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TOPICS IN ARITHMETICAL FUNCTIONS

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NORTH-HOLLAND MATHEMATICS STUDIES

43

Notasde Maternatica (72) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester

Topics in Arithmetical Functions Asymptotic formulae for sums of reciprocals of arithmetical functions and related results

J.-M. DE KONINCK University of Lava1 Quebec, Canada and

A. lVlc University of Belgrade Belgrade, Yugoslavia

1980

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

NEW YORK

OXFORD

North-Holland Publishing Company, 1980 All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright o wner.

ISBN: 0444860495

Publishers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD Sole distributors for the U.S.A.and Canada: ELSEVIER NORTH-HOLLAND, INC. 5 2 VANDERBILT AVENUE, NEW YORK, N.Y. 10017

PRINTED IN THE NETHERLANDS

INT RO DU CT ION

The object of this monograph is to examine several topics in the

theory of arithmetical functions. The principal topic to be studied is that of asymptotic formulae for the sum

where f ( n ) is a non-negative arithmetical function, and the sununation is over those n not exceeding

P

for which f ( n ) * O

.

These sums possess

an intrinsic interest apart from their number-theoretic applications. Two of the most interesting classes of arithmetical functions, namely multiplicative and additive functions, require different techniques for the estimation of (1). Since the reciprocal of a multiplicative function is itself multiplicative, we see that for the case of multiplicative functions f , the estimation of (1) may be regarded as a special case of the estimation of

where g ( n ) is multiplicative. Since an extensive literature already exists for this problem, we will deal with reciprocals of multiplicative functions in Chapter 1 only. There, we use important analytic tools such

J.M. DE KONINCK AND A. IVIC

vi

as the convolution method and the method of complex integration in order to prove the basic Lemma 1.1, which is repeatedly used in later chapters. When f is additive, the sum (1) is much more difficult to estimate, and no significant results were hown for a long time. The first h o r n result is due to R.L. Duncan C11, who proved the inequality

(3)

A general method for estimating reciprocals of additive functions

was introduced by the first named author in his 1972 doctoral dissertation at Temple University, Pa., parts of which appeared in De Koninck C11. This method, which rests on the fact that zf(n)

is multiplicative whenever f

is additive, is explained in Chapter 2 and yields improvements of (3) to

(4)

Z I

n a

l/w(n) = A1x/loglog3c t.. .tA~/(loglogz)'t

O(x/(loglogz)'tl)

,

where IV is arbitrary but fixed, A 1 = l , and the remaining Ails are computable. Subsequent papers, a number of which are due to one or both of the authors, contained further results which give more than enough material for a systematic account of sums of reciprocals of arithmetical functions. This topic seems particularly well suited for a monograph such as this, since it allows treatment in considerable depth without being too wide in scope. Furthermore, this branch of analytic number theory is quite new, and no general self-contained publication has yet appeared on this subject. We have included a number of our hitherto unpublished results, as well as some sharpened asymptotic formulae. In particular, we improve (4)

vii

INTRODUCTION

in Chapter 5 to

where M

is arbitrary but fixed, L .(z) ( j = 1,.. .,M) is a slowly oscil3

lating function admitting an asymptotic expansion of the form

( 6 ) L j (z) a

.(log log z)- '+. . .t aN,3.(log log x)4

1J3

t O(

(log logxrN-l)

,

where all the constants are computable, and N is an arbitrary but fixed integer. Although no claim is made that (5) is the best possible asymptotic formula, it certainly will be difficult to improve. A large number of additive functions arise from the logarithms of

positive multiplicative functions. In Chapter 3 we give asymptotic formulae for reciprocals of logarithms of some of the most important multiplicative functions. In addition we establish asymptotic formulae for reciprocals of many interesting additive functions. The methods we develop are also used t o estimate

(7)

where both g and f are additive; this is the subject considered in Chapter 4. We have devoted Chapter 7 to the study of reciprocals in "short" intervals. These are sums of the type

J . M . DE KONINCK

viii

AND

A. IVIC

where the interval is "short" in the sense that h

o(z)

as x+-

.

The

method used allows us to obtain estimates for other sums as well, such as

(9)

A (z,h) =

4

1 1 , zwath,f(n) =q

where q is a fixed integer, and f is a suitable arithmetical function. This estimate is established by using both general results and methods of analytic number theory. We hope that digressions of this sort will make the book more interesting for the general reader. Even in these digressions, "reciprocals of arithmetical functions" remains the thread which

holds the whole together. Analytic methods are also employed in Chapter 8 to estimate

where f belongs to a certain class of non-negative additive functions, and

9

is a suitable subset of the set of natural numbers. It is only

in Chapter 6 that the analytic approach is abandoned, and special elementary methods are used to deal with large additive functions such as

, B(n)

1

ap

.

Apart from the asymptotic formula (Theo-

Pal In

rem 6 . 2 )

the results are not as sharp as those obtained in other chapters, although we are certain that sharper results (such as those stated as open problems in Chapter 9) can be obtained.

INTRODUCTION

ix

This monograph does not resolve all the major problems connected with asymptotic formulae for reciprocals of arithmetical functions. In fact we give a list of open problems in the last chapter. It is our modest hope that this monograph will induce further research in this interesting field. Although this book is intended primarily for specialists, we think that it will be of interest to the more general reader as well. Apart from a general knowledge of analytic number theory and the theory of arithmetical functions, which may be found in a number of standard texts such as Hardy and Wright C11, Ayoub 111, Grosswald C11, etc., the reader needs only to possess a basic knowledge of calculus and of complex analysis. The text is, with some minor exceptions, completely self-contained; we have tried to make the exposition as clear as possible, without omitting important details, by deleting routine calculations and repetition of similar arguments. Each chapter is followed by a section of Notes, where all necessa-

ry clarifications and discussions are given, together with appropriate references. We have tried to include in these references all the papers which deal with asymptotic formulae for reciprocals of arithmetical functions and related topics. The notation is kept standard throughout and is fully explained in the next section. A number of mathematicians have kindly read the manuscript and have made many useful critical remarks and suggestions. We take this opportunity to thank Dr. E. Brinitzer-Scriba, Professor P. Erdos, Professor

J.M. DE KONINCK AND A. IVIC

X

E. Grosswald, Professor G. Lord, and Professor H. Lord.

We would like to express our appreciation to Facult6 des Sciences et de Ggnie, Universitd Laval, Quebec, and to Mathematical Institute of Belgrade and Repub. Zaj. of Serbia for their financial support of the technical preparation of the monograph. Finally we wish to thank Ms.Louise Papillon who typed the final camera-ready text with considerable care and speed.

Jean-Marie De Koninck

Aleksandar Ivi6

Mpartement de Mathhatiques

Rudarsko-geolozki fakultet

Universitd Laval

Universiteta u Beogradu

Qugbec, G1K 7P4

Djugina 7, 11000 Belgrade

Canada

Yugoslavia

NOTAT I ON

Owing to the nature of this monograph no attempt has been made to secure absolute consistency in the use of notation. The following summary explains some of the most common symbols and functions used; all other necessary notation will be specified within the body of the text.

natural numbers (positive integers).

k , Z,m,n:

p:

a prime number (without exception). the greatest integer not exceeding the real number

[z]:

1

nS?:

n

the empty sum is defined to be equal to zero. : a product taken over all primes p

not exceeding z ; the empty

product is defined to be equal to unity.

1' g ( n ) / f ( n ) :

na

z

n:

a sum taken over all natural numbers n not exceeding

for which f ( n ) z 0

.

a product taken over all primes.

P

(m,n) : the greatest common divisor of m dl n : d divides n

pal In : pa

:

and n

.

.

divides n

n:Z(modk) : k l ( n - Z )

f

.

: a sum taken over all natural numbers n not exceeding z ;

P a

d n

z

, but

pat'

does not.

.

a sum taken over all divisors of n

xi

(including 1 and n ) .

xii

J.M.

f

:

DE KONINCK AND A. I V I C

a sum taken over a l l primes that divide n

P n

d(n)

f 1 : the number of divisors of n

d n

.

.

dk(n) : the number of ways n can be written a s a product of

k

factors.

+(n) =

1 1 : Euler's t o t i e n t function, which represents the msn, (m, n ) =I number of positive integers coprime with n and less than n

.

a ( n ) : the number of f i n i t e non-isomorphic abelian groups with n elements. :

the sum of a l l divisors of

n

a(n)

i

n

: the number of d i f f e r e n t prime factors of

Pn G(n)

.

1

a :

I i:"" .

the number of t o t a l p r i m factors of n

Pal In

p ( n ) : the IGbius function defined by p(n) =

.

1

n=l

(-l)r

n=p,. . .p,,pi's different primes

otherwise

0

A(n) : the von Mangoldt function defined by

a

A(n) =

n=p otherwise

P(n) : the number of unrestricted p a r t i t i o n s of a positive integer n .

n

a(n)

p : the greatest square-free divisor of

n

P In 8(n)

T

: the sum of d i s t i n c t prime divisors of

P n

B(n)

1

up : the sum of a l l prime factors of

n

n

.

.

.

Pal In B1(n) =

1

p a : the sum of d i s t i n c t prime powers t h a t exactly d i -

Pal In

vide n

.

NOTATION

xiii

.

p ( n ) : the greatest prime divisor of n

e(x)

=

$(XI

=

~ ( x )

cm P c A(n) n*

PQ

11

PQ

:

*

the number of primes not exceeding x

.

x : a real variable. : complex variables (Res and

z,s

parts of

s

t = Ims)

.

~ ( s ):

Ims

denote the real and imaginary

respectively; common notation u

Res

and m

Riemann's zeta function defined for Re s > 1 by

~ ( s =)

1 n-' ,

n=l

otherwise by analytic continuation. x ( n ) : character modulo a fixed natural number k

x,(n) : principal character modk ; x,(n)

=

.

1 if (n,k)

1

otherwise.

zero

m

r ( z ) : the Gamma function defined for Rez > 0

by r ( z )

=

J

tz-' e-tat,

0

otherwise by analytic continuation. X

.

expx=e y

:

p

m

Euler's constant, defined as +

y

$(x,y)

c (log(1

P

- Up)

1

t

-

y

J

e logx.dx=O.5772157

-x

.

Up)

1 : the number of positive integers n not ex-

n5xc,p (n)59

ceeding x all of whose prime factors do not exceed y

f(x)

- g(x)

as x

f(x) = O(g(x)) constant

... .

+

xo means l i m f(x)lg(x)

=

.

1

33x

means D O

.

lf(x)

I

5

Cg(x)

Here f(x)

for x a o and some absolute

is a complex function of a real

xiv

J.M. DE KONINCK AND A. I V I e i s a positive function f o r z z o

variable, and g(z)

f(z) <<

g(z) : the same a s

f(x) = O(g(z))

(a,b) : the interval a < z < b

.

Ca,bl : the interval a

.

E

:

5

z

5

b

.

,

an a r b i t r a r i l y small positive number.

f(z) = o(g(z))

If(z) I

means t h a t for each E>O

< Eg(z)

for a l l z a 0

.

there e x i s t s zo such that

TABLE OF CONTENTS

......................................................

V

..........................................................

xi

Introduction Notation

................ .....................................................

Chapter 1: Reciprocals of multiplicative functions Notes

1

21

Chapter 2: Reciprocals of "small" additive functions

. Introduction .......................................... 12 . The method ............................................ 53 . Selberg's result and basic definitions ................ 54 . The main theorem ...................................... 15. Applications of t h e main theorem ...................... 86 . A generalization of t h e main theorem .................. 51

5 7 . Estimates f o r

k

31 32 35

42

46

f o r an a r b i t r a r y positive

............................................ .....................................................

integer Notes

1l/(f(n))k

nsx

29

48

63

Chapter 3: Reciprocals of logarithms of multiplicative functions

....... 1 2 . Functions with main term asymptotic t o Cx/log l o g x ... 53 . Functions with main term asymptotic t o Cx ............ Notes ..................................................... 51

. Functions

with main term asymptotic t o

Cx/logz

65 75

81 89

xvi

TABLE OF ONTENTS

Chapter 4: Suns of quotients of additive functions

. ........................................... 52 . Sums of quotients of "small11additive functions ........ 13. Sums of q w t i e n t s of additive functions which behave "like ~ 1 o g . l ~......................................... Notes ...................................................... 11 Introduction

95 96

100

107

Chapter 5: A sharpening of asymptotic formulae 81

. Introduction ...........................................

