≡ al k l + an−1 k l−1 + · · · + ak + a0

(mod pl ) (k ≥ 0),

(92)

where a0 , . . . , al are integers, and 1, if Bl ∈ Z p and p − 1|m δ(l, m, p) = 0, if Bl ∈ Z p or p − 1 m This is a wide generalization of the von Staudt-Clausen theorem. Using the properties of the Stirling numbers of the second kind, Sun shows also that one can choose a0 , . . . , al so that as s!/ p s−1 ∈ Z p for every s ∈ {0, 1, . . . , l}. As example of (92) we note that (2 − 22k )B2k ≡ 12(2k)2 + 18 · 2k + 1 (mod 27 ), (3 − 32k )B2k ≡ −36k 4 + 108k 3 − 93k 2 + 18k + 2 (mod 35 ), (5 − 54k )B4k ≡ −125k 4 + 125k 3 − 300k 2 − 100k + 4 (mod 54 ) 543

(93)

CHAPTER 5

G. Frobenius [168] proved that for k > 2, 2B2k ≡ 1 − 12m

(mod 32)

and F. T. Howard [219] has extended the Frobenius congruences to 2B2k ≡ 1 − 12k + 16k(1 + k 3 ) (mod 64), 2B2k ≡ 1 − 12k + 16k(1 + k 3 ) − 32k 2 (1 + k) (mod 128), for k > 3, and more generally (1 − 2 )2B2k ≡ 1 − 2k + 2k

l

2

j=1

2j

2k B2 j 2j

(mod 22l+1 )

S. Ramanujan [361] proved that 30B2k ≡ −1

(mod 200),

and that

B4k+2 ≡ 3 (mod 5) 4k + 2 for any k ≥ 1. For new proofs of the Frobenius and Ramanujan theorems, see Howard [219]. A similar congruence is due to J. S. Frame [165]: k−1 30B2k ≡ 1 + 6000 (mod 23 · 33 · 53 ) (94) 2 Recently, E. U. Gekeler [178] has proved that for all k ≥ 4, even 2k 65520 2k + 24 ≡ − Bk+12 Bk 691

(mod 27 · 34 · 53 · 72 · 13)

(95)

This new result is obtained from congruences among Eisenstein series, by representing the Fourier coefficients of certain combinations of Eisenstein series as Iwasawa functions. See also [179]. In 1998 M. Eie (see [149]) proved Kummer’s congruence (83) in a general form: Suppose that m, n are positive even integers and k is a positive integer such that ( p − 1) m for all prime divisors of k. Then, if m ≡ n (mod ϕ(k)), then Bm Bn (1 − p m−1 ) ≡ (1 − p n−1 ) (mod k) m p|k n p|k (Here ϕ is Euler’s totient). For k = p N +1 one obtains (82).

544

(96)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

The method of proof shows that one can derive congruences of the following type for values of Bernoulli polynomials. For a fixed prime p > 3, let k be a positive integer not divisible by p. Suppose further that a, b are positive integers such that 1 ≤ a, b < k and a + jn = bp for some positive integer j with 1 ≤ j ≤ p − 1. Assume that ( p − 1) m and that m ≡ n (mod ( p − 1) p N ) (N ≥ 1). Then 1 a a b b 1 m−1 n−1 −p − p Bm Bm ≡ Bn (mod p N +1 ) Bm m k k n k k (97) A generalization of Voronoi’s congruence (84) is due to A. Simalarides [402]: If p > 3 is a prime, p a and ( p − 1) 2n, then (a − a

p k−1 ( p−2n)

)B(2n−1) pk +1 ≡

p−1 ja p

j=1

j (2n−1) p

k−1

(mod p k ),

(98)

for each k ≥ 1. For k = 1 this gives (84), since by Kummer’s congruence B(2n−1) p+1 = [(2n − 1) p + 1]

B(2n−1) p+1 B2n ≡ (2n − 1) p + 1 2n

(mod p).

See also I. Sh. Slavutskii [410] for a proof using the methods of Voronoi. We note that applying (84) for various values of a, one can deduce congruences for Bernoulli numbers. Put e.g. a = 2, 3, 4 then adding the first two relations, and subtracting the third one obtains: (2 p−2n + 3 p−2n − 4 p−2n − 1)

B2n ≡ 4n

j 2k−1

(mod p),

p>3

(99)

p/4< j< p/3

J. W. Tanner and S. S. Wagstaff [436] proved a similar, but complicated formula which holds for all p ≥ 11: (2 p−2n + 9 p−2n − 10 p−2n − 1)

B2n ≡ (1 + 22n−1 + 32n−1 + 42n−1 ) 4n

+(1 + 22n−1 + 32n−1 + 42n−1 + 122n−1 ) −(22n−1 + 62n−1 )

j 2n−1 − 32n−1

13 p p 120 < j< 9

j 2n−1 − 22n−1

5p 17 p 18 < j< 60

17 p 3p 60 < j< 10

545

j 2n−1 +

p 13 p 10 < j< 120

2p 7p 9 < j< 30

j 2n−1 −

j 2k−1 −

CHAPTER 5

−(22n−1 + 42n−1 + 122n−1 )

−(22n−1 + 42n−1 )

j 2n−1 −

7p 47 p 18 < j< 120

j 2n−1

(mod p)

(100)

47 p 2p 120 < j< 5

Congruences for Tk (n) =

n

j k connected to Bernoulli numbers appear in T.

j=1

Agoh [8], in connection with Fermat and Wilson quotients. For an odd prime p put Tk = Tk ( p − 1). Then Tk ≡ 0 (mod p) for any k ≥ 1 with( p − 1) k,

(101)

but Tm( p−1) ≡ −1

(mod p) for any m ≥ 0

If p ≥ 3 and ( p − 1) k, k even, then Tk ≡ p Bk

(mod p 2 ),

and for p ≥ 5 with ( p − 1)|k, k arbitrary, one has Tk ≡ p Bk

(mod p 2 )

For p = 3 and k even one has if k = 2, k ≡ 0 or 1 (mod 3), 3Bk (mod 32 ) Tk ≡ 2 2 3Bk + 3 Bk−2 (mod 3 ) if k = 2 and k ≡ 2 (mod 3)

(102)

(103)

If p ≥ 5 and k is odd with ( p − 1) (k − 1), then Tk ≡ 0

(mod p 2 )

(104)

If p ≥ 3 and 1 ≤ k ≤ p − 1, then Tk ≡ p Bk

(mod p 2 )

(105)

Recently A. Junod [238] has applied the p-adic mean value theorem of A. Robert [370] to prove that np 0, if ( p − 1) k (mod (105 ) p Bk ≡ Tk ≡ Zp) p − 1, if ( p − 1)|k 2 Robert’s theorem can be stated as follows: Let K be a complete field and let (E, |· |) be an ultrametric Banach space over K . Let f (t) ∈ E[t] with coefficients in E, and 546

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

' ' ' ' suppose that h, a ∈ E with |a| ≤ 1. The Gauss norm on E[t] will be ' ak t k ' max{|ak | : k ≥ 0}. Then, if |h| ≤ | p|

=

E[t] f E[t] ;

, then | f (a + h) − f (a)| ≤ |h|· h2 and if h is odd with |h| ≤ | p|1/( p−2) , then | f (a+h)− f (a)−h f (a)| ≤ f E[t] , 2 with the same conclusion when p = 2, by assuming |h| ≤ 21/2 . Bn+1 (t) Let now f (t) = ∈ Q p [t], when f (t) = Bn (t), f (t) = n Bn−1 (t) and n+1 apply Roberts’ theorem to deduce the above result (which, by the way implies the von Staudt-Clausen and Kummer’s classical congruences). F. T. Howard [219] has proved that if p ≥ 5 is a prime and if pl 2k and ( p − 1) (2k − 2), then p B2k ≡ T2k

1/( p−1)

(mod pl+3 ) (k > 1)

This is related to a result of Carlitz from 1953 [89]: If ( p − 1) p s |2k ( p any prime; k > 1), then (105 )

p B2k ≡ p − 1 (mod p s+1 )

By studying Fermat and Wilson quotients, as a by-product, in [8] the following is derived: For p ≥ 3 and 1 ≤ m ≤ p − 2, p−1 p−1 B p−1−m 1 m Bm + ≡ j (s j − 1) m p−1−m p j=1 s=1

(mod p).

(106)

In particular, −4B( p−1)/2 ≡

p−1 p−1 1 j ( js − 1) (mod p), p j=1 p s=1

j is a Legendre symbol. where p We note that Tk (n) is connected to Bernoulli numbers by a kind of EulerMacLaurin summation formula: k+1 k 1 Tk (n) = Bk+1− j (m + 1) j (107) j j − 1 j=1 ( p − 1)! + 1 be the Wilson quotient. The following famous congrup ences are known: Let p ≥ 3 be a prime. Then Lwt W p =

547

CHAPTER 5

1 − 1 (mod p), (N. G. W. H. Beeger [36], M. Lerch [283]) p W p ≡ B2( p−1) − B p−1 (mod p) (E. Lehmer [275]) p−2 Bk (mod p) (N. Nielsen [335]) Wp ≡ k k=1 (108) If p ≥ 5 and m ≥ 1 any integer, then

W p ≡ B p−1 +

1 − 1 (mod p) (E. Lehmer [275]) p W p ≡ B(m+1)( p−1) − Bm( p−1) (mod p) (T. Agoh [8]) mW p ≡ B(m+1)( p−1) − B p−1 (mod p) mW p ≡ Bm( p−1) +

(109)

We note that the second relation of (109) follows from the first one, by taking into account the well-known relation B2( p−1) − 2B p−1 + 1 ≡

1 p

(mod p).

H. S. Vandiver [462] proved the following two congruences: Let p ≥ 3 and p a (a ≥ 1). Then W p + q p (a) ≡

p−2 p−1 Bk p−1−k 1 + (a − j) (mod p) and a k p j=1 k=1

p−2 p−1 Bk k 1 a + W p − q p (a) ≡ ( ja − 1) (mod p) k p j=1 k=1

(110)

The following theorem was discovered repeatedly by many authors (A. Friedmann and J. Tamarkine [167], M. Lerch [283], N. Nielsen [335], H. S. Vandiver [462]): For p ≥ 3 and 1 ≤ m ≤ p − 2, p−1

q p ( j) ≡ W p

j=1

(mod p);

p−1

j m q p ( j) ≡ −

j=1

Bm m

(mod p).

(111)

a p−1 − 1 ( p a) denotes the Fermat quotient. p Bm , I. Sh. Slavutskii [409] has shown (by strengthening a theFor the fraction m orem of Sylvester and Lipschitz) that for an integer a, and a positive integer m, the number Bm ∗ a [log2 m] +1 (a m − 1) m Here q p (a) =

548

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

is an integer. For related results, we note that Ramanujan [361] observed that 2m (2m − 1)

Bm m

is an integer. In fact, for arbitrary integer k, and any m ≥ 1, k m+1 (1 − k 2m )

B2m 2m

is an integer, see J. Glaisher [193]. Ramanujan proved also that for any m ≥ 0, 2(24m+2 − 1)

B4m+2 2m + 1

and

− 2(28m+4 − 1)

B8m+4 2m + 1

are integers ≡ 1 (mod 30). In 1991 G. Almkvist and A. Meurman [18] have shown that for any m ≥ 1, and any integers a, b, with a = 0, b m − Bm Bm a a is an integer (here Bm (x) is the Bernoulli polynomial). B. Sury [434], K. Bartz and J. Rutkowski [34] have obtained simpler proofs. F. W. Clarke and I. Sh. Slavutskii [114] have deduced the above theorem by first proving that if p is prime and m ≥ 1, then for all integers a, b with p|a, b m ∈ Z( p) , a Bm a where Z( p) denotes the ring of rational numbers whose denominator is not divisible by the prime p. C. Congruences for higher order Bernoulli numbers have been studied by many authors, including L. Carlitz [95], L. Carlitz and F. R. Olson [97], F. T. Howard [223], A. Adelberg [4], P. T. Young [488]. The Bernoulli polynomials of order z, Bm(z) (x) and numbers Bn(z) = Bn(z) (0) were first defined and studied by N. E. N¨orlund [338] in case z = k ∈ N. These are defined by z ∞ t tn xt (z) e = B (x) n et − 1 n! n=0 Then it follows that Bn(z) (x)

=

n n j=0

549

j

n− j B (z) j x

CHAPTER 5

Let An (x, s) = Bn(n+s+1) (x + 1) be the so-called Narumi polynomials. Let p be a prime, and let l, k, a, b be p-adic integers such that l ≡ k (mod p α ) and a ≡ b (mod p α ). If n < p − 1, then (see [4]) An (a, l) ≡ An (b, k) (mod p α )

(112)

In particular, if n < p − 1, and l is a p-adic integer, then Bn(l) is a p-adic integer, and if l ≡ k (mod p α ), then Bn(l) ≡ Bn(k) (mod p α ), while if 0 < n < p − 1 then l|Bn(l) and Bn(l) /l ≡ Bn(k) /k (mod p α ). If 1 < n < p − 1, then Bn(l) /l ≡ −Bn /n (mod l), with sign ”+” if n = 1. (113) For n = p − 1 one has p B p−1 (l) ≡ −l

(mod pl)

(114)

In what follows, suppose that l is a p-adic integer, and put r = [n/( p − 1)]∗ , (z) σ = n − r ( p − 1). Assume that B−1 = 0. Then the following congruences are true (see [4]): (l) r (l) n p Bn σ +r n + r − l Bσ ≡ (−1) (mod p). (115) (−1) n! σ! r If p > 2 and r ≤ 1 then

(−1)n pr Bn(l) ≡ n! r − 1 n + r − l (l−r p) pσ (l−r p) σ +r −1 n + r − l − 1 ≡ (−1) + − Bσ B r −1 σ! r 2 σ −1 p (l−r p+ p) +(−1) p−1 (mod p 2 ) (116) B p+σ −1 ( p + σ − 1)! If p = 2 and n = 1, then (−1) 2

m n

Bn(l) /n!

≡ (−1)

n−1

2n − l − 1 2n 2 + l n n−1

(mod 4)

(117)

If p (n − l) then we can derive stronger congruences mod pl and mod p 2l in terms of low degree polynomials Bm(z) (1). Let Ak ( p, n) = (−1)n pr Bn(n−k) /n! Then for l = n − k, relation (115) can be rewritten as (σ −k−r ) σ +r r + k Bσ Ak ( p, n) ≡ (−1) (mod p) (118) k σ! For small values of σ , (118) contains results due to Howard [223]. We quote also the following theorem by Adelberg: 550

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

If p n or σ > 0, then l| pr Bn(l) /n!; If σ > 0 and p n or σ > 1, then (−1)n pr

(119)

n + r − l Bσ(l) Bn(l) ≡ (−1)σ +r n! σ! r

(mod pl)

(120)

(If σ = 0 and p n, multiply right hand side of congruence by l/(l − r − n)); If p|l and σ > 1, then (l) Bσ n r Bn σ +r −1 n + r − l l (−1) p ≡ (−1) · (mod pl) (121) r n! σ σ! (If p|l, σ = 1 and p n, change sign of right hand side). For a corollary, suppose that σ = s p (n) = sum of digits of n in base p (see Chapter 4) and σ 1, then Bn(l) ≡ (−1)r If p|l and σ > 1, then Bn(l)

r −1

≡ (−1)

n + r − l n! Bσ(l) · r pr σ !