111

. ............................................. 53 . The theorems ........................................... 14 . Applications and remarks ............................... Notes ......................................................

113

5 2 The lemmas

133

141

144

Chapter 6: Reciprocals of "large" additive functions

........................................... for sums of reciprocals ......................... 53 . The functions B B and B1 .......................... Notes ...................................................... . 1 2 . Bounds

147

11 Introduction

.

151

156 167

Chapter 7: Reciprocals i n short intervals 51

. Introduction ...........................................

5 2 . An asymptotic formula f o r

.

i n short i n t e r v a l s

........................ .....................................................

175

..

177

13 Reciprocals i n short i n t e r v a l s

189

Notes

191

Chapter 8: Reciprocals of additive functions r e s t r i c t e d t o p a r t i -

cular sequences of integers

. Introduction ..........................................

11

20 1

TABLE OF CONTENTS

xvii

5 2 . "Small" additive functions and quotients of additive

.

functions

.............................................

13 Reciprocals of logarithms of multiplicative functions

Notes

.....................................................

201

.

21 0 225

Chapter 9: Other estimates and some open problems

. Introduction .......................................... 1 2 . Miscellaneous estimates ............................... 13. Open problems ......................................... 51

References

........................................................

Subject index

.....................................................

229

231 240 251

259

This Page Intentionally Left Blank

CHAPTER 1 RECIPROCALS OF MULTIPLICATIVE FUNCTIONS

An arithmetical function is multiplicative if f(m) = f(rn) f(n)

whenever rn and n are two coprime integers. Hence g(n) multiplicative if f(n) ing

1'

n-

l/f(n)

is

is multiplicative. Thus the problem of estimat-

when f is multiplicative reduces to the general problem

1 g(n) , where

of estimating

l/f(n)

g is a multiplicative function. While

n%

this problem is beyond the scope of these notes, we shall however derive

an asymptotic formula for functions f

1'

nQc

for some well-known multiplicative

l/f(n)

, thereby exhibiting several of the large number of methods

employed in deriving asymptotic formulae for multiplicative functions. We begin by proving Theorem 1.1. (1.11 where

Proof. -

We observe that

1

J . M . DE KONINCK

2

which may be proved by s e t t i n g n t i p l i c a t i v e functions of = logs

t y

+ O(l/x)

Since

$(n)

n

.

pa

AND

A. IVIC

since both sides of (1.2) a r e mul-

Using the elementary estimate

1 l/n

nsx

we obtain

2

4f

we see t h a t

1

n=l

v2(n)

converges, and

the above equation yields t r i v i a l l y

Thus (1.4)

By p a r t i a l summation it follows t h a t (1.5)

and substituting (1.5) and (1.6) i n (1.4) we get (1.1) since

RECIPROCALS OF MULTIPLICATIVE FUNCTIONS

We now turn to the divisor function d k ( n ) natural number 22

, and

a,@")

which gives for Re s

atk-1)

( k-1

=

, where

3

is a fixed

k

form the Dirichlet series associated with

1 nP ( l t -kpS t

2

k(kt1) p 2 s

>1

-..I

t

-S

Ilk

(1 - P 1

(1

-

-S

-l/k

p 1

so that

m

where V ( s )

1

n=l

v(n)n

-S

n

P

(1

-

(k-1)2 p-2s t...) is absolutely 2k2(ktl)

and uniformly convergent for Res >1/2

.

We shall now use (1.7) to derive an asymptotic formula for

1

n a

l/dk(n)

but first we need to develop the concept of the convolution

of arithmetical functions. (More precisely Dirichlet convolution as opposed to other arithmetical convolutions). The convolution h ( n ) of two functions f ( n ) and g ( n ) is defined by

J.M. DE KONINCK AND A. IVIe

4

(1.9)

H(s) m

where F ( s ) =

1

n=l

F ( s ) G(s) m

m

1

, G(s)

f(n)n-s

n=l

g(n)n-S

and H(s) =

1

n=1

h(n)n-’

.

Conversely (1.9) implies (1.8) provided that all the series in question all have finite abscissae of convergence. Equation (1.7) leads us to the study of the arithmetical function generated by C ~ ( S ) I ” ,~ or more generally, of the arithmetical function dz(n)

defined for an arbitrary complex number

z

by the identity

m

(1.10) We shall now derive an asymptotic formula for

1

n-

dz(n)

, and

then make use of

since by (1.7) l/dk is the convolution of u and dllk . Lemma 1.1.

Let

Iz

I

5

1 and N be an arbitrary, fixed positive

integer. Then

+ cN(z)3clog~-~3ct o(210g uniformly in z B.(z) 3

, where

is analytic for Proof. Defining

formula

j - 1 (z)/r(z-j+l)

c.(z) = B

3

IzI s

1

DZ(3c)

.

1

na

dZ(n)

, we

Re Z-N-1

1 ,

(j=1,2,...,N) and each

start from the classical

RECIPROCALS OF MULTIPLICATIVE FUNCTIONS

where c

>

1 is fixed.

=-

-&and s+l and tl = ' Cs(s)l stl n

Res = u

a log tl

2

1-

It

z

2

2

s

, then

0 <

r)

< 1/2

.

I

to

2

1

, then

,

.

Let

~ ( s , z )= Cs(s) (s-1)Iz

t o a fixed number such that i f

Suppose moreover t h a t r

two a r b i t r a r y real numbers s a t i s f y i n g r < T > t

0 be a constant f o r which

>

O(1og Itl)

s(s) =

F(S,Z,X)

q

c-iT

Furthermore, l e t a

(1.14) when

C5(s)lZds , J b

2ai

T-

s+l

c+iT (

1

D z ( t ) d t = lim

(1.13)

5

and

r)

0

<

<

E

and

E

are

a r c t g ?r)

.

If

by Cauchy's theorem the i n t e g r a l on t h e right-hand s i d e of

(1.13) may be replaced by integrals

I1,...,Ig over paths

rl, ..., rg

which a r e defined as follows: a ,1- log T -

rl

is the segment Cc-iT

r2

is the curve described by 1 - + (-T

t

q

-

it, , 1 -

r4 is the segment C1 -

r)

-

i n tg

is the a r c of the c i r c l e -ntE

to

n-E

E

r)

increases from

i n t g ~ l,

re

i E

1 ,

described as

e

increases from

, re-iE

r7 is the segment C1 -

r)

t

,1-

r)

ir)t g €1 ,

t

i r ) t g ~, 1 - n

a r8 i s the curve described by 1 - logt

,

-

,1-

1 t reie

r6 is the segment ~1-

to T

i t as t

-tl ,

to

r3 is the segment C1 -

r5

,

;TI

t

t

it,] ,

it as t

increases from tl

,

J.M. DE KONINCK AND A. IVIE

6

r9 is the segment C1

rl

We note that

on r or

.

E

a -log T t

, r2 ,

,c

iT

r8

iT1

t

rg

and

depend only on T

For T and r fixed, and f o r

the segments C 1-

-

rl

itl,l

-

. E

+

n) and (1 - n , 1 -

, and the I4 , we have

segments w i l l be denoted by c3 and c7 by J 3 and J7 respectively.

0

For

, r3 rl t

, and not r7 become

and

it,]

.

corresponding integrals

l i m l , = lim €+O €4

r6

and since on

If

yr

the argument of

i s the c i r c l e

(1.15)

t

Is)

E+O

T

+

-,

1-r

, then

F(s,z,z)ds

r

.

then both Il and I9 tend t o zero, so that

(1.16) where, for

J

,

YP

which does not depend on the choice of If

2n

excluding the point

Is-11 = P

lim (I4t l5

increases by

s

D, (t)at = 0 < r < rl

,

(PZ

(x) t u(x,2)

These

y

7

RECIPROCALS OF MLJLTIPLICATIVE FUNCTIONS

(1.18)

w(x,z) = J

2

+

J

+

3

J

7

+

J

8

’

m

J

F(1

2ni

8

a - logt t it,z,x) ( i

t

tl

J

3

=

-L 2nz

J c

F(s,z,x)ds.

2si l

3

Using (1.14) we have on

and by fixing 0

J~ =

r2

and

J c7

a )dt

t log2 t

F(s,z,z)

ds

J

.

r8

1 the following estimate holds:

< E <

m

J

2

J

t J 8 <<

x2-a/10gt

t-2 l o g t d t

m

<< x2

J

l o x log t d t << x2 exp(-2A-) exp (-E log t - a L) log t

t-2+E

where A = &

, since

I

from

(m - E )2 0 we have log t

- ~ l o g t- a - + 2 log t

m

and the integral

2

t-2+E log t

.dt

tl

On c3 and

c7

we have

is bounded.

,

,

8

J.M.

DE KONINCK AND A. IVIC

F(s,z,x) << x2-q

and recalling t h a t

is fixed, we obtain

tl

J~ t J

<<

since -n l o g x

tlx2-'

<<

7

<<

szexp(-q logx)

eXp(A2/q) $2 exp(-zA-)

2 A2/q

-

2 A e x

(1.19)

.

x2 e x p ( - Z A G x )

,

Thus we have proven that

w ( x , z ) <<

I t can be shown t h a t f o r

<<

Iz

22

I

5

exp(-ZA=x)

,

~ ~ ( is x )i n f i n i -

1 the function

t e l y differentiable with

(1.20)

QA(X)

-

1

r

(1.21)

1-24

rl

sin T z Qp) = -

r

In the region bounded, so t h a t f o r

where u = v/logx

.

5

n , the function H(s,z)

I z I s 1 and

Is-11 = r

x

0 < r

<

we have

1/2

Cs(s)(s-1)Iz

.

ds

H(s,z)

is

we have

( s - l ) - ' ~ ~<< - ~xr r -1 , so

that

J

-z s-1

Yr

Is-11

For

H(s,z) (9-1)

2ai

(s-1) -2 x s-1 ds

<<

xr

.

RECIPROCALS OF MULTIPLICATIVE FUNCTIONS

We now choose

P

= l/logx

.

9

Then for x sufficiently large, we

have (1.22)

@;(x)

=

o(1)

.

To obtain an asymptotic formula for DZ(x) we refer to (1.16). From the definition of d,(n) , it is seen that we have Id, ( p a ) I

I (-l)a (-: ) I 5 1 for Iz I 5 1 . Therefore by Id,(n) I 5 1 for all n , which gives, for 1 <

=

(1.23)

mltiplicativity 5

<

x/2 ,

I$

'i' 'r X

xt€

1

X

From (1.16),

@A (x)

appears to be a good approximation of Dz (x)

This suggests the following development:

Using (1.19) and (1.22) it follows from (1.16) that X

xrDz(t) dt X

jcrD2(t) dt 0

-

DZ(t) dt 0

.

10

AND

J . M . DE KONINCK

t;@A(x)t 0(52)

A. IVIC

.

a(x2exp(-Um)

t

This implies t h a t \ n Z ( x ) - o;(cc)

(1.24) Letting

t; = x e x p ( - A m x )

(1.25)

1

= @;(x)

rza

t

- .

O(xexp(-Ahogx)

To f i n i s h the proof it remains t o evaluate r

l/logz

QL(x)

.

Recalling

and

(1.26)

j =O

f o r every fixed no..-negative integer functions.

, where

q

the B.(z)ls 3

a r e analytic

Cauchy's c l a s s i c a l inequality f o r coefficients of a power

series implies t h a t R (s,z) = O ( 1 ) 4

(1.27)

.

x2t;-'exp(-ZA=)

t

we obtain

1 a,(.)

D,(x)

5

<<

@i(x)= j 1=O zBj(z)[

.

Substituting (1.26) i n t o (1.20)

m

s i n a (z-j)

J

'TI

W e s h a l l now show that for

uj-z x-udu

r

0

5

j

5

q

RECIPROC4LS OF MULTIPLICATIVE FUNCTIONS

To do t h i s l e t s

1 t reie

1t e

ie

i n the f i r s t i n t e g r a l and

vllogx

u

i n the second i n t e g r a l t o obtain

/logx

-n

I

I

m

(logx)

z- j- 1 s i n

(

'TI (z-

n

j)

vj-z -v e d v t -2 n i

c

1

where C is the c i r c l e If t i o n of

A

l/r(S)

IvI

11

v

1 minus the point

(-mY-l) u C u C-l,-m)

~j-~e~ddv)

-1

.

then by the c l a s s i c a l representa-

by a contour i n t e g r a l we have

1 r 1s )= nz T F j u - s eu du

.

A

On the other hand

\ A

1

u-'

e' du

-

exp(-log (x e - i n ) s) e-x dcc

)

u

e du

C

m

m

-

-s u

t

j

m

ins exp(-log(xein) s) e-x& = ( e

- e

- ~ ' T I s ) e - x x - ~dcc

1

C

1

1

m

= 2isinns

e-xx-Sd;c

t

1

and t h i s proves (1.29) f o r s = z - j

1

u-'eUdu

c

.

u-s e u du

,

12

J.M.

DE KONINCK

Finally, by l e t t i n g u = o / l o g x uniformly f o r

121

2

we obtain f o r 0

5

j 5 q

and

1

since the function uqtl e-' which i s less than -1/2 which means t h a t i f

AND A. IVIe

has t h e logarithmic derivative (qt1) v-' -1 for v

u 2 rl log x

t

n l o g x whenever l o g x 2 2(q+l)/n ,

, where

which implies (1.30). Similarly, n q t l - 2 -u Rq(l-u,z)u x du r

Since B o ( z ) = 1 B

x

t

<<

exp(Z(q+l)/n)

(logs)

, then

Re 2-q-2

, we obtain now (1.12) from (1.25), (1.27), (1.28), (2)

and N = q + 1

.

Now we a r e ready t o prove Theorem 1 . 2 . e x i s t constants bk, l,.

(1.31)

1 l/dk(n) = bk,

n a

Proof.

For every k z 2 and every fixed natural

. ..b k, fl x logl'k-l

depending only on k

xt

. . .t b k, N x logl'k-N

I t follows from (1.11) t h a t

N

there

such t h a t

3:

+ O(x log1/ k-N-1 x)

RECIPROCALS OF MULTIPLICATIVE FUNflIONS For DlIk ( x / n )

.

of x

we use (1.12) with z = l / k

13

and x / n

instead

This yields sums of the type

In the last sum we write

and since v ( s ) converges absolutely for Re s

>

1/2 , we obtain by partial

summation

for every A r O and every

E >

0 , which gives (1.31) after collecting terms

. .

x logl/k-i2 for i = 1 ,2,.. ,N

For another application of the convolution method we now consider a(n)

, the number of non-isomorphic abelian groups with

n elements. This

function is multiplicative and prime-independent, which means that a (pa)

, and not on p tricted) partitions of n , then

depends only on a

a@")

(1.32)

and in particular a ( p ) = 1 Dirichlet series of l/a(n)

t

p-s

P

-2s + ++ t

l

t ____

t

t

a (PIP"

-4s

-3s

t

,

= 2 for every prime p

n (1

n=l n(1 P

is the number of (unres-

.

Thus the

is

1 l/a(n) . n-S

A(s) =

If P ( n )

P(a)

, a(p2)

m

(1.33)

.

t...)

<(s)

1 a (P2)P2s

t.. .)

-2s n (1 - %-+-...) -3s

P

14

J.M. DE KONINCK AND A. IVI6

where u ( s )

is absolutely convergent for Re s

1/3

.

On the other hand,

where v ( s )

is absolutely convergent for Res > 1 / 2

.

Changing

>

we see that (1.34)

A ( s ) = s(s)cs(zs)l-1~2w(s) ,

where

is absolutely convergent for Re s

>

1/3

.

From

we have by the uniqueness theorem for Dirichlet series

whence

s

to 2s

RECIPFWALS OF MULTIPLICATIVE FWCI'IONS

15

so that m

where

1 h ( n ) n-'

n=l

.

= [<(s)]~/~

1 h(n)

ncc

From Lemma 1.1 with <<

x

x

Partial summation gives

2

h(n) <<

n>x n2

I

.

LFz(n) t - 3 d t

<<

log

x

X

I

z =

1/2 we obtain

.

t-2d t

<< x-'

LI:

.

X m

This shows that with

C

1

n=l

n2

we have

Now we obtain from (1.34)

We can write

since w ( s ) converges absolutely for Re s > 113,and partial summation for every

E >

0

gives

J . M . DE KONINCK

16

AND

A. IVId

Since

this means that we have proven Theorem 1.3. There exists a constant D > O such that (1.35)

It may be remarked that from (1.33) D may be evaluated as D = lim A ( s ) / r ; ( s ) s+l t

We derive next an asymptotic formula for log 1 l / a ( n ) na

a(n) =

n

p

is the largest squarefree divisor of n

P In

.

where

The powerful me-

thods of convolution and complex integration that were used in the proofs of the preceding results do not seem well-suited for this particular problem. We shall use instead a classical Tauberian theorem of Hardy and Ramanujan to derive an asymptotic formula for log 1 l / a ( n ) from the n a m

behaviour of

I

(a(n))-ln-s n=l

as

s+O

state the Hardy-Ramanujan theorem as

along positive real values. We

RECIP~CALSOF MULTIPLICATIVE FUNCTIONS

17

m

L e m 1.2. Res>O

Let ~ ( s =)

1 a n n- S

with an

n=l

2

0 be convergent f o r

and l e t f o r some A,a > 0 as o + O t

(1.36)

.

logF(a) - A a-a(logl/o)-B

Then as x + (1.37)

log

1 an

- B(logz)a/(lta)

( l o g l o g z ) -B/ ( l t a )

,

n-

where B = A

I/ (lta) a-a/ ( l t a )

1t- B l+a

(l+a)

Our result is Theorem 1 . 4 .

As

z + m

m

Proof.

Let ~ ( s =)

cative we have f o r

Writing

1

n=l

-1 -s

(a(n)> n

.

Since a ( n )

i s multipli-

Re s > 0

G ( s ) = logF(s)

(1.40)

G(o>

,

we s h a l l show t h a t , as

-u

-1

(logo

>

-1 -1

u+Ot

,

,

and (1.38) w i l l follow d i r e c t l y from (1.37) with A

a = 6= 1

.

Using t h e prime number theorem in t h e form (1.41)

a)(.

=

Jl,","t

+

,

2

where R ( z ) << z l o g

-A

z

f o r every A > 1 , and the S t i e l t j e s i n t e g r a l repre-

18

AND A. IVId

J . M . DE KONINCK

sentation, we obtain G(u)

=

=

1 log(1

P

j

t

p

u (p

-1

m

- 1)-l)

J log(1 + x-l(Xu

=

3/2

log0

t

x -1(xu - 1)-1)- ax log x

t

3/2

i

log(1

t

- 1 ) - l ) d n (5)

x-l(xU- 1 ) - l ) dR (x)

.

3/2

Using integration by p a r t s and (1.41) with A = 2 we have m

m

lOg(1

t

x

-1 u

-1

(X - 1 )

)-

&

log x

ax

lOg(1 t x -1 (Xu -1) -1)1 log2x

t O(

3/2

3/2

and (1.40) (and thus (1.38)) follows from Lenuna 1.3. (1.43)

Let

Ah(U) =

h , u > O and

1

log(1

t

x-l(xU

-

log x

3/2

If (1.44)

h

u

O(u-'(logu -1) -h-1 l o g l o g u - l )

In (1.43) first integrate from

3/2

is s u f f i c i e n t l y small, where we have set

we apply 1 t l / w x 2 3/2

, then

Ah(u) = h -1 u -1

Proof. if

is fixed and u + O t

<

1 / w 2 ( O < w < 1/2)

we have x l o g x > 1 / 2 X(XU

.

& h

1)-l)

with w

and xu

-

-

1) > 0 / 2

z(xU - 1)

1 > u logx > u2

t o x1 = u - l q 2 > 3/2 = (logu

rl

.

,

.

-1 -1

)

, and

Then

since f o r

we obtain

,

which gives log(1 3/2

t

x-l(xU - l ) - l ) log-hxax

l o g ( l / u ) - ~ l o g - ~ z <&< 0 -1 0 h t l

5

.

3/2

For the remaining integral we use log(1

t

y)

5

y (y

2

0)

.

Thus,

19

RECIPROCALS OF MULTIPLICATIVE F W T I O N S for z >1

, log(1

t

u

-1

z (z

-

-1 u 1)-l) < z (z

-

1)-l

<

(zulogz)-l

and eo

m

XdX X

3

1

h-’ u - l vk

k -1 u -1 (logzl)-’

t

1

,

O(0-l nhtl log log u-’)

so t h a t (1.45)

Ah(u)

k -l u - l qh

5

U(0-l vhtl log log 2

t

.

)

For t h e lower-bound estimate we s h a l l use X

’

‘qu)

j

X

where (for

small enough)

u

€or x 2 s x s x

z2

3 9

2

u -1 , z3 = exp((1ogu -1 ) ( h + W h )

Then

3 ’

- 1)

.(XU

Applying

0-l

log(1

t

>

zulogz

t z

2

ulogz2 =

v ) 2 1 - v / 2 (0 < v < 1)

-1

.

with v

we deduce t h a t log(1

f o r z2

.

5 z 5 z

3

+

( . ( X U

and u

- l))-l)

- 1 ) ) - l (1

t ( . ( X U

small enough, and t h a t

u l o g r s ulogz3 =

U(0q

-2

)

o(n)

so that xu

-

- 1 < (1 t

rl)

u logz

.

,

n/2)

(xutl - 2 )

-1

,

J . M . DE KONINCK AND A. IVI6

20

X

3

x-l(logx)-h-lh

X

h-1(log-hx2-log-hx3)

= k - l u h - h -1 IT k t l

2

This gives

which combined with (1.45) proves (1.44). From (1.44) and (1.42) we obtain, as u + O t G(u)

,

u-l(log u -1) -1 t O(o-'(logu -1) - 2 log logo-')

which implies (1.40)

.

As already remarked, Theorem 1.4 follows then direc-

t l y from Lema 1 . 2 and (1.40).

RECIPROCALS OF MULTIPLICATIVE FUNCTIONS

21

NOTES

As remarked in the Introduction, the aim of our monograph is to study asymptotic formulae for

1'

(1.47)

n-

llf(n)

,

where f ( n ) is an arithmetical function of a certain interest. Therefore we restrain from studying

1

n*

f(n)

for multiplicative functions f

,

though this problem is one of the most interesting and important topics in analytic number theory, and we give estimates for (1.47) in the case of several interesting multiplicative functions f only. For the sake of completeness, however, we mention here the following sharpest known asymptotic formulae for some familiar multiplicative functions: (1.48)

(1.50)

The first two of the above formulae are due to A. Walfisz C11, while (1.50) is proven in the forthcoming paper of G. Kolesnik C11 (see Chapter 7 for a more extensive discussion concerning (1.50)).

As a general result concerning summatory functions of multiplicative functions we mention the following result of E. Wirsing C11, which

has wide applications and has initiated subsequent research: Let f ( n ) be a non-negative multiplicative function for which

22

J.M. DE KONINCK AND A. IVI6

for some constants c1

>

0

as x+m for some T T O

.

a 2

2

,0

5

c2 < 2

and all primes p

and integers

, and let

Then, as x+-

,

Formula (1.1) may be found in Montgomery C11. The more general sum

2'

nQ, (n,I ) =1

1/$(n)

is investigated in Halberstam-Richert C 1 1 , Chapter 3 ,

95, and has an application to the so-called Titchmarsh divisor problem of

estimating

I:

l
d(p-I)

.

The right-hand side of (1.2) is a multiplicative function since f ( n ) = P 2 In)

is certainly multiplicative and g(n)=

is multi-

f(d)

d n

plicative if f ( n ) is; a proof may be found in Niven-Zuckerman C11, Chapter 4. In proving (1.5) we have used a partial summation technique. A number of partial summation formulae may be found in Prachar C11; the most comnonly used is the following: Let X1

5

X2 5

...

be a positive sequence tending to infinity,

and let g ( t ) be continuously differentiable for X1 A(t)

1

X15Xn
an

, where

5

t <x

.

Then for

an are arbitrary complex numbers, we have

RECIPROCALS OF MULTIPLICATIVE FUNCl7ONS

23

4

If

and i n (1.51) e i t h e r the series o r the in-

l i m A ( x ) g(x) = 0

x+-

t e g r a l converges, then m

(1.52)

Relation (1.51) i s obvious i f we write the left-hand s i d e a s the Stieltjes integral

r:

A1SAn2T

1

angOn)

s(t)dA(t)

3

1-0

since the right-hand side of (1.51) is then nothing more than the p a r t i a l integration formula f o r the S t i e l t j e s i n t e g r a l . from (1.52) with

hl = x

, An

= n

, an

from (1.2) we e a s i l y obtain A(*) << t

Estimate (1.5) follows

e~, g ( x ) =

= !J2(n)

l/xz

, since

.