(mod pl),

n + r − l n! l Bσ · · r pr σ σ !

(123)

(mod pl)

(124)

We note that (119) and (122) strengthen a theorem of Howard [223] (by weakening the hypothesis in his theorem from p k to p k or ( p − 1) n), while Howard’s theorem improves an earlier result by Carlitz. Another corollary says that if 1 < σ < p − 1 and p|l, then (l) B (l) p+σ −1 ≡ Bσ p ≡ −

l σ

(mod pl)

(125)

In particular, (σ p)

(σ p)

B p+σ −1 ≡ Bσ p ≡ − p Bσ (mod p 2 ), and p ( p) ( p) Bσ p ≡ B p+σ −1 ≡ − Bσ (mod p 2 ) σ

(126)

The following result: If p is an odd prime and p|l, then 2 B (l) p ≡ l /2

(mod pl 2 )

551

(127)

CHAPTER 5

generalizes a Carlitz congruence [95], namely: B (pp) ≡ p 2 /2

(mod p 3 )

(128)

For the values of higher order Bernoulli polynomials Bnr (x) (n, r positive integers) L. Carlitz [83] proved that for r ≥ 1, p odd prime and a ∈ Z p p t (r ) Bn(r ) (a) is a p − adic integer, t (r ) denoting the number of nonzero digits of r in the p-adic expansion (and n ≥ 0). A. Junod [238], by applying the mean value theorem of Robert [370] has recently proved that for all m, n ≥ 0, a ∈ Z p , (r ) (r ) (a) ≡ pr Bm+n (a) (mod pr Bm+np

and (r ) pr Bnp (t) ≡ pr Bn(r ) (t p ) (mod

np Z p ), 2

np Z p [t]) 2

The universal Bernoulli numbers Bn have been introduced in 1989 by F. W. Clarke [112] as follows: Let G(t) be the compositional inverse of F(t) = ∞ ci t i+1 /(i + 1) (a power series over the polynomial ring Q[c1 , c2 , . . . ], ci being i=0

indeterminates). Then

∞ t = Bn t n /n! G(t) n=0

(129)

Note that for ci = (−1)i , F(t) = log(1 + t), G(t) = et − 1, B)n ≡ Bn , the ordinary Bernoulli numbers. The following Kummer type congruences are due to A. Adelberg [5]: If n = q( p − 1) + 1 > 1 (i.e. n ≡ 1 (mod ( p − 1))), then Bn q p q−1 1 − q q−2 1 q q−1 ≡ c1 c p−1 + c1 c p−1 − c p−1 c p + c p−1 c2 p−1 n 2 2 2 1 q 1 − n p q−1 n q−2 q−1 ≡ c2 c p−1 + c1 c p−1 − c p−1 c p + c p−1 c2 p−1 2 2 2

(mod p) ≡

(mod p)

(130)

Similar congruences when n ≡ 0, 1 (mod ( p − 1)) are proved in [6]. In [5] there are defined also universal poly-Bernoulli numbers Bn,k of index k by means of the polylogarithm function, and universal von Staudt-Clausen type congruences are deduced. 552

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

In 1991 I. Dibag [137] considered for a given set S of primes the series L S (x) = (−1)n−1 x n /n, where the prime factors of n belong to S. Let the rational numbers n

(dn ) be defined by x/L −1 S (x)

=

∞ n=0

dn

xn n!

(131)

where L −1 S (x) denotes the functional inverse of L S (x). Dibag’s von Staudt-Clausen theorem states that if p ∈ S and ( p − 1)|2n, then d2n = − 1/ p (mod Z), which p

may be formulated also as pd2n ≡ −1

(mod pZ( p) ) if p ∈ S, ( p − 1)|2n

(132)

where Z( p) denotes the ring of p-local integers. F. W. Clarke [113] proves that if m is even and ( p − 1)|m, and p ∈ S, then (mod p 1+γ p (m) Z( p) ),

pdm ≡ p − 1 otherwise dm ≡ 0

(mod p γ p (m) Z( p) ),

(133)

where p γ p (n) n. For a similar analogue of the von Staudt-Clausen theorem, see I. Dibag [138]. D. For application in the theory of general abelian number fields, generalized Bernoulli numbers belonging to residue class characters were first introduced by N. C. Ankeny, E. Artin and S. Chowla [27] in the special case of a quadratic character, and by H. W. Leopoldt [280], [281] in general. Let χ be a primitive character with conductor f . Then the generalized Bernoulli numbers Bχn are defined by f χ(a)teat a=1

eft − 1

=

∞

Bχn t n /n!

(134)

n=0

If f = 1, i.e. if χ is the principal character (χ (a) = 1 for all a), then Bχn ≡ Bn for 1 all n = 1, and Bχ1 = = −B1 . The numbers Bχn are algebraic numbers and belong 2 to the smallest algebraic number field containing all χ(a). The generalized Bernoulli polynomials are defined by Bχn (x)

=

n n j=1

553

j

Bχj x n− j

(135)

CHAPTER 5

Then we have Bχn (0) = Bχn , and it can be shown that they are connected with the classical Bernoulli polynomials via Bχn (x)

= f

n−1

f

χ(a)Bn

a=1

a+x f

(136)

Kummer type congruences for Bχn were first proved by L. Carlitz in 1959 (see [92]). For von Staudt-Clausen and Voronoi type congruences for Bχn see T. Agoh [10], I. Sh. Slavutskii [407], L. C. Washington [479]. In 1993 K. Girstmair [191] has introduced the Bernoulli numbers Bgn belonging to g, where g is a periodic mapping from Z to C with period m (m positive integer, n nonnegative integer), as follows: m

g(k)ekt t/(emt − 1) =

k=1

∞

Bgn t n /n!

(137)

n=0

When g = χ − a primitive character, then Bgn ≡ Bχn . When g(k) = e2kiπ/m , the cyclotomic Bernoulli numbers of Girstmair are reobtained (see [190]). Girstmair proves von Staudt-Clausen and Kummer type congruences for Bgn . For a Kummer congruence for g = χ = ωk , where ω is the Teichm¨uller character mod p ( p odd prime), see H. S. Gunaratne [203]. If χ is the (unique) quadratic character with conductor f = 4, i.e. χ (1) = 1, χ (3) = −1, χ (2) = χ (4) = 0, then it follows immediately that 2 B n+1 (138) n+1 χ 1 n , this follows from a more general for the Euler numbers E n . Since E n = 2 E n 2 property of Euler polynomials (see B. C. Berndt [45]) En = −

E n (x) = −

1 1−n n+1 2 Bχ (2n − 1) n+1

(139)

Therefore many properties of Euler numbers and polynomials follow also from the corresponding properties of the generalized numbers Bχn and polynomials Bχn (x). For a generalization of the P´olya-Vinogradov inequality, involving Bχn , see S. Kanemitsu and K. Shiratani [242]. For congruences for Bχn belonging to unequal characters, see I. Sh. Slavutskii [406]. E. The sequence of (classical) Bernoulli numbers is periodic after being reduced a modulo n (n ≥ 1). The rational number with b > 0, (a, b) = 1 is said to be nb integral if (b, n) = 1 (i.e. iff the congruence yb ≡ a (mod n) has a unique solution 554

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

y ∈ {0, 1, . . . , n − 1}). By the von Staudt-Clausen theorem B2i is p-integral (where p is a prime) iff ( p − 1) 2i. Let L(n) be the smallest integer > 1 with the following property: there exist m 0 such that for all k, and m ≥ m 0 Bk n − integral and k ≡ m (mod L(n)) ⇒ Bm n − integral and Bk ≡ Bm (mod n)

(140)

L(n) is called the period-length of the sequence (Bk (mod n)), while the smallest m 0 in (140) will be called the preperiod of (Bk (mod n)), and denoted by V (n). The properties of L(n) and V (n) have been studied in detail by W. Herget [216]. Since (n 1 , n 2 ) = 1 ⇒ L(n 1 , n 2 ) = lcm[L(n 1 ), L(n 2 )] and B(n 1 n 2 ) = max{V (n 1 ), V (n 2 )}, it is sufficient to study the case n = p α ( p prime, α ≥ 1). One has: L(2α ) = L(3α ) = 2; V (2α ) = V (3α ) = 2, (141) L( p α ) = p α ( p − 1) for p ≥ 5; V ( p α ) ≤ α + 1 The last relation of (141) can be improved as follows: V ( p) = 2 for p = prime. Let p ≥ 5 be a prime and α ≥ 2 even. Then if Bα ≡ 0 (mod p) and ( p − 1) α, the V ( p α ) = α + 1. Let k be the greatest integer such that for all 0 ≤ i ≤ k one has Bα−2i ≡ 0 (mod p 2i+1 ) and ( p − 1)|(α − 2i). Then V ( p α ) = α − 1 − 2k

(142)

Similarly, if p ≥ 5 and α ≥ 3 is odd, then if Bα−1 ≡ 0 (mod p 2 ), and ( p − 1) (α − 1), then V ( p α ) = α. Let k be the greatest integer such that for all 0 ≤ i ≤ k one has Bα−1−2i ≡ 0 (mod p 2i+2 ) and ( p − 1)|(α − 1 − 2i). Then V ( p α ) = α − 2 − 2k

(143)

The p-divisibility of certain sets of Bernoulli numbers has been studied by S. V. Ullom [459]. For a prime p ≥ 5, let I = {n ∈ N : n even and 2 ≤ n ≤ p − 3}. p−1 is odd, set I (d) = {2k ∈ I : For all even divisors d of p − 1 for which d (2k − 1, p − 1) = ( p − 1)/d}. Then car d{m ∈ I (d) : p|Bm } <

ϕ(d) log log p + ϕ(d) , 2 log p

ϕ being Euler’s totient. See also S. S. Wagstaff, Jr. [472]. 555

(144)

CHAPTER 5

The distribution of Bernoulli numbers modulo primes has been considered by W. L. Fouch´e [164]. Let p ≥ 3 be a prime. Let A be any set of n consecutive integers in (0, p), and suppose that 2 ≤ t ≤ p − 1. Then car d{n ∈ A : B p−t (n) ≡ α

9 (mod p)} ≤ t 1/3 m 2/3 2

for every integer α. Furthermore, p−2 car d n ∈ A : Bk n p−1−k ≡ α k=1

(145)

"

9 (mod p) ≤ m 2/3 2

(146)

It is well known that for every prime p, p B p−1 ≡ −1 (mod p). T. Agoh [9] conjectures that (147) n Bn−1 ≡ −1 (mod n) ⇔ n prime This has been verified up to n ≤ 49999 (see [481]). B. C. Kellner [246] provided a short proof of Agoh’s result on the equivalence of conjecture (147) to a famous conjecture of Giuga (see [192], [58]): If n > 1, then n−1

k n−1 ≡ −1

(mod n) ⇔ n prime

(148)

k=1

The Giuga-Agoh conjecture states that n/ p ≡ 1 (mod n) ⇔ n prime

(149)

p|n,( p−1)|(n−1)

The factorization of Bernoulli numerators goes back to M. Ohm [340]. Then J. C. Adams [3] in 1878 published a table for the first sixty-two numbers of Bernoulli. In 1978 S. S. Wagstaff [473] published the factorizations through N60 and E 42 . Recently [474] he completely factored all N2k for 2k ≤ 152, and also the Euler numbers E 2k for 2k ≤ 112. The results are completed continuously at the web address [475]. F. The prime factors of the Euler numbers determine the structure of certain cyclotomic fields; see e.g. R. Ernwall and T. Mets¨ankyl¨a [157]. Since the Euler numbers are all integers, there is no analogue for them of the von Staudt-Clausen theorem. But Kummer’s congruence has an analogue, due also to Kummer [265]: If n ≥ 1 and p ≥ 3 is a prime, then E 2n+ p−1 ≡ E 2n (mod p) (150) See also L. Carlitz and J. Levine [96] and S. S. Wagstaff, Jr. [474]. The following two theorems have been discovered recently by Wagstaff [474], the first being an 556

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

analogue of Adams’ theorem (83): Let p be an odd prime, n ≥ 1 and N ≥ 0 an integer. Suppose ( p − 1) p N |n. Then (mod p N +1 )

E n ≡ 0 or

(151)

according as p ≡ 1 or 3 (mod 4). For all integers n ≥ 0 and k ≥ 0 we have E 2n ≡ E 2n+2k + 2k

(mod 2k+1 )

(152)

Theorems of type (151) and (152) were probably known to Kummer, too. In 1938 E. Lehmer [275] proved that if p ≡ 5 (mod 8) is a prime, then E ( p−1)/2 ≡ 0 (mod p)

(153)

For an elementary proof, see R. Ernvall [155]. In 1982 R. Ernvall [156] proved that (153) holds true for all p ≡ 1 (mod 4). For connections of congruences of type (153) with Diophantine equations (Pell equations), see L. J. Mordell [314]. In [84] Carlitz proves that none of the following numbers |E 4n |,

|E 8n+6 |,

|E 24n+18 |,

|E 40n+34 |,

|E 40n+26 |

(154)

is a perfect square, and conjectures that the same is true for |E 8n+2 |

(155)

(i.e. |E 2m | is never a square for m > 1). In the proof of (154) various congruences are used, e.g. E 2n ≡ 1 − 2n (mod 8) for all n ≥ 1 (156) The congruence E 4n ≡ −1

(mod 3)

(157)

appears in N. Nielsen [334]. On the other hand E 4n ≡ 1

(mod 4),

(158)

∗ ∗ = (−1)n E 2n , E 2n+1 = 0 More generally, let us define the numbers E n∗ by E 2n ∗ ∗ (n ≥ 0) (see (12 )). Put E m = E 2m (m ≥ 0). Then A. Junod [238] has proved that ∗ ∗ ∗ E m+2 ≡ E m∗ (mod 4), E m+3 ≡ 5E m+1 (mod 36), ∗ ∗ ∗ E m+4 ≡ 14E m+2 − 9E m (mod 576), ∗ ∗ ∗ E m+5 ≡ 30E m+3 − 89E m+1 (mod 14400)

557

(159)

CHAPTER 5

and that m∗ ≡ · · · ≡ E ∗ = 1 (mod 4), ∗ ≡ E E m+1 0 ∗ ∗ ∗ = 5m+1 (mod 36), E m+2 ≡ 5 E m+1 ≡ · · · ≡5m E 2 1, if m is even ∗ ∗ (mod 30) E m+3 ≡ E m+1 ≡ · · · ≡ 5, if m is odd

(160)

These follow from more general results for the Meixner polynomials Q n (x) defined by Q 0 (x) = 1, Q 1 (x) = x, Q n+1 (x) = x Q n (x) − n 2 Q n−1 (x) (n ≥ 1): If p is an odd prime, then Q p (x) ≡ x p − (−1)( p−1)/2 x

(mod pZ p [x])

(161)