A formula similar t o (1.1) could be derived f o r

no nice i d e n t i t y l i k e (1.2) can be e a s i l y found f o r a(n) that the constants appearing i n the formula f o r

l/u(n)

1

n2T

l/u(n)

, which means would be compli-

cated. In t h i s and later chapters we often use the following: i f m

~ ( s =)

converges absolutely f o r

u =

1

n=l

Res > a

ann - S

, then

, but

J . M . DE KONINCK AND A. IVIC

24 f o r any

E

>0

.

This follows from t h e p a r t i a l summation formula (1. S l ) ,

which with A ( t )

1 lanln-",

nst

t"

g(t)

, An

= n

,u

= u

t

since A ( t ) must be bounded i n view of the convergence of

E

F(s)

gives

.

In (1.8) we defined the (Dirichlet) convolution of arithmetical functions.

For other arithmetical convolutions of i n t e r e s t t h e reader i s

referred t o Subbarao C11. Much work has been done recently on unitary convolution (see Cohen C11 and C 2 l ) .

If

f ( n ) and g(n)

metical functions, then t h e i r unitary convolution

U(n)

a r e two a r i t h -

is t h e function

Lemma 1.1 is very deep, and we shall use it i n l a t e r chapters. could be e a s i l y established f o r positive number (instead of

I z I s 1)

the proof, and besides, the range i n l a t e r chapters.

IzI 5 A

A proof with

Iz I

, where

, but

It

is an a r b i t r a r y fixed

A

we d i d not want t o complicate

s 1 is precisely what we shall need

N = l

was given i n Selberg C11, and the

general formula with a r b i t r a r y N was proved by Dixon C11. use complex integration and moreover, i f

xry22

Both proofs

,

(1.53) The proof given here i s from Delange C31.

I t avoids (1.53) and

can be modified t o y i e l d other r e s u l t s which w i l l be needed l a t e r . The inversion formula (1.13) follows f o r example from Theorem B of Ingham C11, Chapter 2 , and (1.14) may be found i n Prachar Cil, Chapter 3.

RECIPROCALS OF MULTIPLICATIVE FUNCI'IONS

Formula (1.31) f o r by Wilson C11.

k

25

2 was conjectured by Ramanujan, and proven

A proof of (1.31) without Lemma 1.1 and complex integration

may be obtained by the Levin-Feinleib method of i n t e g r a l equations; f o r

this p o s s i b i l i t y see I v i i C11. To see that the formula 71 T = r J1 ur s rz

(1.54)

s eu du A

m

I

=

( m , ~ ]

point

u

t = E

a u

.

J

r(s)

holds, note t h a t

,

(E,m)

e

t dt -t -s

, and

consider the contour

a E i s t h e c i r c l e It1

where

minus the

E

Then the i n t e g r a l

J e-tts-ldt

=

I does not depend on

j

m

m

-t s-1 d t e t

E

, and

t

J

(s-1) 2 r i E

letting

e-tts-l dt

J

e-tts-l dt

E + O , we obtain, m

(1.55)

( e 2 n i s - 1) r(s)

( e2nis

-

1)

J e -t ts-1 d t J e-t t s - 1 d t . I

0

Replacing

t = -u

s by

1- s

, taking

E

= 1 and making the s u b s t i t u t i o n

, we obtain

Recalling that

r ( s ) r ( 1 - s)

=

&,

we have (1.54).

For a proof of (1.32) and f o r other properties of f i n i t e abelian groups, see Herstein C11.

Formula (1.35)does not seem t o have been s t a t e d

i n the l i t e r a t u r e before, whereas the problem of estimating

1

n-

a ( n ) has

J.M. DE KONINCK AND A. IVId

26

a long history.

The best known r e s u l t is

1

a(n)

nsx

c

3:

1

t

c x1'2 2

t

c x1'3 3

t

~ (105/40710g x

2 ,

(see Srinivasan C11). Broadly speaking, a Tauberian theorem f o r a Dirichlet s e r i e s m

~ ( s )=

1

n=O

ane

-An s

( t h i s includes ordinary power series f o r z

An = n )

is a r e s u l t about

,

s+Ot

~ ( s ) usually when

1

nsz

e- S

,

an which i s deduced from the behaviour of

.

The inversion formula (1.13) i s sometimes called Perron's formula, a proof of which may be found i n Hardy-Ramanujan [11 o r Knopfmacher [ I ] . Generally speaking, an inversion formula f o r Dirichlet s e r i e s is a r e s u l t about

1 an nsz

, which

is deduced from the properties of the Dirichlet s e r i e s

Perron's formula (1.13) is such an example (giving a r e s u l t about the integral of the sumnatory function of coefficients of Dirichlet s e r i e s ) . A useful variant of t h i s formula (see Prachar [11 f o r a proof) is the fol-

lowing: Let

converge absolutely f o r a XLX

Re s > 1

, where

an

<< A ( n )

, and

A (x) f o r

i s a positive, monotonically increasing function, and l e t , a s

U - t l t O ,

RECIPROCALS OF MULTIPLICATIVE FUNCTIONS f o r some a u

+ b

>

1

>

,7

0

.

If

> 0

real)

u t i v (u,u

w

, then f o r

is a r b i t r a r y ,

27 b

>

0

,

x > 1 not an integer we have

(1.56) t O(A(2x)x 1 - u

T -1 log2x) t O ( A ( Z X ) X ~ - ~ T~ -x~- N I - ~ )

is the integer nearest t o x

where N

.

If x

2

,

1 is an integer, then

btiT

(1.57)

)

b t

O( T(u

t

b

1-u -1 T log

O(A(2x);r

t

h)

.

In the proof of Theorem 1.3 we have used t h e uniqueness theorem f o r Dirichlet s e r i e s .

This elementary, but important r e s u l t (see Prachar C11)

may be s t a t e d as follows: Let f o r every

u = Res

satisfying

m

F(s) =

Then an

bn

1 ann -S , G ( s )

n=l

03

, then

m

m

=

1 b n n-S , F ( u )

n=l

m

< u <

< u

G(u)

.

1 , 2 , ...

for n

Furthermore if u

s

1

n=l

m

lanl n-'

f o r these values of

1

and

n=l

lbnl n

-0

u

where cn = d6=n 1 adb6

f

d n adbn/d

.

converge f o r

28

J.M. DE KONINCK AND A. IVId

Theorem 1.4 was proven by de Bruijn Cll, and we have followed his proof here. By using more powerful Tauberian theorems and subtle analysis,

W. Schwarz

Cll has proven an asymptotic formula for

just for the logarithm of that sum.

where -R(u) = G(H(-u)) + uH(-u) G’(s)

1 lla(n) , and not

nsx

His result is

, and

H(s) is the inverse function of

, where

and it can be shown that, as x+- , R(1og x)

-2J21ogxllog log x

.

Schwarz’s paper contains other interesting results connected with a(n)

, such as the asymptotic formula for

where k > 1 is a fixed real number.

CHAPTER 2 RECIPROCALS OF "SMALL" ADDITIVE FUNCTIONS

11. Introduction

An arithmetical function f is additive if ffmn) = ffm) + f f n J whenever m and n are coprime integers. The problem of finding an estimate for

1

n-

f(n)

, where f is additive, can be approached in the following manner:

29

J.M. DE KONINCK AND A. IVIC

30

For many well-known additive functions f the sum mated, the double sum order than x ( 1 ,

t

c;)

l2

.

This approach allows

1 f(n)

n<x

1, - x

converges, while

1, 1,

can be easily estit

1,

is of lower

to obtain fairly accurate estimates for

LIS

for a large class of additive functions f

, and

so, for example,

we obtain

1 w (n)

2

n 2

log log x

t p

x

0(x/log x)

t

,

(2.3)

where

1 (log(1

y t

P

P

- l/p)

-k

Up)

.

On the other hand, if f is additive, there is no obvious way t o obtain an estimate for sums of the type

1'

nSx

.

l/f(n)

The purpose of this

chapter is to show how one can obtain precise estimates for sums of reciprocals of "small" additive functions (among which we find the functions w

and Q ) . This class of "small" additive functions will be defined more

precisely in 53. Actually our approach will yield an asymptotic expansion of the form

1'

nSx

l/f(n)

x

a

1

Ai/(loglogr)i

t

i=1

O(x/(loglogx)atl)

,

where a is any preassigned positive integer and the Ails are computable constants depending only on f

,

We will also show that our method allows

LIS

to estimate certain

1' g(n) / f ( n ) , where g is multiplicative and f is nsc additive.. Finally we will extend our method to study asymptotic expansions

sums of the form

SMALL ADDITIVE FUNCTIONS for

52.

1'

nsx

l/(f(n))k

31

, for an arbitrary positive integer

k

.

The method.

If f is an arbitrary additive arithmetical function, then, for an arbitrary complex number t * 0 , t f ( n ) is a multiplicative arithmetical function.

If g ( n ) is a multiplicative function, it is, in many cases,

possible to estimate

1g ( n )

using complex integration.

1111:

The idea that we will use is the following: given an arbitrary arith-

Ift f ( n ) for real t~ C0,ll (where nsx the prime in the sum indicates that the sum is taken over all nlz: for

metical function f ( n )

which f ( n ) s 0)

.

, let

Suppose that

(2.4)

F(2,t)

with R ( x , t )

F(z,t) =

O(H(z,t))

G(s,t) t R(z,t)

holds uniformly for t e [0,11

and H(z,t) are integrable functions of t z+-

.

, and

, where

C(z,t)

~(z,t) ~(C(z,t))

as

Then, using ( 2 . 4 ) , we have

One would like to integrate the sum in ( 2 . 5 ) with respect t o t between 0 and 1 and get

J . M . DE KONINCK

32

AND

A. IVI6

Integrating i n a similar way the r i g h t side of ( 2 . 5 ) , one would hope t o get the desired estimate f o r

1'

l/f(n)

n a

.

In most cases, however, the r i g h t side of (2.5) w i l l be unbounded i n the neighborhood of between 0 and 1

t =0

, so

t h a t it i s impossible t o integrate it

. between

To overcome t h i s d i f f i c u l t y , we shall integrate ~ ( z ) and

1

, where

~(z) 0 -f

, as

z-fm

.

We then obtain

If we integrate i n the same manner the r i g h t side of ( 2 . 5 ) and show, by a proper choice of

~ ( z ), that f o r

z

s u f f i c i e n t l y large the second sum i n

( 2 . 6 ) is small compared t o the integrated r i g h t side of (2.5),

obtain the desired estimate of

1'

n a

l/f(n)

we then

.

An estimate l i k e ( 2 . 4 ) is not always possible f o r

F(z,t)

.

But

using a r e s u l t of A. Selberg (Lemma 2.1), we w i l l find a large c l a s s of functions f o r which G ( z , t ) on C0,ll

D ( t ) z logt-'z

, where

D(t)

is continuous

and H(z,t) = ~ l o g ' - ~ z, and t o t h i s class of functions we w i l l

be able t o apply the above method.

13. Selberg's r e s u l t and basic definitions.

Before we define the class o f functions t o which we w i l l apply the method outlined i n Section 2 , we w i l l prove Selberg's r e s u l t .

With our

SMALL ADDITIVE FUNCl'IONS

33

notation, and the r e s t r i c t i o n t o the p a r t i c u l a r case needed here, it may be stated a s follows: m

Lemna 2.1.

Let g ( s , t ) =

1bt(n)/ns

n=l

f o r Res

m

1 Ibt(n) 1n-l log3 2n

be uniformly bounded f o r

n=l

m

(s(s))tg(s,t)

=

uniformly f o r

It

Proof.

1

n=l

I

for

a,(n)/n"

s 1

u>1

.

It

I

5

1

.

u>1

, and

let

Next, set

Then

.

,x2 2

Using Lemma 1.1 with N = 1

, and defining

D,(x)

1dt(n) n a

we obtain

W e have, since

Ibt(n) I 17 (log 2n) 3

n=l

<<

log

t-2

x

.

i s uniformly bounded and

t 2 -1 ,

,

34

J.M.

RE KONINCK AND A. IVI6

On the other hand,

=logt-10

c

.

nsx

bt(n)

(1 t o[*]]

bt ( n )

17 t o(logt-2x) .

=logt-1 x .

nsx

But we have

and this ends the proof of Lemma 2.1. Definition 2.1.

Let

S

denote the set of all arithmetical func-

tions satisfying the following two conditions:

*

1) f ( n ) 2)

nsx

0

=>

1=0(---).

f(n) L 1

, for each positive integer

n

.

X

log x

Definition 2.2. Given

~1

, (which in the remainder of the text

will stand for an arbitrary positive integer unless otherwise stated), we define

Sa

to be the set of those functions in

for which

S

satisfies the conditions of Lemma 2.1 with D(t)

E

catl C0,ll

.

g(n)at(n)

SMALL ADDITIVE FUNCTIONS

Definition 2.3. Given a function f

, let

Sa

E

35

responding function of Definition 2.2; then for t

E

D(t)

(0,ll

be the cor-

, let

D ( t ) (i-1) B . ( t ) = (-)

t

Z

and A i ( t )

=

(-l)i-l B i ( t )

write Ai

for A i ( l )

, for

1,2,.. . ,a+2

i

.

.

We shall occasionally

84. The main theorem.

We first prove two lemmas which we will need in the proof of our main theorem (Theorem 2.4). Lemma 2.2. Given f

E

Sa

, with corresponding functions

holds uniformly for t with some constant M depending only on f

E

,

Ai(t)

(0,ll and lsisat2

,

.

Proof. From Definition 2.3, we have

is an expression containing ~ ( t, D)' ( t ) ,

(t) where Q. 2

distinct positive powers of t ID(j)(t)l s N

constant N

.

.

But D ( t )

holds uniformly for t

E

E

on the function D ( t )

E

[O,ll

and

CatlCO,ll ; hence

C0,ll and 0 s j s a t 1 , with some

Therefore there exists a constant M

holds uniformly for t

... ,D (i-1)(t)

such that

lQi(t)I < M

and l s i s a t 2 , with M depending only

, that is, only on f .

J.M. DE KONINCK AND A. I V I 6

36

Let

Lemma 2 . 3 . xr3

.

1/2

< q s

.

1

Let

E(X)

= (logx)-

2

0

for

Then, f o r x suffic ie ntl y large,

E(X)

Proof.

so that 1 t

Max log t x 1ogqx s t s q tat2 - ,,at2

E(X)

s

Setting h ' ( t ) = 0

t x and suppose t h a t --$-

q

s 1

, we

.

*

lo t x (tloglogx-a-2)

obtain t =

-

minimum a t t = logat2ogx

.

x is large enough

Then

which is s t r i c t l y positive f o r t

, nl

'

h(t) = l o

Let

h'(t)

CE(X)

1

-.

.

at2

-

Moreover

at2

.

Therefore h(t)

has a

and there are no other local maxima or m i n i m a on

Therefore, f o r x suffic ie ntly large, h( t) is decreasing at2 at2 between E ( X ) and loglogx , and increasing between loglogx and n

.

Hence

But log' (E

x

(log"(")x) (log1'2x)

(XI

*

f o r x sufficiently large, since 1 / 2 + ~ ( x 5) n l ogQx s

n

,

<

.

10gqx

J

It i s clear t h a t , since

rl I1

,

SMALL ADDITIVE FUNCTIONS

37

whence f o r x s u f f i c i e n t l y large, t log x - log% ta+2

E.(x)
n

a+2

'

W e a r e now ready t o prove our main r e s u l t . Let f

Theorem 2 . 4 .

Proof.

Since f

=

E

with R ( x , t ) = O(xlogt-2x) Since f E S

Sa

.

Then

a

1

I' nsz

E

Sa

2' i=1

(loglogxf

, we

have

X

uniformly f o r t

O(

E

(log logx)a+l

C0,ll

and D ( t )

E

Ca+'C0,11.

, we have

Hence (2.9)

D(t)xlogt-lxtR(x,t) with R1(x)

= O)-(

(2.9) becomes

3:

log x

.

Dividing by

t

+

R1(x)

, and recalling Definition

2.3,

J.M. DE KONINCK AND A. IVIC

38

(2.10)

Let that

1' tf(n)-'=

n a

~ ( x ) (logx) ~ ( x )< 1)

Since 0 < E (x)

<

.

Bl(t) x 1 o g t - l x

t

1 t R(x,t)

t

1

R1(x)

-mas i n Lennna 2.3 and suppose that Then

1

by the definition of

E

(x)

, this

Therefore (2.11) becomes

Using (2.6) and observing that

i s *(log log x)

.

Hence

. x23

(so

39

SMALL ADDITIVE FUNCTIONS

X

-

(log X I because

=

1

o[

(log log x)a+l

2(a+2)

~ ( x )< 1 and f ( n )

X

2

I

1 i n our r e s t r i c t e d sum, we obtain

Recalling (2.12), it follows t h a t

+

X

O I (log log x)a+l

1.

W e now proceed t o integrate B1

( t )x

log t - l x

Successive integration by p a r t s leads t o

B l ( t ) 1%

-

... +

(-1)

+

(log logx) a

+ (-l)a E (XI

between

~ ( x )and

1.

40

J.M. DE KONINCK AND A. IVIE

W e recall that

Bl(i) (t) = Bitl

(t) and A i ( t )

; therefore

(-l)'-'Bi(t)

(2.14) becomes

(2.15)

jI.(

B1(t)zlogt-lxdt = x

E

{

1

i = 1 (log

M logE

(log log z)

A,+,@) logt%

E (2)

(log log.)at1

11 E (XI

lsisatl ,

Using Lemma 2.2, we can see t h a t f o r

-

*I

a A i ( t ) log*-lx

x

=

- l0g1'2 x (log log z)

o(

(log

From Lemma 2.2 and Lemma 2.3, we have

-- M

log x

Max

E(Z)St
ip !tt 2x 2logx =L logx

Using (2.16) and (2.17) and observing that ~

~

= M

.

~ = o(1) ~ (

,1we )have,

from

SMALL ADDITIVE FUNCTIONS

=x{;

Ai

1

i=1 (log l 0 g x ) i

O I (log

logx)"+l

41

0

'

by the convention made i n Definition 2.3. I t remains t o estimate E

= O(x 1 0 g * - ~ x ) uniformly f o r

X > 0

and E

>

0

, both

f o r a l l x > X and a l l

i t x)

t E C0,ll

independent of t e

=

C0,ll

o(

.

dt

.

.

Recall t h a t

R(r,t)

This means t h a t there e x i s t

t

, such

t h a t IR(x, t )I < B x logt-2x,

Hence, f o r x s u f f i c i e n t l y large,

X

(log log x)

1'

Combining (2.13), (2.18) and (2.19), we obtain

1' ?la

1

fo

which is the desired result.

Q

1

Ai

i=1 ( l o g l o g x )

+.

X

1,

ol(loglogz)Q+~

42

J.M. DE KONINCK AND A. IVId

15. Applications'of the main theorem.

With the help of Lemma 2.1, we will now apply Theorem 2.4 to some well known functions. Theorem 2.5.

1'

n*

where al

1

a

a

i c iJxT i=l (log logz){

1

, a2

= 1- p

X

(log log x)atl),

, and the remaining

3

ails are computable cons-

tants. Proof. It is clear that, for u > 1 and

It I

5

1

,

Let (2.20) 1 seen to be a holomorphic function of s for a > 2 Indeed, if a > l , we have

then

g(s, t ) i s

(where the argument es (2.21).

logg(s,t)

of logg(s, t ) is chosen so that

= t c log

P

- TI

<

.

e s s T ) and

[ 1 -$)+p++&)

SMALL ADDITIVE FUNCTIONS

c

m

= - t C

y 1s + Cp w 1= l

+l

; %=-Iw+l-1)

tw

(_,)V+l

p w=l vp

43

w(ps-l)w

;t

w

p w=2 w ( p

p v=2 wp

ws

t + 1 ; (-l)v+l t v + pI pS(pS-l) p w=2 v(ps-l)w

which converges uniformly for u

2

1+ 2

E

, for any preassigned

E

> 0

Therefore g(s, t ) = elogg(s-'t) represents a holomorphic function of 1 for u > 7 (see Apostol C11 p. 394). As

s

satisfies the conditions of

mentioned by Selberg C11, g(s,t)

t w ( n )and we obtain

L e m 2.1; hence a t ( n )

(2.22)

.

nl?:

uniformly for It1

s

1

.

We can now make use of (2.22) in order to apply Theorem 2.4. We first have to verify that D ( t ) all a

.

It is easily seen that g(1,t)

tion (2.21). Finally, w(1)

Furthermore, l/r(t) 0

, and

w(n)

belongs to CatlCO,ll , f o r

g(l,t)/r(t)

t

Applying Theorem 2.4, we obtain

E

E

CmCO,ll

C"C0,lI

1 for n

f

, from the representa-

, and hence g(l,t)/r(t)~c~rO,lI. 1 . Thus w E Sa , for all a .

J . M . DE KONINCK AND A. IVId

44

a

a

c

i i=1 (log 1 o g x ) i +-

1 1' nsx Jx=

Ol

X

(log logx)atl

with

For al and a2

, we

(2.23)

1

al

have

D(1)

P

D(t)

D(t) m

-tD'(t) t*

t=l

From (2.21) and (2.22),

n

g w = g(1,l)

ew ( t )

m

t

p v=2 vp

t

I

1 1 (1 - -) (1 t -)

P

P-1

-

= D(1)

D'(1)

t=l

, with vtl v

(-1)

t

p

Hence

with

ri(l)

= -y

(for a proof of t h i s , see Landau Cll p . 138) and

= 1

1-D'(1)

.

SMALL ADDITIVE FUNCTIONS

w'(1) = -

1 1 (_,)V+l cp mc ,+cm+c c 7 up p p (p-1) v=2

v=2

1 1 P p(P-1)

1 Ilog(1 - -)p1

t

-11 p

1 { l o g ( l - -) P P

t

-11 P

D'(1) = y

{ l o g ( l - -) 1 P

P

=

45

t

~

-

;r n 1

.

Therefore t

P

t

-11 P

p

and (2.24)

a2

= 1 - p

.

W e w i l l now give a second application of Theorem 2.4.

denotes the t o t a l number of prime f a c t o r s of

n

, then,

If

n(n)

proceeding as i n

the proof of Theorem 2.5, we obtain Theorem 2.6. a

1

.(

X

(log log x ) a + l

where b, = 1

, b2

= 1 - 0-

1 e7 , and the remaining p(P-1

b i t s a r e com-

putable constants. From the preceding two theorems, we have the following corollary. Corollary 2.7. 1

n s iJ3i-J--

L

1

cx -@T= (loglogx)2

3

+ o~(loglogx)~

J.M. LE KONINCK AND A. IVIC

46

with

C =

1-P k1- 1 .

Proof. The proof is immediate from Theorem 2.5 and Theorem 2.6.

86.

A

generalization of the main theorem. Definition 2.4.

Let

g(n)

tf(n) = a,(n)

be the set of all ordered pairs of arith-

S;

which satisfy the following four conditions:

metical functions (g,f)

4)

,

satisfies the conditions of Lemma 2.1, with

~ ( t= )g(l,t)/r(t)

E

.

c~+~co,~I

From this definition, we observe that if (g,f)

uniformly for It I s 1

.

E

S;

, then

Therefore to each ordered pair (g,f)

E S;

we

can associate the function D ( t ) , and, using this function, we define the functions B i ( t )

and A i ( t ) , 1 s i s CL t 2 , as in Definition 2.3.

now state the following theorem. Theorem 2.8.

Let @,f)

E

S;

.

Then

We can

47

SMALL ADDITIVE FLINrnIONS

Proof: Since (g,f)

with R ( z , t )

E

Si

, we have

U(zlogt-2z) uniformly for t

E

C0,ll

, D(t)

E

Cat1CO,11

and

Now

We observe that

-.(

2

)

= o (

3:

(log log z)

(log-)2

lJ

and the proof of the rest of the theorem is entirely similar to the one of Theorem 2.4 and will therefore be omitted.

As an application of Theorem

2.8,

we now state Theorem 2.9, without

proof since its proof follows essentially the same lines as that of Theorem 2.5.

J.M. DE KONINCK AND A. IVId

48

Theorem 2.9.

where ~ ( n )is the Gbius function, d,

= 6/n2

6

, d,

= -

11 - p

n2

t

1

P

and the remaining d i t s are computable constants.

17.

Estimates for

1 l/(f(n))k

for an arbitrary positive integer k

nsx

In this section, we obtain estimates for

, and

f E S a

k

2' l/(f(n))k

n-

is a fixed positive integer, k s a

.