Junod conjectures that Q np (x) ≡ Q np (x) ≡ (x p − (−1)( p−1)/2 x)n

(mod npZ p [x]),

(162)

which would imply ∗ ∗ E m+np ≡ (−1)n( p−1)/2 E m+n (mod npZ p ) and ∗ ∗ E m+np ≡ E m+n (mod npZ p )

for any m, n ≥ 0, and p = 2 a prime. The higher order Euler polynomials are defined by r ∞ 2 tn xt (r ) e = E (x) n et + 1 n! n=0

(163)

(164)

where n, r ≥ 0 are integers. For r = 1, E n(1) (x) = E n (x), as given by relation (14). By generalizing results by M. Zuber [489], A. Junod [238] has recently proved that (r ) E np (t) ≡ E n(r ) (t p ) (mod npZ p [t])

(165)

for all integers n, r ≥ 0, and an odd prime p. For r = 1 this gives a Honda type congruence for the classical Euler polynomials. Further, if a ∈ Z p , then (r ) (r ) E m+np (a) ≡ E m+n (a) (mod npZ p )

(166)

for all n, m, r ≥ 0. Particularly, the sequence (E n(r ) (a))n≥1 is p − 1-periodical (mod pZ p ): (r ) (r ) E m+ (167) p−1 (a) ≡ E m (a) (mod pZ p ) As for the variant of Euler numbers defined in (159), there are also other ”Euler numbers” definition in the literature. However, we wish to use always the classical 558

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

definition (11) for the Euler numbers. For example, in [380] the Euler numbers are defined by ∞ xn = tan x + secx (168) En n! n=0 Then clearly, E 2n = (−1)n E 2n (called also as ”secant numbers”), and E 2n+1 = B2n+2 (−1)n · 4n (4n+1 − 1) (called also as ”tangent numbers”). In [420] R. Stanley 2n + 2 has shown that the volume of the convex polytope determined by xi ≥ 0 (i = 1, n), and xi + xi+1 ≤ 1 (1 ≤ i ≤ n − 1) is given by E n . (169) In fact, E n counts the number of permutations π = a1 a2 . . . an in Sn (the symmetric group of all permutations of {1, 2, . . . , n}) that alternate, i.e. a1 < a2 > a3 < . . . , due to D. Andr´e [24] (see also [116]). There are many known q-analogs of Euler numbers or its variants. See e.g. a paper by L. Carlitz [82] from 1954. The divisibility properties of q-tangent numbers have been studied e.g. by G. Andrews and I. Gessel [26], D. Foata [162] etc. Let q ≥ 1, and as usual put [k] = [k]q = 1 + q + · · · + q k−1 . The q-Euler numbers E n|k (q) can be defined (see Sagan and Zhang [380]) by the recurrence E (n+1)|k (q) =

∗ [n/k]

m=1

n q n−mk+1 E (mk−1)|k (q)E (n−mk+1)|k (q)+ mk − 1

+χ (k n)E n|k (q),

(n ≥ 0)

(169)

and E 0|k (q) = 1, where χ (P) is the characteristic function of property P (i.e. is 1 if n [n]! ([n]! = [n][n − 1] . . . [1]) is P is true, and 0 if it is false), and = [k]![n − k]! m the Gauss q-binomial coefficient. Sagan and Zhang have proved the following divisibility theorems: Let p be a prime, and 1 ≤ i ≤ p − 1. Then E (np+i)| p (q) is divisible by [ p]n

(170)

E (np+i)| p (q) is divisible by [ p][ p]q 2 [ p]q 3 . . . [ p]q n

(171)

Further, For p = 2, i = 1 these were proved in [26]. Let E n|k = E n|k (1). Then (170) gives: E (np+i)| p is divisible by [ p]n = p n 559

(172)

CHAPTER 5

for all 1 ≤ i ≤ p − 1. I. Gessel and G. Viennot [189] have shown that np− j ∗ np E (np− j)| p p p−1 j

(173)

where [·]∗ denotes the integer part. For p = 2, j = 1 this reduces to the well known fact that 22n−1 |2n E 2n−1 (174) where E 2n−1 = E 2n−1|2 is a (classical) tangent number. In 1911 F. H. Jackson [235] defined the q-variant of sin, cos and tan by sinq (x) =

∞ (−1)n x 2n+1 n=0

[2n + 1]!

,

cosq (x) =

∞ (−1)n x 2n n=0

[2n]!

(175)

1 tanq (x) = sinq (x)/ cosq (x), secq (x) = cosq (x) Then the q-Euler numbers E n (q) may be defined by n≥0

where (x, q)n =

E n (q)

xn = tanq (x) + secq (x) (q, q)n

(n ≥ 0)

n−1 (1 − xq l ), for n ≥ 1 and (x, q)0 = 1. l=0

More generally, J. D´esarm´enien [133] defined sinq (u; x) = sin u cosq (x) + cos u sinq (x) = =

(−1)(m+n−1)/2 m,n

um x n m!(q, q)n

(m, n ≥ 0, m + n = odd)

and cosq (u; x) = cos u cosq (x) − sin u sinq (x) = =

(−1)(m+n)/2 m,n

um x n m!(q, q)n

(m, n ≥ 0, m + n = even)

tanq (u; x) = sinq (u; x)/ cosq (u; x),

secq (u; x) =

1 cosq (u; x)

He defined the numbers E m,n (q) by m,n≥0

E m,n (q)

um x n = tanq (u; x) + secq (u; x). m!(q, q)n 560

(175 )

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

One has E 0,n (q) = E n (q), E m,n (1) = E m+n . The following congruences are true: Let m, k, a, b be integers, m, a, b ≥ 0, k ≥ 1, Then: i) if k is even, and m + ka + b is odd, with a ≥ 1, then E m,ka+b (q) ≡ 0

(mod k (q)).

If m + ka + b is even, then E m,ka+b (q) ≡ (−1)(k+2)a/2 E m,b (q) (mod k (q))

(175 )

ii) If k is odd, then E m,ka+b (q) ≡ (−1)(k−1)a/2 E m+a,b (q) (mod k (q)), where k denotes the kth cyclotomic polynomial (see Chapter 3). For m = 0, relation i) of (170 ) is due to G. Andrews and I. Gessel [26], see also [162] and [25]. For k = 2, q = 1 these were proved by Kummer in 1850. In 2000 H. Prodinger [356] as variations to Jackson’s definitions, has defined the q-trigonometric function sinq∗ (x) =

∞ (−1)n x 2n+1 n=0

[2n + 1]!

q n , cosq∗ (x) = 2

∞ (−1)n x 2n n=0

[2n]!

2

qn , (176)

tanq∗ (x) = sinq∗ (x)/ cosq∗ (x) and deduced among others divisibility properties of these tangent numbers. For continued fraction expansions, see M. Fulmek [169]. For another q-analogue of the Euler numbers, see G.-N. Han, A. Randrianarivony and J. Zeng [212]. G. Several nonequivalent definitions of q-analogues of Bernoulli polynomials and numbers are known in the literature. In 1948 L. Carlitz [91] defined the qBernoulli numbers βk (q) by β0 (q) = 1,

k k j+1 1, k = 1 q β j (q) − βk (q) = 0, k≥2 j j=0

(177)

In 1954 he introduced the q-Euler numbers Hk (u, q) (associated to u) by (see [82]) H0 (u, q) = 1,

k k j q H j (u, q) − u Hk (u, q) = 0 (k ≥ 1) j j=0

561

(178)

CHAPTER 5

Note that if q → 1, then βk (q) → Bk and Hk (u, q) → Hk (u), where (Bk ) are the ordinary Bernoulli numbers, while Hk (u) are defined by ∞ 1−u tk = H (u) k et − u k! k=0

(179)

J. Satoh [392] defined a q-Riemann ζ -function by ∞ ∞ qn qn 2−s ζq (s) = (1 − q) + s−1 [n]s−1 n=1 [n]s n=1

(s ∈ C)

(180)

and proved that ζq is analytic in the whole complex plane. For k ≥ 1, integer, one has   qβ1 (q) if k = 1 (181) ζq (1 − k) = β (q)  − k if k ≥ 2 k Further, Satoh defined l(s, u, q) =

∞ u −n [n]s n=1

(s ∈ C)

and proved that l(s, u, q) is analytic in C, and for k ≥ 0 integer    1 if k = 0 u−1 l(−k, u, q) = u   Hk (u, q) if k ≥ 1 u−1

(182)

For a construction of q-Bernoulli type numbers using Stirling numbers, see J. Satoh [390]. The q-Bernoulli polynomials βk (x, q) can be defined inductively by βk (x, q) = (q x β + [x])k with replacing β i by βi (x, q). N. Koblitz [257] defined the generalized q-Bernoulli numbers βk,χ (q), where χ is a primitive Dirichlet character with conductor f : f a f k−1 a ,q (183) χ (a)q βk βk,χ (q) = [ f ] f a=1 Sato [392] has constructed the generalized q-Euler numbers Hk,χ (u, q) by f k f −a f a f (184) u χ(a)Hk u , , q Hk,χ (u, q) = [ f ] f a=1 562

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

where the q-Euler polynomials Hk (u, x, q) are defined inductively by Hk (u, x, q) = (q x H + [x])k

(k ≥ 1)

with replacing H i by Hi (u, x, q). Further, Sato introduced a q − L-series L q (x, χ) by ∞ ∞ q n χ (n) q n χ (n) 2−s L q (s, χ) = + , (185) (q − 1) s−1 [n]s−1 [n]s n=1 n=1 and a function lq (u, s, χ) by lq (u, s, χ) =

∞ u −n χ (n) n=1

[n]s

(186)

These functions interpolate the q-Bernoulli and q-Euler numbers as follows: L q (1 − k, χ) = −βk,χ (q)/k, 1 lq (u−, −k, χ) = Hk,χ (u, q), 1−uf

(187)

for k > 1 integer. The values L q (k, χ) for k ≥ 2 integer have been evaluated by H. Tsumura [453]: f 1 a(2−k) (−1)k (k) −a [a] f · ,q , q q χ (a)G L q (k, χ) = (k − 1)! [ f ]k a=1 [f]

(188)

where

1 G(x, q) = x − 1+q

log x − x +

∞ (−1)k+1 1 βk+1 (q) q k(k + 1) x k=1

for |x| > 1/(1 − |q|)2 (with |q| < 1). Since ∞ ∞ (−1)k k qn qn 2−k G (x, q) = (1 − q) + (1 + x − q x) (k − 1)! k−1 (x + [n])k−1 (x + [n])k n=0 n=0

for k ≥ 2 integer, one has a q-analogue of the classical formula for the Euler gamma function: ∞ (−1)k d k 1 log (x) = k (k − 1)! d x (x + n)k n=0 Therefore, G(x, q) can be considered as a q-log- function. 563

CHAPTER 5

n (q) by In 1991 H. Tsumura [389] defined the modified q-Bernoulli numbers β Fq (t) =

∞

n

n (q) t , β n! n=0

(189)

where Fq (t) is determined as a solution of the following q-difference equation: Fq (t) = et Fq (qt) − t,

Fq (0) =

q −1 log q

t n (1) = Bn , which is the ordinary , and β −1 Bernoulli number. For 0 < q < 1 the following series representation is valid: Moreover, we let F1 (t) =

et

∞ q − 1 t/(1−q) −t q n e[n]t e log q n=0

Fq (t) =

Thus, Fq (t) is continuous as a function of (q, t) on (0, 1] × {t ∈ C : |t| < 2π }. n (x, q) are defined by The modified q-Bernoulli polynomials β Fq (q x t)e[x]t =

∞

n

n (x, q) t β n! n=0

(190)

Note that Fq (q x t)e[x]t =

∞ q − 1 t/(1−q) e −t q n+x e[n+x]t log q n=0

J. Satoh [389] considered the q-series Z q (s) =

∞

q n /[n]s

n=1

k (q)) at non-positive integers for a complex number q with which interpolated (β |q| < 1, and where [x] = (1 − q x )/(1 − q). Let Z q (s) = Z q (s) + (1 − q)s /((1 − s) log q). Note that if q → 1, then Z q (s) → ζ (s) for Re s > 1. Satoh proved that  k (q)  β − if k ≥ 2 Z q (1 − k) =  k −β1 (q) − 1 if k = 1 564

(191)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

In [452] H. Tsumura considered Z q (s) at positive integers s. Particularly, we quote the following result: 2k Z q (2k + 1) − 2k

∞ q n cos([n]π )

[n]2k+1

n=1

=

k−1 (−1) j−1 π 2 j j=1

(2 j)!

−π

∞ q n sin([n]π )

[n]2k

n=1

=

π (1 − q)2k+1 (2k − 2 j) Z q (2k − 2 j + 1) − cos + log q 1−q +(−1)k+1 π 2k

∞ (−1)m π 2m β2m (q), (2k + 2m)! m=0

(192)

which for q → 1 gives a result of D. Cvijovi´c and J. Klinowski [126]: ( k−1 (−1) j−1 j ζ (2 j + 1) (−1)k (2π )2k + ζ (2k + 1) = · k(2k+1 − 1) j=1 (2k − 2 j)! π2j ∞

ζ (2m) (2m)! · 2m + (2m + 2k)! 2 m=0

) (193)

The modified q-Hurwitz ζ -function is defined by ζq (s, x) =

∞ (1 − q)s q n+x /[n + x]s for x > 0. + (1 − s) log q n=0

Then for k ∈ N, one has k (x, q) β ζq (1 − k, x) = − k The modified q − L-series and generalized q-Bernoulli numbers are given by f a −s χ (a)[ f ] ζq f s, L q (s, χ) = f a=1 and k,χ (q) = [ f ]k−1 β

f

k χ(a)β

a=1

a f ,q f

(k ≥ 0)

k,χ (q) = Bχn , the generalized Bernoulli numbers given by (134). Note the lim β q→1

The following formula holds true: k,χ (q) β L q (1 − k, χ) = − k 565

(k ≥ 1 integer)

(194)

CHAPTER 5

For (194) and related results, see H. Tsumura [455]. By applications of methods of [455] the following result of M. Katsurada [245] (who used Mellin transform) can be reobtained: Let n be a positive integer, x a real number with |x| ≤ 1, and let f ∞ τ (χ) = χ(a) exp(2πia/ f ) be the Gauss sum. Let L(m, χ) = χ (γ )γ −m be γ =1

a=1

the Dirichlet L-function. If χ (−1) = 1 and χ ≡ 1, then n L(2n + 1, χ) − n

∞ χ(l) cos(2πlx/ f )

l 2n+1

l=1

= (−1)n

2π x f

2n n−1 k=1

−

∞ χ(l) sin(2πlx/ f ) πx = f l=1 l 2n

∞ (−1)k−1 k L(2k + 1, χ) τ (χ ) (2k)!L(2k, χ ) 2k x + 2k (2n − 2k)!(2π x/ f ) f k=1 (2n + 2k)!

"

If χ (−1) = −1, then L(2n, χ) −

∞ χ (l) cos(2πlx/ f )

l 2n

l=1

=

2n−1 n−1

(−1)k−1 L(2k, χ) + (2n − 2k)!(2π x/ f )2k−1 k=1 " ∞ (2k)!L(2k + 1, χ) 2k+1 2iτ (χ) + x f (2n + 2k)! k=0

= (−1)

n

2π x f

(195)

For q-analogue of the p-adic L function of Kubota and Leopoldt [264], and connections with q-Bernoulli and Euler numbers, see N. Koblitz [257], H. Tsumura [456]. For the q-extension of Morita’s p-adic gamma function, see N. Koblitz [258], Y. S.Kim [250]. For p-adic log-gamma functions, see e.g. J. Diamond [136], T. Kim [249]. See also Y. Morita [315]. For the q-gamma functions of F. H. Jackson, see R. Askey [29], D. S. Moak [313], M. Badiale [31], A. Gupta [204], A. B. Olde Daalhuis [341], K. Ueno and M. Nishizawa [458], H. Alzer [21], J. W. Son and D. S. Jang [415]. For p-adic functions attached to the Lubin-Tate groups, see K. Shiratani [400], K. Kozuka [127] and H. Tsumura [454]. By means of p-adic integrals with respect to the q-analogue of the p-adic measure defined by T. Kim [249], recently J. W. Son and M. S. Kim [252] have defined other p-adic q-analogs Bn∗ (q) and Bn∗ (x, q) of Bernoulli numbers and polynomials. The numbers Bn∗ (q) can be defined by ∞ tn q − 1 log q + t ∗ = B (q) , (196) n log q qet − 1 n! n=0 566

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

while the q-analogs Hn∗ (q) of Euler numbers are defined by ∞ tn q −1 ∗ = H (q) n qet − 1 n! n=0

(197)

∗ ∗ They defined also Bn,χ (q) and Hn,χ (q), where χ is a Dirichlet character, proved ∗ Kummer type congruences for Hn,χ (q), and some relations between Bn∗ (q) and Hn∗ (q). H. The Euler numbers (E n ) should nor be confused with another kind of numbers A(n, k), introduced by Euler in 1755, and called Eulerian numbers. These numbers may be defined recursively via

A(n + 1, k) = (n − k + 2)A(n, k − 1) + k A(n, k),

(k ≥ 2, n ≥ 0)

(198)

A(n, 1) = 1 for n ≥ 0 and A(0, k) = 0 for k ≥ 2. These numbers are generated by G = G(t, u) =

n t n k−1 1−u A(n, k) = 1 + u et (1−u) − u n! k=1

It is easy to verify that (u − u 2 )

∂G ∂G + (tu − 1) + G = 0, ∂u ∂t

and by letting the coefficient of u k−1 t n /n! equal to 0 in this differential equation, (198) follows at once. Some authors define the Eulerian numbers a(n, k) which satisfy a(n, k) = (n − k)a(n − 1, k − 1) + (k + 1)a(n − 1, k) (n ≥ 1, k ≥ 1)

(199)

with a(n, 0) = 1 for n ≥ 0 and * n +a(0, k) = 0 for k ≥ 1. is also applied. (We note that some authors use also The notation a(n, k) = k *n + ). These numbers a(n, k) count in fact the number of permutations π A(n, k) = k of the symmetric group Sn of {1, 2, . . . , n} having exactly k ascents (an ascent is an occurrence of π( j) < π( j + 1) for 1 ≤ j ≤ n − 1), see e.g. [116]. However, the numbers A(n, k) and a(n, k) are connected by A(n, k) = a(n, k − 1)

(k ≥ 2)

(200)

and they have the symmetric property A(n, k) = A(n − k + 1),

a(n, k) = a(n, n − k − 1) 567

(201)

CHAPTER 5

Therefore, any property for A(n, k) can be written via (200) to a property of a(n, k) and vice-versa. Two fundamental properties of A(n, k) are given by: k j n+1 A(n, k) = (k − j)n , (−1) j j=0

(202)

and the Worpitzky identity from 1883 (see [116] and [487]) x = n

n k=1

x +k−1 A(n, k) n

(203)

The Eulerian polynomials An (x) are defined by An (x) =

n−1

A(n, k + 1)x k+1 , n ≥ 1

(A0 (x) = 1)

k=0

While no closed formula is known for the Eulerian polynomials, their generating function is given by (see e.g. D. Foata and M. P. Sch¨utzenberger [163]) ∞ 1−x An (x)t n = and n! 1 − xe(1−x)t n=0 ∞ n=0

−1 An (x) tn = · n+1 (x − 1) n! 1 − xe−t

(204)

giving the recurrence relation An (x) = x(1 − x)

n−1

+

n−1 n i=1

i

Ai (x)x(1 − x)n−1−i

The Eulerian numbers A(n, k) and a(n, k) occur in many fields. For example, F. Schmidt and R. Simion [395] deal with a set of probability problems that lead to the computation of volumes of regions obtained by dissecting a polytope P with hyperplanes. They consider in particular the case when P is an n-dimensional symplex. The regions formed by dissecting a symplex by certain pencils of hyperplanes through a point have volumes proportional to the Eulerian numbers a(n, k). R. Ehrenborg, M. Readdy and E. Steingrimsson [148] give a combinatorial interpretation for the mixed volumes of two adjacent slices from the unit cube in terms of a refinement of the Eulerian numbers. M. Dash, L. Samantaray and R. Panda 568

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

[127] study the interpolation of band limited signals using Eulerian filters, and establish a relation of different derivatives of a sigmoid function to the standard Eulerian numbers. There exist many identities involving Eulerian numbers. The following show a connection with the Stirling numbers of the second kind, and with Bernoulli numbers. The identity n n j−1 2 A(n, j) = j!S(n, j), (205) j=1

j=1

has been proposed in [483], and generalized by R. Tauraso (see [483]) to: n j=1

A(n, j)

n ∞ x n+1− j xj = j!S(n, j) = jnx j j+1 (1 − x)n+1 (1 − x) j=1 j=1

(206)

∞ 1 for |x| < 1. For x = , (206) gives (205) with a common value equal to j n /2 j . 2 j=1 The identity n 2n+1 (2n+1 − 1)Bn+1 (−1) j a(n, j) = n+1 j=1

(n ≥ 1)

(207)

is well known. For a proof, see J. Stopple [429] (who states that (207) may have been known to Euler). Since ζ (−n) = −Bn+1 /(n + 1), (207) can be written also in terms of ζ (−n). There are a lot of generalizations or extensions of the Eulerian numbers. For example, in 1954 L. Carlitz [82] defined the q-Eulerian numbers Bn,k (q) and polynomials Bn (x) by Bn,k (q) = [k + 1]Bn−1,k (q) + q k [n − k]q Bn−1,k−1 (q),

(208)

with B0,k (q) = 1 if k = 0; 0, otherwise, and Bn (x) =

n

Bn,k (q)x k

k=0

Here q ≥ 1 and [k + 1] = [k + 1]q is the standard q-notation. For combinatorial and statistical properties, see e.g. L. Carlitz [81] and M. Skandera [404]. Another q-Eulerian polynomial An (x, q) has been introduced in 1971 by R. P. Stanley [417] (see also [418]). As usual, the q-ascending factorial is defined by (u, q)0 = 1, (u, q)n = (1 − u)(1 − uq) . . . (1 − uq n−1 ) (n ≥ 1). Let (u, q)∞ = 569

CHAPTER 5

∞

(1−uq n ), and put e(u, q) =

n=0

u n /(q, q)n = (u, q)−1 ∞ . Let desπ denote the num-

n≥0

ber of descents of π of the permutation group Sn (i.e. the number of i ∈ {1, . . . , n−1} with π(i) > π(i + 1)), and invπ the number of inversions of π (i.e. the number of pairs (i, j) with 1 ≤ i < j ≤ n such that π(i) > π( j)). Then An (x, q) is defined by x 1+desπ q invπ An (x, q) = π ∈Sn

Then the generating functions of these polynomials are given by (see [418]) tn 1−x , (208 ) An (x, q) = (q, q) 1 − xe((1 − x)t, q) n n≥0 so that this is equivalent to the recurrence relation An (x, q) = x(1 − x)

n−1

+

n−1 n i=1

i

Ai (x, q)x(1 − x)n−1−i

We note that for q = 1, An (x, q) = An (x) =

x 1+desπ , so we get the recur-

π ∈Sn

rence from (204) in this case. For q = −1 one obtains the signed Eulerian polynomial A− n (x) = An (x, −1), introduced by J. L. Loday [289] in connection with his study of the cyclic homology of commutative algebras. Loday conjectured that − A− 2n (x) = (1 − x)A2n−1 (x),

− − A− 2n+1 (x) = (2n + 1)x A2n (x) + x(1 − x)(A2n ) (x)

or equivalently n A− 2n (x) = (1 − x) An (x),

n A− 2n+1 (x) = (1 − x) An+1 (x),

and this has been proved by J. D´esarm´enien and D. Foata [135]. The q-Eulerian polynomials Am,n (x, q) (m, n ≥ 0) with two indices were introduced by J. D´esarm´enien [134] via their generating function tn sm 1−t Am,n (x, q) · = (q, q)n m! 1 − xe((1 − x)t, q)e(1−t)s m≥0,n≥0 Then A0,n (x, q) = An (x, q), Am,0 (x, q) = Am (x). He proved the following congruence property (see also [135] for a detailed proof): Let n and k be two positive integers and let n = ka + b (0 ≤ b ≤ k − 1) be the Euclidean division of n by k. Then Am,ka+b (x, q) ≡ (1 − x)(k−1)a Am+a,b (x, q) (mod k (q)), 570

(208 )

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

where k is the kth cyclotomic polynomial. For k = 1 we then have Aa,1 (x, q) = Aa+1,0 (x, q) = Aa+1 (x); for k = 2, b = 0, 1 one has A2a+b (x, q) = A0,2a+b ≡ (1 − x)a Aa,b (x, q)

(mod (q + 1)) ≡

≡ (1 − x)a Aa+b (x) (mod (q + 1)), − so Loday’s conjecture on A− 2n and A2n+1 is a particular case of (208 ). For an involution for signed Eulerian numbers, see M. Wachs [477]. Recall that these numbers A− (n, k) satisfy

A− (2n, k) = A− (2n − 1, k) − A− (2n − 1, k − 1), A− (2n + 1, k) = k A− (2n, k) + (2n − k + 2)A− (2n, k − 1), implying analogs of the Worpitzky formulas 2n − 1 + i 2n + i − n A (2n, k − i) = k , A− (2n − 1, k − i) = k n i i i i F. Chung, P. Diaconis and R. Graham [108] have considered the generalized Eulerian numbers a(n; k, l) defined by a(n; k, l) = (l + 1)a(n − 1; k, l) + (l + 1)a(n − 1; k − 1, l + 1)+ +(n − k − l)a(n − 1; k − 1, l) + (k − l + 1)a(n − 1; k, l − 1)

(209)

where a(0; 0, 0) = 1 and a(m; n, p) = 0 if m, n or p < 0. P. L. Butzer and M. Hauss [71] have introduced the Eulerian functions a(α, k), α ∈ R, which for α = n ∈ N are given by (199). The a(α, k) satisfy recursion formulae, they are monotone in k, and as functions of α, are arbitrarily often differentiable. A counterpart of the Worpitzky formula (203) is given, as well as connections to the fractional Stirling numbers S(α, k) are pointed out (see relation (98) of 5.2.). For Eulerian numbers of higher order, see J. F. Dillon and D. P. Roselle [145]. For ”Pseudo-Eulerian” numbers, see [100]. In [227] Hsu and Shiue, by using the Dickson-Stirling numbers S(n, k, α) (see (116) of 5.2.) have defined the Dickson-Eulerian polynomial by An (x, α) =

n

k!S(n, k, α)x k (1 − x)n−k ,

k=0

and the Dickson-Eulerian numbers by An (x, α) =

n k=0

571

A(n, k, α)x k

CHAPTER 5

The numbers A(n, k, α) have many properties, e.g. n k−i n − i S(n, i, α) i!(−1) A(n, k, α) = n−k i=0 They also have introduced the degenerate Eulerian numbers A(n, k, λ|θ ), which satisfy e.g. Worpitzky type formulas. m ˇ Starting from the identity (202), Z. Sami [381] has introduced the numbers ak,n where n ≥ 1, j ≥ 0, m ∈ {0, 1, . . . , n − 1}, by defining k n m (k + 1 − j)n−m−1 (k − j)m ak,n = (−1) j (210) j j=0 0 = A(n−1, k +1) and An−1 with the convention 00 = 1. Therefore ak,n k,n = A(n−1, k), m so the numbers ak,n can be considered as generalization of the Eulerian numbers A(n, k). m These numbers ak,n satisfy the recurrences m m m ak,n = (k + 1)ak,n−1 + (n − k − 1)ak−1,n−1 (0 ≤ m ≤ n − 2) m−1 m−1 m−1 m ak,n = ak,n − ak,n−1 + ak−1,n−1 (1 ≤ m ≤ n − 1),

where k ≥ 1 and n ≥ 2. m Further, if k ≥ n, then ak,n = 0 and 0, if 0 ≤ m ≤ n − 2 m an−1,n = 1, if m = n − 1

(211)

(212)

and have the symmetric property n−m−1 m ak,n = an−k−1,n for all n ≥ 1; m, k ∈ {0, . . . , n − 1}

(213)

The following identities are true: n−1

m ak,n = A(n, k + 1) (k ≥ 0, n ≥ 1),

m=0 n−1

m ak,n = (n − 1)! (n ≥ 1, 0 ≤ m ≤ n − 1),

k=0

n−i m n−1 −1 k m n i m 2 (−1) ak,n = 2 (−1) Bn−i i n−i k=0 i=0 (n ≥ 3, 1 ≤ m ≤ n − 2) 572

(214)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

Let Sk,n =

n−1

(n ≥ 1, k ≥ 0)

i ak+i,n

i=0

Then Sk,1 = 0 if k ≥ 1; = 1 if k = 0, and Sk,n = Sk,1 +!n −

n−1 k

A(r, j)

(215)

r =0 j=0

with !n =

n−1

k! denoting the left-factorial function of Kurepa [266]. Applying (215)

k=0

for k = 0 and k = 1, one gets n−1

i ai,n =!n,

i=0

n−2

i ai+1,n =!n − n

(216)

i=0

From (216) it follows that for p ≥ 3 prime, !p ≡

p−1

(i + 1) p−i−1 i i

(mod p),

i=0

!p =

p−2

(217) (i + 1) p−i i i−1

(mod p)

i=1

Then Kurepa’s conjecture that ! p ≡ 0 (mod p) ∀ p ≥ 3 is equivalent to the following: p−2 (i + 1) p−i i i−1 ≡ 0 (mod p) ∀ p ≥ 3 (218) i=1

An identity as the second one of (214) is valid for the classical Eulerian numbers, too (due to Worpitzky [487]) n

A(n, k) = n!