.

, where

We first make a defi-

nition and prove two lemmas that will be used in the next theorem. Definition 2.5. For t = Bi(t)

'Bi(t)

1,2,

for i

(0,11 , set 1A i ( t ) =

E

...,a t 2

; with A i ( t )

~ ~ (and t )

and B i ( t )

as in De-

finition 2 . 3 . Next, define 2

i-1

1

Bi(t)

(i-j-1)

j=1

and 2 s k

2

~ ~ =( (-l)i-2 t ) 2Bi(t) 5

a

and k A i ( t )

for i = 2 , 3 , ..., a t 2

.

More generally, for

, set

= (-l)i-k* B i ( t )

for i = k , k + l ,...,a t 2

sionally write k ~ i for k~i(l)

.

.

We shall occa-

49

SMALL ADDITIVE FUNCTIONS

Lemma 2.10. k s

CL

.

.

(0,ll

and

5

E

(OylI and Proof:

, depending

7

tz

be a positive integer, kAi(t)

5

p t 2

, such

only on f

that

uniformly f o r

k5iscit2 ;

2 ) there e x i s t s a constant N

t

k

Then

I kA i ( t ) I E

let

the corresponding functions

1) there e x i s t s a constant M

t

, and

f E S a

, associate

To f

k s i s a t 2

Let

ksisa

, depending

only on f , such t h a t

.

This lemma i s a generalization of Lemma 2 . 2 , and it follows

hnnediately from Definition 2.5. The following lemma is a generalization of Lemma 2.3.

Lemma 2.11. positive integer.

Proof:

Let

2 2

3

, and

let

~ ( 3 : )5 u

E(Z)

and

Let

u

1

.

Let

B

be a

Then

h(t) =

t t B

.

From the proof of Lemma 2 . 3 , it is

e a s i l y seen that the only two possible maxima of

at

5

, Therefore

h(t)

in

[ED)

ul

are

J.M. DE KONINCK AND A. IVIC

50

W e now e s t a b l i s h a general formula which w i l l help

Let f

Theorem 2.12. t r a r y positive integer,

I'

(2.25)

uniformly f o r u Proof:

E

f (n)

c

, 11

1

t

.

Let k be an arbi-

E

(x))

1

(logx)-20

The proof is by induction on k

and R1(x)

3

0(x(log log x)

, where ~ ( x =)

~ ( x 2) v s 1

(2.10) holds and we have, f o r

R(x,t)

x

P k + l(log logx)cL+l

CE(Z)

2

(1oglogx)i

i=k

3: logU-

let x

Then

=

(f(n))

n*

1

U

.

ksa

, and

Sa

E

us find an es-

.

Since f

E

Sa

, equation

,

a r e the functions defined i n the proof of Theorem 2.4.

R ( x , t ) = O(xlogt-2x)

uniformly f o r

Since 0 < ~ ( x 5) v

1

dt t

5

1

E

[0,11

, and

~ ( x ,)

R1(x)

,

logv - logE(x) = o(l0gE

E (XI

From the definition of

t

-1

(x))

.

0(-1g :2

*

SMALL ADDITIVE FUNCTIONS

51

Hence

v

I

(2.27)

O(x logt-2

T h a t R(x, t )

e x i s t a constant B IR(x,t)

I

<

>

0

Bx logt-2x

x) uniformly for t

and an X > 0

, for

, both

.

O(ZE(2))

E

C0,ll

means that there

independent of

a l l x > X and t

E

C0,ll

.

Therefore, i f

is s u f f i c i e n t l y large,

<

Max

Bxlog-lx(U-E(x)) E

1< Bx 1 logx E ( x )

(x)
1

'20 B x (log 5) log x

O(ZE(2))

.

Hence using ( 2 . 2 7 ) and ( 2 . 2 8 ) , we have, from (2.26),

Since f

E

S

,

t , such t h a t x

J.M. DE KONINCK AM) A. IVId

52 Hence (2.2 9) becomes

By repeated integration by p a r t s , a s i n the proof of Theorem 2 . 4 , we obtain, with

Ai(t)

= (-l);-'

Bi(t)

,

V

V

i=1 (log log z ) i

E (XI

Using Lemma 2 . 2 we obtain

Aatl ( v ) 10gV-lz

(log log z),+l

logV- x = o l v a f 2 ( l o g log z)a+l

and V

0";:p)

t O(E(X))

;

1.

53

SMALL ADDITIVE FUNCTIONS

therefore

=

log0-1z o( v a t 2 (log log z ) a t l

)

t O(E(X))

*

From these estimates, (2.31) becomes V

Bl(t) zlogt-' zdt

a =

3:

i.1

E (XI

uniformly for v

E

[ ~ ( z ),

11

1

.

Ai(v)

logv-' x

(log log z ) i

Using this relation and equation (2.30),

we obtain

+ o(

uniformly for v

E

[ ~ ( z ),

11

vat2 (log log z)utl

1

, and the proof will be completed

.

Assume that (2.25) holds for k hypothesis we have

v- 1

.

Therefore formula (2.25) holds for k by using induction on k

xlOg

= m

,m

< a

.

By the induction

54

with

J.M.

W(z,v)

and W1(x)

DE KONINCK AND A. I V I C uniformly f o r v

0

CS(X) , 11 ,

E

.

O ( x ( l o g l o g ~ ) ~ - ~ ~ ( xDividing )) equation (2.32) by v

, we

obtain

(2.33)

f (n1-1

a

V 1' x, 1 n- ( J Y ~ ) ) ~ i =rn

rn A i ( v ) logv - 1 x . (log log 2) ,L

W(x v )

+

Integrating both sides of (2.33) from E(X) t o u

A V

t

w1

-

, ~ ( x )s

(XI

.

V

u s 1

, we

have

U

Now, f o r x

4.2

s u f f i c i e n t l y large,

E

(XI

Bx logv - 1 x vcitrnt2 (log log x)atl

f o r some positive constant B

dv

independent of

v

.

This last integral i s

equal t o

Bx (logx) (log log x y + l

5

B x (u-E (x) )

(logx) (loglogx)"tl

1

~

vatrnt2

dv

E (XI

log" x Max atrnt2 E(X)
*

SMALL ADDITIVE FUNCTIONS

55

By Lemma 2.11, t h i s is smaller than Bx (logz) (log log z)a+l

-

B z logU-lz

U

atmt2

B z logE(+l

t

X

(E(X))atm+2 (log log z) a+1

(log log z)

*

But c l e a r l y

and hence

On the other hand, since 0

70

E

GI

so that

(2.36)

Finally

(2.37)

< ~ ( z )5

= l o g u - logE(x)

u < 1

,

O(1OgE-l (z)) = O(l0glogz)

,

J.M. DE KONINCK AND A. IVIE

56

and

Therefore, using (2.35), (2.36), (2.37) and (2.38), we can write (2.34) a s

+

x l o gu- 1x oIuatm+2

(log logx)"+l

1

t

O(x(l0g log x)m E (z))

.

Our theorem w i l l be proven i f we can show t h a t

We observe t h a t the l a s t term i n t h e sum on the l e f t s i d e of (2.39) is

(log log 1 2) a

E

1%

(XI

logv-lxdv

,

and with i n t e g r a t i o n by p a r t s , Lemna 2.10 (with i = CL) and Lemma 2.11 produce 1

(log log x)

1%

E (XI

logv-lxdv = 0

lu

:"Pk,

CL+2

u-1 1:gx,

) I .

57

SMALL ADDITIVE FUNCTIONS

Comparing t h i s with ( 2 . 3 9 ) , w e observe t h a t our theorem w i l l be proven i f we can show t h a t

(2.40)

a-1

1 i 1 i = r n (log log x)

= r=rntl c

7-

l o p x dv

E (XI

rnt 1 A,(u) 10gu-l

(log log x l r

o[ ua t 2

log

u-1

x

(log logx)atl

Integrating by p a r t s , as i n the proof of Theorem 2.4, we obtain f o r each rnsi
therefore

a-1 a-i

i:m j=1

v-1

log

Ic

[

1

J . M . 1IE KONINCK AND A. IVId

58 a-1

4 i=m

a-itl

(-1) (log logs)at1

l m ~ < v ) (a-itl) 10gV-lx dv

= It12tI

3

1

, I2 , and

W e now estimate Il

I3

.

-

Change of summation gives

From Definition 2.5,

(2.42)

c r=mt

(-l)r-m-l

a

I1 =

where R(E(x)) =

(log log zf

1

mtl

A

0

1

mtl

(u) logU-'z

1".

E

1'

mt 1

Using Lemma 2.10, we have

(2.43)

I?(€($))

= 0

1

log€

-1

(log logx)r

I

(I.

SMALL

=

o(

ADDITIVE FLINCTI0N.S

logE(x3-1xlo

Recalling the d e f i n i t i o n of

mx

I

(log log X)r

r n t l ~ a t (l u )

59

O(E(X))

.

,

we have

v- 1 - log x (log log x)atl

m

i =m

rn Ba (V) using Lemma 2 . 1 0 , we see t h a t m t 1Batl(v) and - a r e both, i n V absolute value, less than o r equal t o Hence

Finally, r e c a l l i n g the d e f i n i t i o n of

a+2 , uniformly f o r V

mtl

Bat2(v)

,

we have

u

(0,11

.

J.M. DE KONINCK AND A. IVIC

60

+

rn BaU t1

logV-lx du) ;

from Lemma 2.10, it follows that

<- M V

at2

'

each of the three inequalities holding uniformly for u

P3I

1 (log log

p

E

1&

(XI

E

(0,11

u-1 log xdu

.

Hence

,

U

for some positive constant Q ; and, using Lemma 2.10, we have

P3I <

= O

=

[

OI

Q

1 (loglogx)atl

10gU-lx

uat2 (log log x p + l

uat2(log log x)atl

logx

1 )

*

Max l0gU3c E(X)
(XI

-1 X

(E(x))at2(loglogx)"t1

t

o(E(X))

1

I

The estimates (2.42), (2.43), (2.44) and (2.45) yield (2.40) and so our theorem is proven.

From Theorem 2.12, we easily obtain the final desired result:

SMALL ADDITIVE FUNCXIONS Theorem 2.13. Let f

61

Sa ; then for an arbitrary positive integer

E

k s a ,

Proof: The proof is immediate from Theorem 2.12 by the substitution of

u=1

in (2.25) and the use of

x ( l 0 g log 3

p E (x)

0

I

X

(log log x y l

1.

We now outline two applications of Theorem 2.13. Theorem 2.14.

1'

(2.46)

1 x

n-

where e 2

= 1

Let a

&(n)

, e3

= 3

-

then

e

a

c

i (loglogx)

i=2

2p

2 ;

2

X

(log l o g x ) a + l

, and the remaining

eits

are computable

constants. Proof: Taking f ( n ) (2.46)

with ei

= ~ ( n )and

2Ai , for 2

e

2

5

= 2A2 =

i

5

a

.

k

in Theorem 2.13, we obtain

2

To evaluate e 2

BT(l) = al

1

, by

Now e3

=

2A3 = (-1)3 - 2 2B 3 ( l )

, we observe that

(2.23)

.

62

J.M. DE KONINCK AND A. IVId

-

= Bi(1)

,

= 3-2p

B;(1)

1 1

where m2 = 1

B1(l)

-

2B;(1) = al

t

2a2 = 1 t 2(1-p)

m

z

i a i = 2 (loglogx)z

n2(n)

3-2p

m3

a 2 2 ; then

Let

- =1x

nsx

B1(l)

by (2.24).

Theorem 2.15.

(2.47)

t

-

1

2

X

(loglogx)atl and t h e remaining mils

are com-

putable constants. Proof:

As i n the previous theorem, we let f ( n ) = n(n) and

k = 2 i n Theorem 2.13. 2 s i s a

.

Thus m2 = LA2 = B1(l)

Theorem 2.16.

(2.48)

W e then obtain (2.47) with mi

Let a

a P 2 b ) = 2’ I! nsx u 2 ( n ) i = 2

where q, = 6/r2

q,

2

1

= 1

, and we

for

see t h a t

2 ; then

qi (loglogz)i

6 (3- 2 p T

b

2Ai

t

2

I

P

X

(log log x)atl 1

and the remaining qils

are computable constants. Proof:

Relation (2.48) follows from an obvious generalization of

Theorem 2.8 and Theorem 2.13.

A simple computation shows t h a t

SMALL

q2 =

B1(l)

q3

B1(l)

ADDITIVE FUNCTIONS d,

-

,

6/n2

ZBi(1)

63

d, + 2d2

NOTES

For proofs of asymptotic formulae ( 2 . 2 ) and (2.3) see Hardy and

Wright C 11.

1' l/f(n) nsx R.L. Duncan Cll, C21 for the case f Sums of the type

additive) were first studied by

(f w

.

Duncan proves

(2.49)

essentially using the well-known Tur&-Kubilius

inequality

J . M . DE KONINCK AND A. IVI6

64

where f

is additive,

A(N) =

1

1

f(p)lp , @(N)

p
If(pa) I2/p"

, and

p%N

C i s a positive absolute constant. The r e s u l t s of t h i s chapter were obtained by J.-M.

De Koninck i n

h i s doctoral d i s s e r t a t i o n (Temple University, Philadelphia, 1972) , p a r t s of which were published i n De Koninck 111. derably sharpens (2.49))

(Note t h a t Theorem 2 . 5 consi-

.

A concise proof of Lemma 2 . 1 is given i n Selberg C11. follows from the very strong Lemma 1.1. The r e s t r i c t i o n

Our proof

t real (assumed

throughout t h i s chapter) is not necessary, and one may a l s o replace

It1

5

1 with

[ti

5

A

, where A

A detailed computation of

> 0

i s a r b i t r a r y , but fixed.

al and a2

i s given i n the proof of

Theorem 2.5, and similar computations of corresponding constants i n Theo-

rem 2.6 and 2.9 a r e therefore omitted. In Theorem 2 . 9 we could have written

For t h i s , and other elementary properties of the Mobius function

u ( n ) see Grosswald C11, o r Hardy and Wright C11.

CHAPTER 3 RECIPROCALS OF LOGARITHMS OF MULTIPLICATIVE FUNCTIONS

51. Functions with main term asymptotic to Cx/logx

.

In this chapter we shall be concerned with asymptotic formulae for nQ

1'

nsx

l/logf(n)

, where f is a positive, multiplicative function, and

denotes summation over those n not exceeding x for which f(n)

>

1.

Most well-how multiplicative functions will be seen to satisfy one of the following:

b)

(3.2)

n a

l/log f( n )

- Cf x/log logz

,

or

f is a positive constant depending on f .

where C

In this section we shall give an asymptotic formula of the type (3.1) f o r a large class of arithmetical functions which includes the functions g ( n ) and o ( n )

.

We begin by making the following

Definition 3.1. Let

be the class of multiplicative functions

f with the property 65

66

DE KONINCK AND A. IVI6

J.M.

, where

f o r all positive integers n

tisfying 0

1 and y

8 s

c

> a

.

, 8 ,y

For f

nf = rninh: f(m)

(3.5)

Q

let

F

E

are positive numbers sa-

1 for all m

>

2

.

nl

Because of multiplicativity, it is enough to verify (3.4) for n = p k , in order to establish that a given multiplicative function f

belongs to

.

$

number such that

$ , let

a=8

, and recall that

k

To see that 4 y >

u = 1

E

= 1 , let

4(p )

k

p (1

-

y

be any

Up)

.

Then

(3.4) reduces to

which is obvious since One

o,(n) =

and

y

i

dn

(1

-

l/p)Y s 1

y >

1

.

- l/p s

(1

-

l/p)-y

,

can also show that the generalized sum of divisors function

, belongs to 3

dX

any number such that

for X > 0

y >

max(1,X)

.

.

To see this let a = 8 X

If n = p k then

and (3.4) therefore reduces to (1

-

l/pX)Y s 1

.t

upX

t

...

.t

u pk X s (1

-

l/pX)-y

.

The inequality on the left is obvious, and the right-hand side follows from 1 because Y

>

1

.

t

l/pX t

. .. t u p k X

1/(1

-

UPh)

5

1/(1

- l/P3 -y

L0C;ARITHMs OF MULTIPLICATIVE FUNCTIONS

67

Now w e formulate Theorem 3.1.

1'

13.6)

n-

, then

If

f

E

l/logf(n)

=

5

.

(-1) m-1 F ( m - 1 )

1

F ( t ) = (at t 1)-l

Proof.

Let f

E

$

, and

for -l/y

nP (1 - l / p ) ( 1 .

(O)

(a log zlm

m=l

where the 0-constant depends on M

(3.7)

f o r every positive integer M

For - l / y

t

,

o(5/logMt15)

It I0

m

t

5

1 p - m ~ ( p m ) -am p )t) .

m=l

t

define

I0

(3.8)

If

E

> 0

, then

f o r some positive constants

el = el(€)

and

c2 = c ~ ( E , B ) we have

(3.9)

l/pB

t

l/(pB - 1)

I c 2 p-'+'I2

f o r p s el

,

f o r p > c1

.

and (3.10)

l/pB

t

l / b B-

1)

Now f o r every integer m

(1 - l/pB)Ylt

I

5

2

I p-BtE'2

0p(prn)p -am)Itl

so that we obtain from ( 3 . 8 ) , f o r m ht (P")

(3.4) yields

0

(f (pm, p - 9 -

2

It'

1

I

(1 - l / p B) - Y l t l

, - (f(pm-l) P - a ( m - l ) ) - It I

,

68

J.M. DE KONINCK AND A. IVIE

ax

where we have used that f(x) that

yltl s

-

-x is non-decreasing for a

a

>

1

,

, (3.9), and (3.10).

A lower bound for ht(pm) may be proven similarly; the multiplica-

tivity of ht

where a ( n ) =

now gives, for some

np.

,

c3 = c 3 ( ~ , $ )

From Theorem 1.4 it follows that for every 6

P In

and therefore for 6 = d 2 we obtain

Partial summation gives m

By using the kbius inversion formula, we obtain from (3.8)

> 0

L 0 G A R J . m OF MULTIPLICATIVE FLJN(TI0NS

69

Using (3.12) and (3.13) we have

Suppose that -l/y

5

t

5

.

~ ~ m a x ( O , B t a l y - l ) Then we obtain uniformly i n

t

0

m

Since ht

is multiplicative and

1 ht(n)/n

n=l

i s absolutely convergent, t h i s

may be written a s

I ,

where F ( t )

is given by (3.7).

W e w i l l now deduce (3.6) from (3.15).

By (3.5) we have

O(1)

and theref ore 0

0

,

J.M. DE KONINCK AND A. IVId

70

To estimate the last sun observe that, for n

2

no

, f(n)>n

a/2

by (3.4), so that (3.15) yields

Using (3.15) we obtain

1' l/logf(n)

(3.16)

7

= x

nsx

F(t)xatdt

+

O(xl-BtE

.

t x

-l/Y

Since F ( t )

1

integration of

, partial

is infinitely differentiable on C-l/y,Ol F(t)zatdt

in (3.16) gives (3.6), for

E

>

0 suffi-

-l/y

ciently small. We now present another class of multiplicative functions for which

1'

ne

behaves asymptotically as c x/log x

l/log f (n)

f

.

Our approach this

time will be of a more elementary nature, and the obtained asymptotic formula (Theorem 3.2) will not provide as sharp an estimate as Theorem 3.1. Definition 3.2. A multiplicative function f belongs to the class

&I

if for every prime p

numbers a l J k

,

a2,k

9

*

and every positive integer k there exist

*.$a

kJ k

such that

(3.17)

where -1

5 aiJk 5

K uniformly in i and k with some K > 0

.

LOGARI'IHMS OF MULTIPLICATIVE FtbiCI'IONS From t h i s d e f i n i t i o n it is obvious that

f(n)

71

is s t r i c t l y p o s i t i v e

s integers ( i f t h e ai,k's and t h a t f ( n ) i s an integer i f the u ~ , ~ ' are were allowed t o take integer values l e s s than -1 then f

would not

always be p o s i t i v e ) .

$(n) and ~ ( n ,) it i s easy t o check t h a t

In addition t o

8

contains other well-known multiplicative functions such as: t h e Dedekind's function

$(n)

n

(1

t

, the

l/p)

P In function

$*(n)

n(-l)w(d)/d 1 dln, (d,n/d)=l

sum of divisors function

u*(n)

r e l a t e d t o the functions

o(n)

a r e a l s o contained i n function f

unitary analogue of Euler's t o t i e n t

3:

1 d d In, (d, n / d ) =I

unitary analogue of the

, and some other functions

.

and a*(n) While a l l of these functions

6

of Definition 3.1,

a l s o contains the

defined by k k f(p ) = p - p k - 1

f(2k)

, since

which does not belong t o f o r n = 2k

, the

and every integer k

which i s c l e a r l y impossible f o r k

t

-...- p-1 1 and (3.4) would give then

1

+

, since

the left-hand s i d e is a

p o s i t i v e constant.

On the other hand 5

means t h a t neither Theorem 3.2.

%

If

does not contain

5

nor

f

E

b

, and

Q

o,(n)

for

x

s 1

, which

.

ak,k 2 - 1 / 2

for k r k o

, then

72

J.M. DE KONINCK AND A. IVId

(3.18)

1'

nQ

= 5

l/logf(n)

log x

log log log x

The proof of Theorem 3 . 2 is based on the following elementary

8,

Lemma 3.1. If f E and

C2

then there exist positive constants

and a natural number nl

(3.19)

f(n)

C1

such that

< Cln(loglogn)K

for n

t

nl

,

and (3.20)

f(n)

2

C2rn/loglogm for rn

>

1

,n

= 2k r n , rn

odd,

where K is the constant appearing in Definition 3.2. Proof of Lemma 3.1.

If r = w(n) and p

T

since f is multiplicative

= n

and

(it-) K P-1 P In

,

is the r-th prime, then

LOGARITHMS OF MULTIPLZCATIVE FUNCTIONS W e now use the elementary estimates pn s n3’2 and log P

n

2

P In

p)

(valid f o r n t 3)

(which i s v a l i d f o r n

log ~ ( n )5 21og log n

follows from n

73

t

5

,

,

and which

t o obtain, f o r n t n 1 ’

s n e ~ p ( l o g ( ( 3 E )( l~o g l o g n ) K

which proves (3.19) f o r C1 = (3B)

( 3 ~ ) ~ (log n . 1ognIK

,

.

K

To prove ( 3 . 2 0 ) we note t h a t by (3.17)

f@k)

pk

- pk-l- ...-p- 1 ,

so t h a t f(p k ) = 1 can occur only f o r p

2 ; otherwise,

f(pk)

>

1

,

and we have

Since we have

log

log 1 / ( 1 -x)

1

x

+

x2 f o r

(1 - h 1 1 = f log(1 P Im P m

5

for m > l

5

and C4

1 - 1-P-1

log(C3 log logm)

s u f f i c i e n t l y large, so t h a t

0 s x -1

5

,Im[

+ o(1)

1/2

, we

1

p-l

obtain 1

+

02)

s log(c, log logm)

J.M. DE KONINCK AND A. I V I d

74 -1

where C2 = C4

which, when combined with (3.21)

)

proves (3.20).

1 l / p and p n would lead t o explicit psx C1 and C2 but C1 and C2 would still depend on K . Taking

Sharper estimates of values of =

Pi P2

nl '*

)

)

primes we find t h a t t h e

t o be the product of the f i r s t k

'Pk

bounds of (3.19) and (3.20) a r e actually attained, Proof of Theorem 3.2. there a r e O(1ogx)

1

the fact t h a t

numbers not exceeding x l / l o g n xflogx

292.2

1'

(3.22)

U l o g f(n)

2

%X

2

1 can occur only f o r n = 2k

Since f ( n )

lfl / l o g n t O

1'

nCx

t

O(x/log2 x)

, and

log2 x

)-

.

Using

(3.19) we obtain

1;x

log2 x

which gives the necessary lower-bound inequality. bound note t h a t i f

1

I/(log n t log c1 t log log log n)

xlogloglogs

n<x

f o r which f ( n )

k

k > k o and f(2 ) z 1

)

To obtain the upper

then f(Zk)

3/2 by

2

(3.17) and

.