(219)

k=1

Now, N. Robbins [368] proved the followinf congruence for A(n, k): If p is prime and m ≥ 1, 1 ≤ k ≤ p m − 1, then 0 (mod p), if k ≡ 0 (mod p) m A( p − 1, k) ≡ (220) 1 (mod p), if k ≡ 0 (mod p) 573

CHAPTER 5

p−1

So A( p − 1, k) ≡ 1 (mod p) for 1 ≤ k ≤ p − 1, and by (219), ( p − 1)! = A( p − 1, k) ≡ p − 1 ≡ −1 (mod p). Therefore (220) and (219) imply the

k=1

Wilson theorem: ( p − 1)! ≡ −1 (mod p). For other arithmetic properties of Eulerian numbers, see e.g. L. Carlitz and J. Riordan [98], R. A. S´aenz Casa and J. E. Nymann [374], or K. Knopfmacher and N. Robbins [256]. For example, in [374] one can find that for all primes p and all 1 ≤ k ≤ p one has A( p, k) ≡ 1 (mod p) (220 ) Let P(n, k) = k!S(n, k) (see Part D. of section 4 of 5.2., e.g. relation (194)). Then (220’) is equivalent to the following: P( p, k) ≡

1 (mod p) if k = 1, 0 (mod p) if 2 ≤ k ≤ p

(220 )

These congruences follow in fact from the inversion formulas P(n, k) =

k−1 n−k +i i

i=0

A(n, k) =

k−1 n−k +i P(n, k − i) (−1)i i i=0

combined with the known relation (−1)

4

A(nk − i),

k−1

p−1 ≡ 1 (mod p) for 1 ≤ k ≤ p. k−1

Estimates and inequalities

Estimates and inequalities for Bernoulli or Euler (and Eulerian) numbers have a central importance in many areas connected with related fields. ∞ 1 via B2n . A A. The important property (7) gives the values of ζ (2n) = n 2k k=1 similar property for the Euler numbers is ∞ k=0

(−1)k π 2n+1 n = (−1) E 2n (2k + 1)2n+1 (2n)!22n+2

574

(221)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

Relations (7) and (221) yield various inequalities for the Bernoulli or Euler numbers. For example, Ch. Jordan [237] showed that

and

2(2n)! 2(2n)! π 2 < |B | ≤ · 2n (2π )2n (2π )2n 6

(222)

24(2n + 1)(2n + 2) B2n+2 (2n + 1)(2n + 2) < < (2π )4 B2n (2π )2

(223)

Various authors have given improvement to these inequalities, e.g. D. J. Leeming [273], A. Laforgia [268], D’Aniello Ciro [110], H. Alzer [20]. For example, in [110] one can find the inequalities 1 2(2n)! 1 2(2n)! · < |B2n | < · 2n −2n 2n (2π ) 1−2 (2π ) 1 − 21−2n

(224)

We note that the upper bound can be found also in Abramovicz-Stegun [1] from 1965 (where the lower bound of (222) is included, too). Leeming proved that √ n 2n √ n 2n 4 πn < |B2n | < 5 π n (n ≥ 2), (224 ) πe πe and Alzer obtained the sharp bounds 2(2n)! 1 2(2n)! 1 · < |B2n | ≤ · 2n α−2n 2n (2π ) 1−2 (2π ) 1 − 2β−2n

(n ≥ 1)

(224 )

where the best possible constants are α = 0 (i.e. the lower bound in (224)), and log(1 − 6/π 2 ) β =2+ = 0.6491 . . . log 2 It can be shown that (224”) sharpens also the above bounds (224’) given by Leeming. By remarking that π 2 (1 − 21−2n )4 ≥ 1 for all n ≥ 2, the upper bound of (224) and the lower bound of (222) imply (4n)!(4n − 2)! B4n B4n−2 (225) ≤ (n ≥ 2), 4 (2n!)4 B2n which was a problem due to L. I. Nicolaescu [326]. By using (224), the left side of (223) can be improved too: (2n + 1)(2n + 2) B2n+2 (2n + 1)(2n + 2) < < , (226) xn (2π )2 B2n (π )2 575

CHAPTER 5

where xn = (22n+2 − 8)/(22n+2 − 1). By (224) and (226) one obtains |B2n | ∼ and

2(2n)! (2π )2n

(227)

2 B2n+2 ∼ n as n → ∞ B π2 2n

(228)

For estimates of type B2n 2 −2n (2n)! − (2π )2n = O((4π − ε) ) as n → ∞, (ε > 0), see A. M. Odlyzko [339]. Similarly, (221) implies 4n+1 (2n)! |E 2n | ∼ π 2n+1 and

(229)

2 E 2n+2 4 2 E ∼ π n as n → ∞ 2n

(230)

and similar inequalities to (222)-(226) for |E 2n | can be deduced. Remark that for |B2n | a similar formula to (221) holds true: ∞ (−1)k−1 k=1

k 2n

= (1 − 21−2n )ζ (2n) = (22n−1 − 1)

|B2n | 2n π , (2n)!

(231)

and depending on the number of summands one can get various lower bounds. By using the Euler-Maclaurin summation formula (see e.g. [338]) one can deduce the asymptotic expansions: N −1 ∞ π2 1 B2m 1 − ∼ + = 6 k2 2N 2 m=0 N 2m+1 k=1

=

1 1 1 1 1 + + − + − ... 2 3 5 N 2N 6N 30N 42N 7

(232)

and [N /2] ∞ (−1)k−1 E 2m 1 5 61 1 π = −2 ∼ − 3 + 5 − 7 + ... 2m+1 2 2k − 1 N N N N N k=1 m=0

576

(233)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

The above infinite series are clearly divergent, the correct interpretation of their asymptotics is ∞ K (2K + 1)! B2m B2m , = +O N 2m+1 m=1 N 2n+1 (2π N )2K +1 m=1 (234) ∞ K E 2m E 2m (2K + 1)! = +O 2m 2m N N (π N )2K +1 m=1 m=1 where K is a fixed positive integer. By using the Boole summation formula (see e.g. [237], [338]), the following more precise result is valid (see J. M. Borwein, P. B. Borwein and K. Dilcher [59]): For positive integers n and M we have 4

M ∞ (−1)k 2E 2k = (−1)n + R1 (M), 2k + 1 (2n)2k+1 k=n k=0

(235)

where

2|E 2M | (2n)2M+1 By similar methods, for the tangent numbers given by (9) (where, in fact T (0) = n n k 2 T (n − k) + T (n) = 0 for n ≥ 1) one can deduce: 1, k k=0 |R1 (M)| ≤

∞ (−1)k+1 = (−1)n+1 k k=n+1

M 1 T (2k − 1) + 2n k=1 (2n)2k

" + R2 (M),

(236)

with

|E 2M | (2n)2M+1 For a generalization of the Euler-MacLaurin and Boole summation formula, see Berndt [45]. B. The sequence of Eulerian numbers (A(n, k)) is unimodal in k for any fixed n: |R2 (M)| ≤

A(n, k − 1)A(n, k + 1) ≤ (A(n, k))2

(0 ≤ k ≤ n),

(237)

see L. Comtet [116] and V. Gasharov [177]. For a new (combinatorial) proof of this result, see M. B´ona and E. Ehrenborg [54], where the log-concavity of (R(n, k))k is proved, too (R(n, k) = number of permutations of {1, 2, . . . , n} with k runs). More generally, the sequence (A(n, k)), 0 ≤ k ≤ n, is a P´olya frequency (PF in short) sequence, in the following sense: 577

CHAPTER 5

An infinite sequence a0 , a1 , . . . of nonnegative numbers is called a P´olya frequency sequence, if the matrix (ai− j )i, j≥0 is a totally positive (TP - for short) matrix, i.e. if all minors of this matrix have nonnegative determinants (suppose ak = 0 if k < 0). A finite sequence a0 , . . . , an is PF-sequence if the infinite sequence a0 , a1 , . . . , an , 0, 0, 0, . . . is PF. (Thus, a PF sequence is necessarily log-concave), The following theorem due to M. Aissen, I. J. Schoenberg and A. Whitney [16] gives a characterization of finite PF-sequences (see also Curry and Schoenberg [125]): A finite sequence a0 , . . . , an of nonnegative numbers is PF if and only if its genn ai x i has only real zeros. (238) erating function (polynomial) i=0

The fact that (A(n, k))k is PF, can be found in [116]. L. H. Harper [213] proved this fact for the Stirling numbers of the second kind (S(n, k))k , while L. A. Sz´ekely [435] for his holiday numbers. J. C. Ahuja and E. A. Enneking [14] proved the PFproperty of a sequence (L m (n, k))k associated with the Lah numbers. Recently Y. Wang and Y.-N. Yeh [478] have studied a general sequence (X (n, k)) with nonnegative terms satisfying the recurrence X (n, k) = (r n + sk + t)X (n − 1, k − 1) + (an + kb + c)X (n − 1, k) (n ≥ 1, 0 ≤ k ≤ n) where X (n, k) = 0 unless 0 ≤ k ≤ n. They have proved that if the (given) real numbers r, s, t, a, b, c satisfy the following conditions: r b ≥ as,

(r + s + t)b ≥ (a + c)s,

then for each n ≥ 0, (X (n, k))0≤k≤n is a PF-sequence. For PF-sequences, see also J. Pitman [350]. L. Lesieur and J. L. Nicolas [285] have proved the inequality A(n, k − 1) <

n−k n−k+2

(239)

n−2k+2 A(n, k)

(n ≥ 2k − 1, k ≥ 2)

(240)

which enabled then to show that the sequence (M2n /(2n)!)n≥1 is decreasing, where Mn = max A(n, k). 1≤k≤n

578

(241)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

Further, by using Cauchy’s integral formula and the Laplace method, they have deduced , 3 3 M2n−1 1− (n ≥ 3), ≤ (2n − 1)! πn 40n M2n−1 ≥ (2n − 1)!

,

13 3 3 − 1− (n ≥ 1), πn 40n 4480n 2

(242)

M2n+5 M2n M2n+3 ≤ ≤ (n ≥ 1) (2n + 5)! (2n)! (2n + 3)! See also [286]. For the Eulerian numbers a(n; k, l) given by (209), Chung, Diaconis and Graham [108] prove: −l −k 4 a(n; k, l)2 = O(n!cn ) (243) 3 k,l where 0 < c < 0.68528 . . . The asymptotic normality of Eulerian numbers have been studied e.g. in L. Carlitz, D. C. Kurtz, R. Scoville and O. P. Stackelberg [79], and recently in L. H. Y. Chen, T. N. T. Goodman and S. L. Lee [105]. In [105] the following general theorem is proved: Let rn, j ( j = 1, 2, . . . , n) be complex numbers lying in the sector π | arg z| ≤ , and define the numbers an,k (n ≥ 1, k = 0, 1, . . . , n) by 3 n

n an,k z = (z + rn, j )/(1 + rn, j ) k

k=0

j=1

Assume that the numbers rn, j are bounded away from −1, the the coefficients an,k are real, and that σn2 =

n

rn, j /(1 + rn, j )2 → ∞ as n → ∞

j=1

Then

and if

n k=0

k − µn −1/2 max σn an,k − G = O(σn ), σn k=0,n

(244)

an,k z k is reciprocal, then k − µn −2/3 max σn an,k − G = O(σn ) σ k=0,n n

579

(245)

CHAPTER 5

1 2 Here G(x) = √ e−x /2 , µn = µ(m n ), where m n (n ≥ 1) denote the discrete 2π measures defined by m n ({k}) = an,k (k = 0, 1, . . . , n). √ When an,k = A(n, k) (i.e. Eulerian numbers), then σn = π(n + 1)/6, and we get from (245) the order of convergence O(n −1/3 ), which improves O(n −1/4 ) of [79]. In 1965 L. Carlitz [78] considered the problem of expanding G m (λ) = zn n n−m in terms of λ = n n−1 z n n!. He proved that n! n≥1 n≥1 G −m (n) =

n

n+m

n≥1

m λ zn = E(m, k)λk n! (1 − λ)2m+1 k=0

(246)

The coefficients E(m, k) satisfy the recurrence E(m, k) = (k + 1)E(m − 1, k) + (2m − k − 1)E(m − 1, k − 1) For combinatorial interpretations, see J. Riordan [366] and I. Gessel and R. Stanley [188].** In++the book [200] of Graham, Knuth and Patashnik, the notation m is applied, where these coefficient are called as the ”second E(m, k) = k order Eulerian numbers”. See also L. M. Smiley [412]. There is an interesting connection between E(n, k) and the associated Stirling numbers of the second kind b(n, k) (see section 4. of 5.2.) which count the ways in which n distinct objects may be partitioned into exactly k true heaps (a true heap has more than one object). One has the identities n k=0 n

E(n, k)x k =

n

b(n + k, k)x k−1 (1 − x)n−k , and

k=1

E(n, k)(1 + x)n−k−1 x k =

k=0

n

(247) b(n + k, k)x k−1 ,

k=1

see L. M. Smiley [413]. We note that here Smiley uses the notation E(n, k) = n, k and b(n, k) = {{n, k}}. C. As we have seen (see e.g. in connection with (112)) the higher order Bernoulli polynomials can be generalized to ”complex orders”, i.e. by defining Bnµ (x) for all x, µ ∈ C by µ ∞ t tn µ zt Bn (z) = e , |t| < 2π n! et − 1 n=0 For µ = 1 we have the Bernoulli polynomials Bn (x). For x = 0 we get the Bernoulli numbers Bnµ of order µ, which for µ = 1 give the classical Bernoulli 580

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

numbers Bn . As we have seen also, the classical Stirling numbers of the first and second kind are in fact higher order Bernoulli numbers, since n n−1 n B −m Bn−m , S(n, m) = s(n, m) = m n−m m−1 In 1961 N, E. N¨orlund [337] has shown that if ρ and x are independent of n, then ( ) ρ (−1)k Bnn+ρ+1 (z) (−1)n (log n)ρ m−1 −m (248) Ak (z) + O(log n) ∼ n! nz k (log n)k k=0 as n → ∞. The coefficients of Ak (z) are derivatives of the reciprocal gamma function: 1 dk Ak (z) = k dz (1 − z) When ρ ≥ 0 is an integer, then (248) reduces to a finite number of terms (because the binomial coefficient vanishes when k > ρ). In particular, when ρ = 0, one has Bnn+1 (z) = (z − 1)(z − 2) . . . (z − n). Therefore ∞ Bk1−z Bnn+1 (z) (n + 1 − z) (−1)k = (−1)n ∼ · z+k , n! (1 − z)(n + 1) k!(1 − z − k) n k=0

which is a well-known result for the ratio of two gamma functions. For fixed values of n N¨orlund gives the expansion 1

2 cos π 2z + µ − n 1 Bnµ (z) 2 µ−1 1+O =n n! (µ)(2π )n n Since Bnµ can be represented as a Cauchy type integral µ t dt n! µ zt Bn (z) = e , t n+1 2πi C e −1 t

(249)

(250)

where the contour C is a circle of radius less than 2π around the origin, N. M. Temme [442] recently has generalized Bnµ (x) for complex n, via µ t dt (ν + 1) µ zt Bν (z) = e (251) t 2πi e −1 t ν+1 C with Re z > 0. Because of the algebraic singularity of t ν+1 at the origin, we assume that the contour of integration runs like a Hankel integral from −∞ around the origin 581

CHAPTER 5

to −∞ (i.e. from −∞, arg t = −π encircles the origin in positive direction, terminates at −∞, with arg t = +π ). We assume that all zeros of et − 1 (except t = 0) are not enclosed by the the contour, and initially, the many valued function t ν is real for real values of ν and t > 0. By using analytic continuation, the Bernoulli polynomials B(z) (i.e. when µ = 1 in (251)) can be defined in the sector | arg z| < π , for any ν ∈ C. By using (251) it follows that the two basic properties of Bernoulli polynomials remain valid: Bν (z + 1) = Bν (z) + νz ν−1 (ν ∈ C, | arg z| < π), d (252) Bν (z) = ν Bν−1 (z) (ν ∈ C, | arg z| < π) dx Generalizations of this type have origins in the classical papers by A. Jonqui`ere [236] and P. B¨ohmer [53]. See also P. L. Butzer, M. Hauser and M. Leclerc [68], [69] (who have overlooked the Jonqui`ere and B¨ohmer papers) for generalizations of Bernoulli and Euler polynomials, and related results. n n n−k z The well known property Bn (z) = Bk has now the following anak k=0 logue: ∞ ν ν−k z Bk as z → ∞, | arg z| < π (253) Bν (z) ∼ k k=0 It follows that, when Re ν < 0, then Bν (z) → 0 as z → ∞ in | arg z| < π

(254)

Further, the following Maclaurin series expansion is valid (see [442]): Bν (z) = −νz ν−1 +

∞ 1 (−z)k , (k + 1 − ν)ζ (k + 1 − ν) (−ν) k=0 k!

(255)

where |z| < 1. This reduces to the finite polynomial expansion of Bn (z) when ν = n. The analytic continuation of Bν (z) from the half plane Re z > 0 into the left half of the complex plane also follows from the first equality of (252). Also in this way we cannot reach the negative z-axis. For the definition of Bν (x) for ν > 0, remark that this depends on the way we approach the negative z-axis: from above or from below. Taking the average of the two values obtained so, we define (see [442]) Bν∗ (x) = lim

y→0

Bν (x + i y) + Bν (x − i y) , 2

Then

x <0

1 ∞ sin 2π kx − νπ 2 Bν∗ (−x) = cos π ν Bν (x + 1) + 2(ν + 1) sin π ν ν (2π k) k=1 582

(256) (257)

STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS

for x > 0, Re ν > 1; and Bν∗ (x) satisfies the following difference property: if x ≥ 0 νx ν−1 , ∗ ∗ Bν (x + 1) − Bν (x) = ν−1 −ν|x| cos π ν, if x < 0, Re ν > 1 For another definition of Bν (x) for x < 0, see Butzer et al. [68]. For asymptotic results on Bνµ , via the saddle point method, see Temme [442]. The method is based of Stirling asymptotics from [439]. We quote only the following result. Assume that ν is large and positive, and the µ is a real parameter. Let φ(t) = (µ − ν) log t − µ log(et − 1). Then Bνµ =

(ν + 1) µ · µ−ν 2πi

C

eφ(t)

et dt , et − 1

d so to calculate the saddle points, i.e. the solutions of φ(t) = 0, we have to solve dt the equation µ −t (258) λ= 1 − e = λt µ−ν When λ < 0, the equation has an infinite number of complex solutions, which occur in complex conjugate pairs, and one pair {t + , t − } can be used for the saddle point method. Locally at t = t + or t = t − we can approximate φ(t) up to the quadratic term of its Maclaurin expansion, and we obtain the asymptotic result: ( ) − + eφ(t ) eφ(t ) (ν + 1) µ − √ as ν → ∞ (259) Bν ∼ √ √ t − −φ (t − ) t + −φ (t + ) 2πi The uniform asymptotics of Bν (z) for large values of z is based on earlier work on asymptotic expansions of Laplace integrals, see F. W. J. Olver [343], N. M. Temme [440], [441], R. Wong [486].

583

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[13] J. C. Ahuja, Concavity properties for certain linear combinations of Stirling numbers, J. Austral. Math. Soc. 15(1973), 145-147. [14] J. C. Ahuja and E. A. Enneking, Concavity property and a recurence relation for associated Lah numbers, Fib. Quart. 17(1979), 158-161. [15] M. Aigner, Kombinatorik I, Springer Verlag, Berlin, 1975 (English ed. by Springer, New York, 1979). [16] M. Aissen, I. J. Schoenberg and A. Whitney, On generating functions of totally positive sequences, I, J. Analyse Math. 2(1952), 93-103. [17] I. Aldanondo, Generalizaci´on del concepto de sumas y differencios finitas y algunas aplicationes de las mismas, Memorias Mat. Inst. ”Jorge Juan” - Consejo Sup. Investigs Cientificas - Madrid, 1948. [18] G. Almkvist and A. Meurman, Values of Bernoulli polynomials and Hurwitz’s zeta function at rational points, C. R. Math. Rep. Acad. Sci. Canada 13(1991), 104-108. [19] H. Alzer, Ungleichungen f¨ur Mittelwerte, Arch. Math. 47(1986), 422-426. [20] H. Alzer, Sharp bounds for the Bernoulli numbers, Arch. Math. (Basel), 74(2000), no. 3, 207-211. [21] H. Alzer, Sharp bounds for the ratio of q-gamma functions, Math. Nachr. 222(2001), 5-14. [22] H. Alzer, Ein Duplikationstheorem f¨ur die Bernoullischen Polynome, Mitt. Math. Ges. Hamburg 11(1987), no. 4, 469-471. [23] H. Alzer and S. L. Qiu, Inequalities for means in two variables, Arch. Math. (Basel), 80(2003), no. 2, 201-215. [24] D. Andr´e, Sur les permutations altern´ees, J. Math. Pures Appl., 7(1881), 167184. [25] G. E. Andrews and D. Foata, Congruences for q-secant numbers, European J. Combin. 1(1981), 283-287. [26] G. Andrews and I. Gessel, Divisibility properties of the q-tangent numbers, Proc. Amer. Math. Soc. 68(1978), 380-384. [27] N. C. Ankeny, E. Artin and S. Chowla, The class number of real quadratic number fields, Annal of Math. 56(1952), 479-493. 586

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618

Index (A, r )-powerful, 114 (m, k)-perfect, 41, 42 (α, β)-multiamicable numbers, 71 3x + 1 problem, 190 A-analogue of Smith’s determinant, 266 A-convolution, 112, 278 A-multiplicative, 114 A-rational function, 114 Ak -convolutions, 113 B-convolution, 116 B-splines, 471 b-additive, 404 e-multiperfect, 52 e-perfect numbers, 52 e-superperfect numbers, 53 f -expansions, 432 f -generated number, 386 f -iterative sequence, 427 f -self-number, 386 G-ary digital expansion, 414 G-convolution, 121 G-multiplicative functions, 121 jth iterate of the Carmichael function, 200 K convolution of incidence functions, 143 K -convolution, 114, 115 K -iterate of an arithmetical function, 115 k-ary divisor, 136

k-Eulerian posets, 148 k-free part, 134 k-full, 236 k-full part, 134 k-harmonic number, 44 k-hyperperfect numbers, 50 k-M¨obius function of a poset, 147 k-multiperfect numbers, 30 k-perfect number, 32 k-perfect numbers, 33 k-reduced residue system, 276 k-Smith number, 383 k-th power of an arithmetic function, 108 k-unitary perfect number, 46 kth cyclotomic polynomial, 561 kth iterate of f , 196 l p norm ( p ≥ 1) of matrices, 274 M-void analogue of the Euler totient, 290 m-bi-unitary perfect numbers, 57 m-bi-unitary superperfect numbers, 58 m-integral, 539 m-multisuperperfect numbers, 55 m-perfect numbers, 55 m-superperfect numbers, 39, 56 m-unitary (k, l)-perfect number, 57 m − e-perfect numbers, 58 m − e-superperfect numbers, 58 n-dimensional symplex, 568 619

INDEX

n-permutations with η-colored head, 480 n-th prime pn , 180 nth prime, 234 nth symmetric mean, 471 p-adic L function, 566 p-adic analysis, 541 p-adic expansion, 498 p-adic gamma function, 566 p-adic integers, 550 p-adic integrals, 566 p-adic log-gamma functions, 566 p-adic mean value theorem, 546 p-adic method, 494 p-group, 155 p-local integers, 553 p-Stirling numbers, 472 p, q-log-concavity, 515 p, q-Stirling numbers, 484 Q-additive function, 389 Q-multiplicative, 406 q-additive, 387, 401 q-additive functions, 388 q-analog of log-concavity, 513 q-analogs of Euler numbers, 559 q-analogue of a positive integer, 481 q-analogue of Bell numbers, 486 q-analogue of Jordan’s totient, 286 q-analogue of power series, 486 q-analogue of the p-adic measure, 566 q-analogue of the classical Euler totient, 286 q-analogue of the classical Ramanujan sum, 286 q-ascending factorial, 569 q-Bernoulli numbers, 483, 561 q-Bernoulli polynomials, 562 q-binomial coefficient, 471 q-binomial coefficients, 480, 506

q-Euler function, 286 q-Euler numbers, 561 q-Euler polynomials, 563 q-Eulerian numbers, 569 q-Eulerian polynomials, 570 q-exponential polynomial, 483 q-generated number, 384 q-Hurwitz ζ -function, 565 q-log- function, 563 q-multinomial coefficients, 506 q-multiplicative, 401, 405, 413 q-Niven number, 382 q-Riemann ζ -function, 562 q-Stirling numbers, 481, 504 q-tangent numbers, 559 q-trigonometric function, 561 q − L-series, 563 r -Stirling numbers, 469 S-perfect numbers, 55 t-chain, 370 Z -perfect numbers, 55 α-adic representation, 419 γ -perfect numbers, 54 ψ-multiperfect numbers, 41 ψ-superperfect numbers, 40 ψ ◦ σ -perfect numbers, 40 σ ◦ ψ-perfect numbers, 40 ϕ-amicable pairs, 71 ϕ-convergence, 187 ϕ-partitions, 223 R-multiplicative, 407 R-system, 407 U-Stirling numbers, 485 ”breeding” method by Borho and Hoffman, 63 Abelian minimal normal subgroup, 155 abelian number fields, 553 abstract prime number theorems, 154 620

INDEX

analytic continuation, 582 and Mellin-Perron summation formula, 423 aperiodic elements, 490 Apostol function, 129 applet, which computes the values ϕ(n) and λ(n), 193 applications of M¨obius inversion formulae, 156 applications of the M¨obius functions, 139 Approximation theory, 469 arithmetic process, 343 arithmetic progression, 221 arithmetic progressions, 136, 151, 220, 229, 254 arithmetical products, 285 arithmetical semigroup, 149 arithmetical semigroups, 292, 331 arithmetical triangles, 502 ascents, 567 associated Stirling numbers, 502, 580 asymptotic analysis, 511 asymptotic behavior of the Stirling numbers, 516 asymptotic density, 363 asymptotic development of the Stirling numbers of the first kind, 515 asymptotic expansions of Stirling numbers of the second kind, 520 asymptotic normality, 416 asymptotic normality of Eulerian numbers, 579 asymptotic prime divisor, 341 asymptotic sign fluctuations, 517 asymptotically Poisson distributed, 518

abundant numbers, 17, 43 action of the cyclic group, 490 adding prime totatives, 248 adding composite totatives, 248 additive analogue of Euler’s totient, 291 additive analogue of the σ , 292 additive arithmetic function, 366 adjacent slices, 568 Airy asymptotic, 524 algebraic function fields, 161 algebraic independence of real valued arithmetic functions, 185 algebraic integers, 293, 421 algebraic number field, 159, 263, 419 algebraic number fields, 193 algebraic number theory, 160 algebraic numbers, 553 Algebraic topology, 162 algebraic topology, 499 algorithms, 36, 419 aliquot sequence, 72, 74 all-Niven, 383 Alladi’s totient, 282 almost all sums, 340, 342 almost amicable pairs, 69 almost perfect numbers, 37, 38 almost superperfect numbers, 41 amicable k-tuples, 70 amicable numbers, 36, 60, 63 amicable numbers in special sequences, 67 amicable triples, 69 analogue of Adams’ theorem, 557 analogue of Euler’s formula, 62 analogue of Lehmer’s problem, 213, 214 analogue of Smith’s determinant, 266 analogue of Thabit’s formula, 62 621

INDEX

binary recurrence sequence, 240 binomial coefficient, 49, 107, 126, 371 binomial coefficients, 243, 460, 474 binomial convolution of sequences, 117 binomial measure, 375 biological analogies, 17 bipartite graphs, 348 birational transformation, 192 block-additive, 410 block-multiplicative, 408 Bohr almost-periodic function, 136 Bohr-Fourier spectrum, 413 Bolyai-R´enyi expansions, 432 Bombieri’s sieve, 227 Boole summation formula, 577 Boolean lattice, 513 Borel σ -algebra, 344 bounded below set, 148 bounded variation, 103 Brauer-Rademacher identity, 144, 180 britannic number system, 407 Brownian motion, 344 Brun’s sieve, 235 Brun-Titchmarsh theorem, 216, 227 Buchstab function, 357 Burnside basis theorem, 293

asymptotically unit exponential variables, 341 automorphic numbers, 431 average number of blocks, 512 Baker’s method, 418, 430 Barrucand numbers, 494 base q palindromes, 430 basis, 481 Bassily-K´atai theorem, 403 Bauer’s theorems, 245 BDE method, 61, 63 Bell number, 461 Bell numbers, 493 Bell polynomials, 484, 495 Bell sequence, 512 Benford law, 430 Bernoulli and Euler polynomials, 529, 582 Bernoulli and Stirling numbers, 259, 467 Bernoulli number, 461 Bernoulli numbers, 243, 244, 495, 500, 518, 525 Bernoulli numbers of fractional order, 474 Bernoulli numbers of higher order, 500 Bernoulli numbers of order k, 468 beta-expansions, 432 betrothed numbers, 68 Beurling’s generalized primes, 207 bi-unitary amicable numbers, 67 bi-unitary convolution, 111 bi-unitary divisor, 46 bi-unitary multiply perfect numbers, 47 bi-unitary perfect numbers, 47 bi-unitary quasiperfect, 47 bilinearly recurrent orthogonality, 476

C++ programming language, 217 cancellation property, 120 canonical basis, 393 canonical number system, 418 canonical number systems, 404 Cantor representations, 412 Carlitz congruence, 552 Carmichael conjecture, 228 Carmichael’s λ function, 193 Carmichael’s theorem, 190 Cashwell-Everett totient, 280 Catalan’s conjecture, 73 622

INDEX

composition of some arithmetic functions, 39 computational evidence, 42 computer algebra, 486 conductor, 553 congruence of Kummer, 541 congruence of Voronoi, 541 congruence properties, 477 congruence properties of σ (m) (n), 42 congruence properties of Euler’s totient, 201 congruence property, 188 congruences for Genocchi numbers, 541 congruences for higher order Bernoulli numbers, 549 congruences for Stirling numbers, 488 conjecture, 42, 47, 51–53, 59, 195, 229, 235, 241, 348, 382, 400, 494, 499, 556, 558, 573 conjecture of Sierpinski, 229 conjectured, 46, 352 conjectures, 199, 216, 237, 557 conjugacy classes, 157 conjugate Bernoulli numbers, 538 conjugate Bernoulli polynomials, 538 consecutive divisors, 342 consecutive prime factors, 337 constants, 181 continued fractions, 432, 537, 561 continued powers and roots, 432 convex polytope, 559 convolution of incidence functions, 139 convolution on the set of sequences, 537 convolutions with unbounded unity, 128 core, 208

Cauchy product, 117, 122, 128 Cauchy’s integral formula, 579 central factorial numbers, 479 central limit theorem, 344, 402, 423 chain, 72, 369 chains, 141, 144 change of the sum of digits by multiplication, 378 character group, 151 characteristic equation, 240 characteristic polynomial, 414 characters of finite abelian groups, 151 Chebysheff’s theorem, 207 circle of convergence, 415 circulant matrix, 183 circular permutations, 192 classical M¨obius function, 99, 128, 135 code, 179, 396 coefficients of cyclotomic polynomials, 256 Cohen totient ϕ S , 290 combinatorial and statistical applications, 480 combinatorial applications, 107, 477 combinatorics on words, 390 common generalization of Liouville’s function and the M¨obius function, 137 commutative group, 106 commutative ring, 110, 121 commutative semigroup, 117, 118 commutator subgroup, 158 complete asymptotic expansion, 520 complete symmetric functions, 471 complex bases, 417 complex variables, 381 composition of ϕ and other functions, 234 623

INDEX

derangements and cycle indicators, 507 derived k-cycle, 337 derived sequence, 337 descents, 570 determinant, 269 determinants, 265 determinants involving q-Stirling numbers, 483 Dibag’s von Staudt-Clausen theorem, 553 Dickman function, 359 Dickson’s conjecture, 221 Dickson’s theorem, 30 Dickson-Eulerian numbers, 571 Dickson-Stirling numbers, 477 difference operator, 123 differential equation, 567 differential equations, 469 differential geometry, 390 digits of Fibonacci numbers, 430 Diophantine approximations, 263 Diophantine equations, 28, 61 Diophantine Equations, 107 diophantine equations, 219 direct factor set, 135 directed graph, 183 Dirichlet convolution, 112, 117, 127 Dirichlet density, 152 Dirichlet inverse, 120 Dirichlet inversion, 123 Dirichlet product, 106, 109, 119, 132 Dirichlet series, 100, 131, 243, 423 Dirichlet set, 206 Dirichlet’s theorem on arithmetical progressions, 210 discrete analog of integration by parts, 145 discriminator, 211

core-reduced residue system, 290 core-reduced totient of J. S´andor and R. Sivaramakrishnan, 290 cosets, 151 counterexample to Carmichael’s conjecture, 228 Criptography, 15 cross-convolution, 113 cumulative distribution function, 397 cycle, 72, 427 cycles, 73, 75, 183, 462, 479, 491 cyclic homology, 570 cyclotomic Bernoulli numbers, 539 cyclotomic field, 539 cyclotomic fields, 556 cyclotomic polynomial, 27, 138, 139, 203, 251, 256, 506, 571 cyclotomic polynomials, 24, 261 Daykin’s M¨obius inversion formula, 110 Dedekind arithmetic function, 39 Dedekind arithmetical function, 231, 284 Dedekind domains, 191 Dedekind zeta-function, 293 deficient numbers, 17, 43 degenerate Eulerian numbers, 572 degenerate weighted Stirling numbers, 476 degree of resolution, 367 dense, 331 density, 221 density of e-perfect numbers, 52 density of k-perfect numbers, 33 density of amicable pairs, 65 density of friendly pairs, 70 density of harmonic numbers, 44 density of multiamicable numbers, 71 density of odd perfect numbers, 30 624

INDEX

Euclidean norm of a matrix, 274 Euler factor, 23, 30 Euler functions of finite groups, 293 Euler gamma function, 563 Euler l.c.m. sequence, 211 Euler maximal function, 210 Euler minimum function, 210 Euler numbers, 527 Euler quotients, 538 Euler totient, 40, 496 Euler totient of matrices, 278 Euler’s divisibility theorem, 188, 538 Euler’s function, 22 Euler’s gamma function, 523 Euler’s methods, 61 Euler’s theorem, 24 Euler’s totient, 67, 71, 106, 134, 137, 179, 184, 497, 544, 555 Euler’s totient function, 269, 272 Euler-MacLaurin summation formula, 547 Euler-Maclaurin summation formula, 576 Eulerian functions, 571 Eulerian numbers, 567 Eulerian numbers of higher order, 571 Eulerian polynomials, 477, 568 Eulerian posets, 146 even functions (mod n), 184 even perfect number, 19, 21, 22, 32, 54 even perfect numbers, 20, 22, 23, 44, 59 even-odd amicable pair, 65 exponential analogue µ(e) of the M¨obius function, 118 exponential Bell polynomials, 477 exponential convolution, 117 exponential cycles, 75 exponential divisor, 51

distance geometry, 486 distinct divisors, 347 distribution (mod 1) on divisors, 366 distribution function, 242, 345, 351, 353, 402, 524 distribution functions, 355 distribution in residue classes, 206 distribution of Bernoulli numbers modulo primes, 556 distribution of primes, 211 distribution of totatives, 246 divided differences, 469 divisibility algorithm, 191 divisor density, 366 divisorial graph, 369 divisors in residue classes, 363 divisors of factorials, 354 Dobinski’s formula, 461 double series, 101 double-factorial, 483 dynamical system, 367 Egyptean fractions, 16 eigenspace, 394 eigenvalue, 393, 409 eigenvectors, 393 eingenvalues, 393 Eisenstein series, 544 elementary Abelian, 156 elementary abelian, 159 elementary symmetric function, 510 epimorphisms, 158 equidistributed (mod 1), 366 Erd¨os-Kac theorem, 388 Erd¨os-Tur´an-Koksma inequality, 405 Erd¨os-Wintner theorem, 401 ergodic theory, 412 Euclid’s ”Elements”, 16 Euclid-Euler theorem, 21 Euclidean division, 570 625

INDEX

exponential inverse, 118 exponential operator, 108 exponential sum of digit counting function, 379 exponential sums, 261 exponential totient function, 119, 288 exponentially A-multiplicative functions, 119 exponentially multiplicative functions, 118 extended Nagell totient, 132 extension of Nagell’s (and Alder’s) totient, 279

Fibonacci sequence, 22 Fibonacci, or Lucas numbers, 204 finite p-group, 293 finite alphabet, 161, 374 finite automata, 390 finite field, 192, 286 finite group, 154, 490 finite permutation groups, 193 finite poset, 154 finite recurrences, 414 flag, 481 formal languages, 161 formal Laurent series, 488 formal power series, 162, 426 Fourier analysis, 415 Fourier coefficients, 184, 544 Fourier expansions, 531 Fourier transforms, 103 fractal dimension, 367 fractional parts of the Bernoulli numbers, 540 Frattini chief factor, 155 free abelian group, 151 frequency of a block, 421 friendly pair, 70 Frobenius congruences, 544 Frobenius’ classical theorem, 156 function λ, 189 function by Jacobsthal, 247 functional limit theorems, 416 fundamental domains, 421

factor-closed set, 267 factorable function, 143 factorable functions, 127 factorial base, 386 factorial numbers, 478 factorial ring, 106 factorization methods of integers, 256 factorization of Bernoulli numerators, 556 falling factorial, 478 falling factorial power, 460 Fermat number, 223 Fermat numbers, 254 Fermat prime, 27, 191, 239, 527 Fermat primes, 225, 236 Fermat quotient, 548 Fermat quotients, 530 Fermat’s ”little theorem”, 15 Fermat’s divisibility theorem, 179 Fermat’s little theorem, 488 Fermat’s quotient, 211 fermionic oscillators, 487 Fibonacci and Lucas numbers, 182, 498 Fibonacci numbers, 398 Fibonacci or Lucas sequences, 239

Galois field, 481 Gauss norm, 547 Gauss sum, 566 Gauss’ theorem, 193 Gaussian integer, 263 Gaussian integers, 32, 404 Gaussian limit law, 399 Gaussian prime, 26, 263 626

INDEX

generalized Thabit’s formulae, 61 generalized totient function, 269 generalized totient functions, 137 generating function, 460, 461, 480, 483, 505, 578 generating functions, 474, 476, 484, 570 Genocchi numbers, 532 Genocchi numbers of order k, 533 geometrical investigations of congruence theorems, 189 Gill’s complexity class, 36 golden number, 486 Golomb-Guerin totient function, 288 Golubev totient functions, 289 good lattice points modulo composite numbers, 345 graceful graphs, 369 graded poset, 146 Gram’s determinant, 420 gravaritm, 201 greatest common divisor, 263 greatest prime divisor of m, 255 greatest prime factor, 356 group of units of the ring Zn , 249 groups acting on functions, 490 groups acting on graphs, 490 Guerin analogue of the M¨obius function, 122

GCD matrices associated with arithmetical functions, 273 gcd-closed matrices, 270 gcd-closed set, 270 gcud-closed, 271 Gegenbauer’s theorem, 192 general theorem of log-convexity, 511 generalization of Euler’s theorem, 191, 192 generalization of Euler’s totient, 246 generalization of finite differences, 475 generalization of M¨obius’ function, 115 generalization of the M¨obius function, 137 generalizations of Euler’s totient, 291 generalizations of Klee’s totient, 278 generalizations of Ramanujan’s sum, 277 generalizations of Ramanujan’s sum to matrices, 278 generalizations of the M¨obius function, 139 generalized q-Bernoulli numbers, 562 generalized q-Euler numbers, 562 generalized arithmetical functions, 140 generalized Bernoulli numbers, 553 generalized Eulerian numbers, 571 generalized harmonic numbers, 488 generalized Liouville function, 114 generalized M¨obius function, 107, 108, 135, 138 generalized M¨obius functions, 134 generalized M¨obius inversion, 104 generalized primes, 150 generalized Ramanujan sum, 114 generalized Stirling numbers, 469

H-Hypothesis by Schinzel, 229 Hall π -subgroup, 158 Hankel contour, 523 Hankel integral, 581 Hankel matrix, 483 Hardy-Littlewood circle method, 261 Hardy-Littlewood conjecture, 219 Harmonic analysis, 415 harmonic number, 466 harmonic numbers, 43, 44, 472 627

INDEX

inequalities for Stirling numbers, 508 inequalities for the r -Stirling numbers, 470 inequalities for the Bernoulli or Euler numbers, 575 inequality for det(S) f , 273 infinitary amicable numbers, 69 infinitary convolution, 124 infinitary divisor, 48 infinitary divisors, 124 infinitary harmonic numbers, 50 infinitary M¨obius function, 124 infinitary perfect numbers, 49 infinitary sociable numbers, 75 infinitary totient, 287 infinite chess games, 390 infinite dimensional matrix, 127 infinite product, 182, 424 infinite recurrences, 415 infinite sums, 278, 423 infinitude of primes, 180 inner product, 185 integer partitions in dominance order, 148 integer-perfect numbers, 53 integral ideal, 293 integral ideals, 159 integral product, 128 integral words, 151 intermediate prime divisors, 341 intuitionistic fuzzy theory, 148 inverse problems in Physics, 146 inversion formula, 153, 160 inversion formulae, 100 inversion formulas, 462 inversion statistic, 481 inversion theorem, 126, 145 inversion theorem of M¨obius type, 120 inversions, 570

Haukkanen M¨obius function, 114 Hausdorff dimension, 368 high-speed computers, 73 higher order Bernoulli polynomials, 580 higher order Euler polynomials, 558 highly abundant, 347 highly powerful numbers, 334 Hilbert space, 185 Hilbert transform, 538 Hilbert’s inequality, 354 hoax number, 384 Hoheisel-Ingham theorem, 380 Holiday numbers, 488 holiday numbers, 578 homogeneous symmetric polynomial, 189 homotopy and homology of finite lattices, 148 Hooley’s function, 365 Hurwitz zeta function, 423 hyperelliptic equations, 507 hypergeometric functions, 525 hyperharmonic numbers, 471 hyperperfect numbers, 50 hyperplanes, 568 hypothesis H , 42 Hypothesis H of Schinzel, 222 H¨older’s theorem, 259 identities, 181, 464 incidence algebra, 145 Incidence algebras, 139 incidence function, 140 incidence functions, 267 incidence matrix, 268 incidence ring, 140 independent random variables, 511 indicator, 179 inequalities for n (x), 261 628

INDEX

Lagrange’s theorem, 488 Lah numbers, 578 Lah’s numbers, 478 Laplace integrals, 583 Laplace method, 579 Laplace’s rule, 259 large prime divisors, 341 lattice, 141 lattices of subgroups of a group, 139 law of iterated logarithms, 364 lcm-closed sets S, 271 least common multiple, 348 least nonnegative residue, 543 least prime divisor, 355 left inversion, 123 left-factorial function, 573 Legendre formula, 371 Legendre symbol, 250, 396, 547 Legendre totient, 283, 287 Legendre totient of u arguments, 285 Legendre-Jacobi symbol, 183 Legendre-Stirling numbers, 488 Lehmer ψ-product, 128 Lehmer’s conjecture, 212 Lehmer’s totient function, 289 Lehmer, McCarthy, and Erd¨os theorems, 247 Leudesdorf’s theorem, 244 limit distribution mod m, 389 limiting distribution, 540 linear forms in logarithms, 508 linear groups, 186 linear independence over Q, 263 linear recurrences, 414 linear transformations, 185 Liouville function, 116, 137 ”Little theorem”, Fermat, 19 local limit law, 416 locally distributive, 143

involutions of permutations, 507 irrational number, 411 irrationality result, 254 irreducibility of polynomials, 253 irreducibility of the cyclotomic polynomial, 252 irreducible polynomial, 262, 362 irreducible polynomials, 192 isomorphic classes of objects, 186 iterated Stirling numbers, 493 iterates, 75 iteration, 42 iteration of s(n), 429 iteration of ϕ, 195 Iwasawa functions, 544 joint distribution, 403 Jordan block, 409 Jordan curve, 519 Jordan totient, 235, 275 Jordan totient function, 182, 214 Jordan’s totient, 106, 133, 280 Jordan’s totient function, 259 Jordan-curve, 473 Jordan-Nagell totient, 132, 275, 276 Kanold’s method, 30 Kaprekar numbers, 22 Kaprekar routine, 429 Kesava Menon’s totient, 281 Klee’s method, 228 Klee’s M¨obius function, 278 Kloosterman-sums, 250 Kubilius class, 344 Kummer type congruences, 552, 554, 567 Kurepa’s left-factorial function, 495 l-c product, 126 l.c.m.-product, 119 Lagrange theorem, 245 629

INDEX

Mellin transform, 566 Mersenne prime, 21, 22, 234 Mersenne primes, 15, 19, 41, 42, 222 metabelian group, 159 methods of Voronoi, 545 Miller-Rabin and Lucas tests, 218 minimal polynomial, 420 minimal, normal, and average order of Carmichael’s function λ, 193 Minkowski product, 104 modern computers, 24 modified q-Bernoulli numbers, 564 modified q-Bernoulli polynomials, 564 modified Stirling numbers, 500 modular group, 272 modular idempotent numbers, 431 Monica sets, 384 monoids with finite decomposition property, 148 multidimensional Smith’s determinants, 266 multiperfect numbers, 32, 35 multiplicative inverse of t, 250 multiplicative semigroup, 105 multiply perfect numbers, 32, 35, 71 M¨obius categories, 148 M¨obius function, 153, 154, 264, 268, 490, 496 M¨obius function µG , 128 M¨obius function associated with the dual poset, 157 M¨obius function generated by the B-convolution, 116 M¨obius function of n arguments, 109 M¨obius function of P, 140 M¨obius function of a direct product, 158

locally finit poset, 142 log-concave function, 524 log-concavity, 577 logarithm operator, 108 logarithmic and identric means, 486 logarithmic convexity, 471 logarithmic distribution function, 397 logarithmic Fourier series, 398 logarithmic polynomials and numbers, 533 logarithmically concave, 470 lovely numbers, 69 lower closed sets, 267 lower density, 236 Lubin-Tate groups, 566 Lucas sequence of the second kind, 240 Lucas sequence of the first kind, 240 Lucas sequences, 36 Lucas’ totient, 277 Mac Mahon’s Master theorem, 161 magic squares, 184 Magma system for computational algebra, 258 Maillet’s theorem, 207 Makowski-Schinzel conjecture, 235 Marriage theorem, 348 matched asymptotic expansions, 516 matrix-generated convolutions, 127 maximal generalization of Fermat’s theorem, 190 maximum and minimum exponents, 330 maximum of the exponents, 355 mean value, 401 meet semilattice, 267 meet-closed sets, 273 meet-closed subset, 268 meet-matrix, 268 630

INDEX

M¨obius function of a generalized integer, 125 M¨obius function of a set of primes, 105 M¨obius function of a trace monoid, 162 M¨obius function of geometric lattices, 148 M¨obius function on a subset of a power set, 154 M¨obius functions of arbitrary direct factor sets, 135 M¨obius inversion, 102–104, 148, 490 M¨obius inversion formula, 180, 251 M¨obius inversions, 482 M¨obius system, 104 M¨obius type inversion theorems, 150 M¨obius’ function, 184 M¨obius’ function of an arithmetical semigroup, 148 M¨obius-type function, 193 M¨obius-type inversion formula, 135 M¨obius-type inversion formulae, 125

noncototients, 208 nontotient, 230 nonunitary k-cycle, 76 nonunitary aliquot sequences, 76 nonunitary amicable pairs, 68 nonunitary divisors, 48 nonunitary perfect numbers, 48 nonunitary totient function, 287 normal order, 332 nowhere differentiable, 373, 375, 392, 400, 417 nowhere differrentiable, 399 number of decimal digits, 431 number of digit blocks of 11, 426 number of distinct prime divisors, 33, 110, 180 number of distinct prime factors, 28, 243, 250, 265, 283, 519 number of divisors, 35, 180, 264 number of odd digits, 425 number of solutions of the congruence, 279 number of square-free divisors, 130 number of terminating nines, 426 number of unitary divisors, 284

Nagell’s totient, 133, 278 Narkiewicz M¨obius function, 144 Narkiewiecz M¨obius function, 113 Narumi polynomials, 550 negative integer bases, 417 Newton’s formulae, 258 Nielsen polynomials, 467 nilpotent elements, 118, 290 Niven number, 381 no unitary m-superperfect numbers, 57 non-central Stirling numbers, 479 non-commuting lifting, 162 non-crossing set partitions, schuffle posets, 148 non-divisibility property, 202

occurrences of the block w, 425 odd amicable pairs, 64 odd perfect number, 25, 28, 29, 31, 33, 34, 41 odd perfect numbers, 19, 23, 26, 31 open problem, 203, 209, 222, 272, 349, 416 open problems, 31, 43, 76, 203 open question, 67, 209 operatorial theory, 469 optimization theory, 339 orbit, 490 orbits, 481 order of a modulo n, 189 631

INDEX

positive definite, 273 power series, 100 power-harmonic number, 44 powerful, 52 practical numbers, 59, 359 preferential arrangements, 496 primality criterion, 254, 256 primary pseudoperfect number, 45 prime k-tuple conjecture, 218 prime factors of n (a, b), 255 prime factors of ϕ ( j) (n), 198 prime factors of the Euler numbers, 556 prime ideal, 293 prime ideal theorem, 160, 161 prime number theorem, 149, 150 prime-counting function π(n), 180 prime-power groups, 139 primitive character, 553 primitive divisors of Lucas sequences, 216 primitive element, 420 primitive root mod n, 251 primitive root of unity, 258 primitive roots, 189 primitive roots of unity, 138, 251 primitive solutions, 197, 198 primitive substitution, 374 probabilistic group theory, 364 product of all e-divisors, 58 product of all divisors, 55 product of digits, 383 product of divisors, 202 product set, 104 pseudoperfect numbers, 43 pseudoprime, 30 P´olya frequency sequence, 578 P´olya-Vinogradov inequality, 554

order of the group, 202 orthogonal projection, 394 orthogonality, 476, 477, 485 orthonormal basis, 185 oscillatory asymptotic, 524 other analogs of Lehmer’s problem, 215 overlap, 390 palindrome, 391 parallelotope, 420 parametric M¨obius inversion, 145 Parry’s α-expansion, 415 partial p, q-Stirling numbers, 486 partial ordering, 141 partial product, 128 partially ordered sets, 105 partition lattice q-analogs, 481 partitions, 461 Pascal’s triangle, 379 Pell sequence, 67 perfect number, 369 perfect totient number, 240 periodic mapping, 554 periodic perfect numbers, 17 periods of the Stirling numbers, 492 Phibonacci number, 223 Pillai arithmetic function, 182 Piltz divisor function, 106, 107 Pisot number, 415 Plato’s Republic, 17 pointwise product, 119 Poisson process, 341 poly-Bernoulli numbers, 501 polylogarithm function, 552 polynomial ring, 552 polynomial sequences, 402 poset, 139 poset-theoretic generalization of Smith’s theorem, 268 632

INDEX

residue theorem, 478 resolution function, 367 resultant of cyclotomic polynomials, 262 reversing the digits, 429 Riemann hypothesis, 102, 129, 130 Riemann Hypothesis, 187 Riemann hypothesis, 262 Riemann zeta function, 99, 187, 263, 330, 510, 527 Riesz representation theorem, 185 right inversion, 123 right inversion formula of M¨obius type, 123 rising factorial power, 461 roots of unity, 184, 393 Roth’s theorem, 426 Rudin-Schapiro sequence, 396 R´edei’s generalization of Euler’s theorem, 208

quadratic character, 554 quadratic field, 285, 293 quadratic fields, 32, 252 quadratic residue, 216 quadri-amicable numbers, 70 quasi-amicable pairs, 68 quasi-multiplicative functions, 270 quasi-multiplicativity, 116 quasi-orthogonality, 476 quasi-superperfect, 41 quasiperfect numbers, 36 queues, 464 Ramanujan expansions, 278 Ramanujan identities, 536 Ramanujan sum, 184, 258, 277 Ramanujan sums, 125, 137 Ramanujan sums on semigroups, 277 Ramanujan theorems, 544 Ramanujan’s sum, 265 ray method, 516 reciprocal GCD and LCM matrices, 272 recursive algorithm, 223 recursive set, 32 refined Stirling numbers, 487 regular S-representation, 285 regular arithmetic convolution, 144 regular convolution, 112 regular convolutions, 183 regular polygon, 188, 239 relatively k-prime, 136, 275 relatively prime amicable pairs, 65, 66 relatively prime amicable pairs of opposite parity, 66 remarkable identities, 282 representations of real numbers, 432 repunit number, 383 residue classes, 387 residue classes modulo n, 249

saddle point method, 516, 583 Salem numbers, 415 Schemmel totient, 184 Schemmel totient function, 229 Schemmel’s totient, 133, 216, 276 Schemmel-Nagell totient, 133, 276 Schinzel’s H -Hypothesis, 380 Schinzel’s conjecture, 218 Schinzel’s Hypothesis H, 198, 221 Schinzel-Szekeres function, 356 Schmidt’s subspace theorem, 418, 508 Schr¨odinger operators, 391 Schur’s theorem, 260 secant numbers, 559 second order Eulerian numbers, 580 Selberg’s identity, 160 Selberg’s upper bound sieve, 226 self-number, 384 semi-multiplicativity, 116 633

INDEX

semi-unitary divisor, 281 semi-unitary greatest common divisor, 271 semi-unitary product, 128 semigroup approach to the Fermat and Euler’s theorems, 191 semigroups with unique factorization, 150 semisimple rings, 333 sequence (Bk (mod n)), 555 sequence of Eulerian numbers, 577 sequences of Fibonacci, Pell, and Lucas, 59 set of totients, 206 Shapiro-Pillai function, 200 Shintani zeta functions, 536 Siegel-Walfisz theorem, 248 sieve of Eratosthenes, 338 Sieve theory, 103 sieve-theoretical problems, 355 sigmoid function, 569 signed Eulerian polynomial, 570 signless q-Stirling numbers of the first kind, 481 signless Bernoulli numbers, 525 Silverman constant, 182 simple groups, 157 simple pole, 415 Sita Ramaiah’s M¨obius function, 113 small prime divisors, 341 small sieve, 359 smallest prime totative, 248 Smarandache function, 22, 210, 237 Smith brothers, 384 Smith number, 383 Smith-Jones number, 384 sociable number of order k, 75 sociable numbers, 72 soluble group, 155, 157

soluble groups, 154 special cyclotomic polynomials, 252 spectral norm of a matrix, 274 square-reduced residue system, 289 square-totient of R. Sivaramakrishnan, 289 squarefree integers, 33, 207, 265 squarefree number, 35, 46 squarefree numbers, 136 squarefull, 209 squarefull integers, 331 staggered sums, 429 statistic on permutations, 479 statistical convergence, 331 statistical limit, 335 Stevens totient, 280 Stevens’ totient, 280 Stieltjes transforms, 525 Stirling functions of the first kind, 473 Stirling functions of the second kind, 473 Stirling numbers associated to a matrix, 475 Stirling numbers of complex argument, 487 Stirling numbers of complex arguments, 523 Stirling numbers of fractional order, 472 Stirling numbers of the first kind, 462, 479 Stirling numbers of the first kind st (n, k) associated to the sequence t, 474 Stirling numbers of the second kind, 460, 578 Stirling numbers of the third kind, 464 Stirling polynomials, 466, 469 Strassen functional, 343

634

INDEX

strong log-concavity, 508 strong regular A-convolution, 119 strongly q-additive, 411 strongly q-log-concave, 514 subadditive functions, 344 subblock occurrences, 422 subgroup, 151, 157 sublime numbers, 22 submatrix, 269 subring, 142 subspace, 185 sugcd-closed, 271 sum of jth powers of totatives, 242 sum of digits, 371, 499, 551 sum of digits of primes, 379 sum of digits with prime indices, 390 sum of divisors, 180, 264 sum of prime digits, 390 sum of totatives, 242 sum-product numbers, 383 super-lovely pair, 69 super-multiplicative, 355 supermultiplicative, 346 superperfect number, 58 superperfect numbers, 23, 38, 69 Suzanne sets, 384 Sylow p-subgroup, 155 Sylow subgroups, 158 symmetric signed digit expansion, 422 symmetric system, 422 symmetries and dualities, 476

the characteristic, 511 The exponential A-convolution, 118 theorem by J. Touchard, 29 theorem by Touchard, 35 theorem of Adams, 541 theorem of Howard, 551 theorem of Sylvester, 25 theorem of Sylvester and Lipschitz, 548 theoretical physics, 375 theory of M¨obius function, 139 Thue-Morse sequence, 379 Thue-Morse sequence, 390 total number of prime divisors, 180 total number of prime factors, 26, 133, 137 totatives, 179, 242 totient functions, 125 totient-free residue classes, 206 Touchard’s congruence, 495 trace languages, 161 trace monoids, 161 transcendence, 391 transcendental, 426 transitive orientation, 162 triangular matrices, 265 triperfect numbers, 34 twin dragon, 421 twin primes, 234 twin-prime constant, 219

tangent numbers, 527, 559, 577 Taylor expansion, 488 te Riele’s trick, 63 Teichm¨uller character, 554 Thabit’s rule, 17, 61 Thacker’s function, 243 Thacker’s totient function, 289

ud-closed, 271 ultrametric Banach space, 546 unambiguous lifting, 162 unambiguous M¨obius function, 162 unequal characters, 554 uniform asymptotic, 524

635

INDEX

unitary hyperperfect numbers, 51 unitary multiply superperfect numbers, 48 unitary M¨obius function, 111, 272 unitary perfect numbers, 45 unitary product, 110 unitary product set, 111 unitary quasi superperfect numbers, 47 unitary quasi-sociable numbers, 75 unitary quasiperfect numbers, 47 unitary Smith determinants, 271 unitary sociable numbers, 74 unitary superperfect numbers, 47 unitary totient function, 213, 266 universal Bernoulli numbers, 552 universal generated, 385 unsigned Lah numbers, 464 unsigned Stirling numbers of the first kind, 462 unsolved problems on generalized integers, 125 upper density, 345

uniform asymptotic expansion, 518 uniform limit distribution mod m, 389 uniform probability measure, 343 uniform summability, 413 uniformly distributed mod m, 389 uniformly distributed mod 1, 410 uniformly distributed modulo 1, 397 uniformly periodic, 427 unimodal, 577 unique factorization domains, 32 unitary m-perfect numbers, 57 unitary abundant numbers, 46 unitary aliquot sequences, 74 unitary almost perfect numbers, 47 unitary almost superperfect number, 47 unitary almost-sociable numbers, 75 unitary amicable numbers, 67 unitary analogue of Euler’s totient, 242 unitary analogue of Legendre’s totient, 284 unitary analogue of Nagell’s totient, 281 unitary analogue of Ramanujan’s sum, 267 unitary analogue of Schemmel’s totient, 281 unitary analogue of Smith’s determinant, 266 unitary analogue of the Carmichael’s conjecture, 229 unitary analogues of generalized Ramanujan sums, 277 unitary convolution, 112, 127 unitary divisor, 46 unitary harmonic numbers, 49, 50

valence function of ϕ, 226 vector space, 145, 286, 420, 481 von Mangoldt function, 160, 259 von Staudt-Clausen and Voronoi type congruences, 554 von Staudt-Clausen theorem, 245, 539 von Sterneck and the Cohen totients, 275 Waring’s problem, 421 weak convergence of distribution functions, 343 weakly-composite, 340 weird numbers, 43, 44 well-distributed mod 1, 411 636

INDEX

Worpitzky type formulas, 572 wreath products, 490

Weyl derivative, 474 Weyl’s lemma, 421 Wieferich prime, 212 Wieferich primes, 540 Wiegandt convolution, 139 Wiener measure, 344 Wilson quotient, 540, 547 Wilson theorem, 574 Wilson’s theorem, 191, 488, 542 Worpitzky identity, 568

Zeckendorf sum-of-digits function, 398 Zeckendorf Thue-Morse sequence, 399 zero divisors, 118 Zsigmondy-Birkhoff-Vandiver theorem, 202

колхоз 11/14/06

637

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