5-1/2 If m 2 3 i s odd, then from k, k l/logm - l / ( l o g C2 t log m - log logm) = O(log log logrn/log2 m)

a

1'

n2

l / l o g f ( n )
t

k

1 l/logf(m)

2 m5x

We have

2

k

2

1 Wogm

t

t

O(1ogx)

summation (m i s odd)

1l/log m

msx

= x / 2 log x t o(xAog2 x)

, we

obtain

o x log log log z log2 x

m5z

U l o g n = 1 l / l o g 2k rn k 2
1

)

)

I

s o by p a r t i a l

LOGARITHMS OF MULTIPLICATIVE FUN(TTI0NS

75

Thus

k

1

(l/logm - l / l o g 2 k m)

5

2 m a

k

1

k l o g 2/log2m

2 m5.z

<< z

1k

<<

2

1 k 2-k log-2(z 2-k k 2 3

t

k

ma/2

QC

1) << z/log2 z

1

k

l/log2m

.

I t follows t h a t

(3.23)

which, with ( 3 . 2 2 ) proves Theorem 3 . 2 .

§2.

Functions with main term asymptotic t o

Cz/loglogz

.

This class of functions contains, amongst others, the d i v i s o r funct i o n d k ( n ) f o r which we prove Theorem 3 . 3 .

(3.24)

For every integer

1’ l/logdk(n)

n a

Uf ( n )

2

2 we have

= z/(logk. loglogz) t 0(z/(loglogz)2)

where the 0-constant depends only on Proof.

k

k

,

.

Since t h e function f ( n ) = ( l o g d k ( n ) ) / l o g k is additive,

is multiplicative, and therefore, f o r Res > 1 and

IuI

5

1

,

76

DE KONINCK AND A. IVId

J.M.

where

(3.25)

9(s,u)

n P

=

-s u

( I - ~

(log(k2

(ltup-stu

t

k ) /2) /log k - 2 s

. . .)

P

is absolutely and uniformly convergent f o r Res > 1 / 2

.

Theorem 2.4 cannot be applied here since f ( n ) does not belong t o sa

.

To see t h i s , note t h a t f o r u near zero l/r(U)

.

u t o(u2)

This implies t h a t

with c 2 =

1 Up2

f

P

, so

0

that

D ( u ) k C3C0,11 because

k)/Z)/log k

<:

2

However, we can s t i l l take u

E

defined since

(log(k2

t

, while

Df"(0)

i s un-

~ " ( 0 ) is f i n i t e (and

depends only on k ) .

obtain, uniformly in u

C'

(3.27)

nsx

to 1

, we

, (noting that f ( n )

uf(n)-l

where F(u) = D(u)/u

.

, and

(0,ll

xF(u) 10gu-l

2 t

0

2

1 for n

if and only if

O(z/(u l o g x ) )

Integrating ( 3 . 2 7 ) over u from

have that t h e left-hand s i d e becomes

since f ( n )

apply Lenuna 2 . 1 t o

>

1

.

n

1)

,

E(Z)

log

-1/2

z

LOGARITHMS OF MULTIPLICATIVE FUNCTIONS

77

Integrating the right-hand side we see t h a t the i n t e g r a l of t h e

, and

e r r o r term i s O(x log l o g x /logx)

/

x

(3.29)

E

F ( u ) logu-'x.

xlogx *

du =

(XI

]

X

log x E

since F ' ( u )

so integration by p a r t s yields

log log x

U

F ' ( u ) log x . du

(XI

(log log x)2

is bounded i n

C0,ll

.

~ ~ ( l ) / l o g l o gt xO(x/(log logx)2),

Further repeated integration by p a r t s

would not yield more main terms since F " ( 0 ) 1 and

F(1)

l/f(n)

(logk)/logdk(n)

is undefined.

Noting t h a t

we obtain ( 3 . 2 4 ) from ( 3 . 2 8 ) and

(3.29).

Recalling t h a t t h e logarithm of a multiplicative function is additive, i.e.

for

(m,n) = 1

, we

now see t h a t t h e problem of estimating

is a special case of t h e more general problem of estimating where g

is additive.

1'

l/logf(n)

1'

l/g(n)

n~

nsz

,

W e have already d e a l t with t h i s more general pro-

blem i n Chapter 2 , where we used an a n a l y t i c a l method. I t is i n t e r e s t i n g t o see t h a t t h e problem of finding asymptotic ex-

pansions f o r sums of reciprocals of additive functions can a l s o be investigated i n the l i g h t of p r o b a b i l i s t i c number theory. t h i s theory, we w i l l prove t h a t

More precisely, using

78

J . M . DE KONINCK AND A. IVIC

1'

(3.30)

nSx

llf(n)

x(1

o(1)) r f ( 2 ) loglogx3-1

f

f o r a large class of prime independent additive functions

,

f(n)

.

Further-

more we s h a l l indicate the significance of obtaining a second term on the r i g h t s i d e of (3.30)

. Recall t h a t a

The general idea of the approach is the following.

c l a s s i c a l theorem of Hardy and Ramanujan (see Hardy and Wright C11, p. 356) s t a t e s t h a t "for almost all" n the number of

n

, not exceeding x , f o r which

1 w(n) -

(3.31)

is o(x)

f o r every positive

function

n(n)

extensions.

.

-

, ~ ( n ) log log n , o r more precisely t h a t

log log n 1 > (log log n)

6

.

1/2t6

The same r e s u l t actually holds f o r t h e

Moreover the theorem of Hardy and Ramanujan has several

This allows us t o s t a r t with a large c l a s s of functions.

Our

class of functions f(n) includes, a l l functions f o r which f(n)

- log log n

f o r "almost a l l "

because the function l o g l o g n

n

.

i s slowly varying i n the sense t h a t

loglogn = loglogx

(3.32)

We can take such a l a r g e c l a s s

t

0(1)

, XI'*

5

n

5

x

,z+tm .

Although a precise d e f i n i t i o n of a slowly varying (or slowly o s c i l l a t i n g ) function w i l l be given i n Chapter 5, only property (3.32) is a l l t h a t i s required i n t h i s section.

A t t h i s point, it is a l s o i n t e r e s t i n g t o note

t h a t a r e l a t i o n similar t o (3.32) f a i l s t o hold for functions growing too rapidly t o

tm

, such

as the function

logn.

This i s why estimates of t h e

type (3.1) cannot be f u r t h e r improved by our p r o b a b i l i s t i c approach.

79

JBGARITHMS OF MULTIPLICATIVE FUNCTIONS

Let

be a given arithmetical function and l e t R(x)

f(n)

positive function tending t o x+tm n

.

and s a t i s f y i n g R(x)

+m

Furthermore l e t A(x)

not exceeding x

, for

f a i l t o hold. those values of

o(1og logx)

as

denote t h e number of p o s i t i v e integers

which the i n e q u a l i t i e s

log l o g n - R(x)

(3.33)

be a

S

f ( n ) < log l o g n

I n view of ( 3 . 3 2 ) , i f we assume t h a t n where f(n)

*

0

t

R(x)

f(n)

2

1 for all

we have

and

2

z - ~ " ~ - A ( x ) t A ( x1 / 2 ) i l t o ( R(x) log log x log log x

I f moreover A(x) = o(x/log logx) f(n)

-

f o r t h e given arithmetical function

it follows from ( 3 . 3 7 ) and ( 3 . 3 8 ) t h a t

The functions A(x) can be shown t o be

and R(x)

associated w i t h the function f ( n ) = w(n)

o(xAog l o g z )

and

o(1og logx)

respectively,

80

J.M.

DE KONINCK AND A. IVIC

allowing us t o s t a t e t h a t r e l a t i o n (3.36) holds with f(n)

w(n)

replaced by

*

This approach can be s l i g h t l y extended by using t h i s t i m e a large deviation theorem due t o Kubilius (C11, p. 161) which w i l l provide us with smaller upper bounds f o r the functions A(x) theorem of Kubilius t o the function f ( n ) / f ( 2 )

and R(x)

.

Applying the

instead of f ( n )

w e obtain

as x + m

A (x) = o (x (log log x) - 2 )

(3.37) if

R(x) = (log logx) 1/2+6

(3.38)

where

0 < 6 < 1/2

is a fixed constant.

Therefore i f

conditions of Kubilius's theorem and f ( n )

2

Y

f

s a t i s f i e s the

1 for f(n) z 0

, then we

obtain

1' l/f(n) nsx

(3.39)

X

f(2) loglogx

+

X

O I (log logx)

3/2-6

I

'

This estimate i s somewhat stronger than (3.36), since it specifies the e r r o r term.

The function w(n) clearly s a t i s f i e s the conditions of

Kubilius's theorem, and hence (3.38) holds f o r f = w

and Theorem 2.5 gides

a much b e t t e r r e s u l t .

The error term i n (3.38) can be improved on t h i s l i n e of attack, but the order of magnitude x/(log logx) 3 / 2 e a s i l y seen from the asymptotic expansion of

cannot be attained, as i s

81

L O G A R I W OF MULTIPLICATIVE FZINCTIONS (see Kubilius C11, p. 61 and the inequality in Galambos C11).

Essential

new information can be therefore obtained by determining the exact order of magnitude of the second term on the right-hand side of (3.39).

The

finite asymptotic expansion obtained in Theorem 2.4 goes further than this, but it uses deeper analytic results, and this is why the probabilistic approach is of interest.

13. Functions with main term asymptotic to Cx

.

We shall now present a theorem giving an asymptotic formula for

1'

&a:

l/logf(n)

when f belongs to a certain class of multiplicative, po-

sitive, prime-independent functions (we recall that an arithmetical funcfor all primes p

tion is called prime-independent if f(pv) = g(v) v =

1,2.. .)

.

and

This class of functions includes, amongst others, the func-

tions a ( n ) and d ( e ) ( n )

, which represent the number of non-isomorphic

abelian groups of order n , and the number of exponential divisors of n

,

respectively. None of the previous estimates can be applied to this class of functions. O u r result is Theorem 3 . 4 .

Let f ( n ) be a multiplicative arithmetical function

such that for all primes p

where g(l) = 1

, g(v)

> 1

, and for v for v

= 1,2,..

2

2 and lim inf g(w) > 1

1

(C(t)

w

have

1'

(3.41)

l/logf(n) = x

6 X

C(t)

f(p") = g(v)

.

,

Then we

- 6/r2) d t + O ( r ~ l ' l~og1'*,)

,

-m

n (1 k =12 ( g t ( k ) P m

where

. , we have

t

- g t ( k -1)) p - k ) , and

over those values of n for which f ( n )

>

1

.

1'

denotes summation

J.M. DE KONINCK AND A. IVId

82

Proof.

We note t h a t f ( n )

2 1

, and

f(n)

is square-free, o r , equivalently, i f and only i f p(n)

1 i f and only if n 1 - p2(n) = 0

, where

i s the K b i u s function.

We define

(3.42)

W e then have

where C

is a positive constant.

1 ft(n)

We w i l l now estimate

m

1f $ ( n ) Y T S =

n=l

n-

for t S O

.

I t is clear that for

nP ( l + p - S t g t ( 2 ) p - 2 s + g t ( 3 ) p - 3 s t . . . )

m

where g(s, t ) =

1 h(n,t) n-', n=l

Since $ 1 0 , Ih(n,t)

h(n)

I

1

u(n)

i s multiplicative and

, where

LOGARITHMS OF MULTIPLICATIVE FUNCTIONS

0 if there is a p

83

such that pI In

u(n) =

1 otherwise, so that

With w = g t ( 2 )

, further factoring yields

where for t s 0

m

Thus if

1 c(n, t) n-'

n=l

= u(s,t)

, then for every

E

>

0

, and uniformly in

since, uniformly in t 5 0 , the generating function u ( s , t )

is absolutely

convergent for Res > 1 / 3 . The above formula then gives, by partial summation,

(3.45)

J.M. DE KONINCK AND A . I V I d

84 m

For

1 b(n,t) n-'= cw-l(2t3) n=l

which gives, uniformly i n t s O

, then, by Lemma

2.1, we obtain

,

(3.46)

From (3.44) it follows t h a t

m

Since

1

1 Ic(n, t) 1 n-2

n=l

converges we have

so that

since f o r

t s O we have w s l

.

P a r t i a l summation gives

85

MGARITHMS OF MULTIPLICATIVE FLINCI'IONS

We define y

by y = x/logx

and

2

by

z

logx

.

Since

we have

where

s3

O(zy4 l o g - l x ) = O ( x l log-f x)

,

and

C ( t ) 3: + O ( x y - f l o g - l y) t O(y$

We thus obtain, uniformly i n

(3.47)

where

t

,

C ( t ) 2 + O ( d log-'x)

.

86

J.M. DE KONINCK AND A. IVId

Using (3.47) i n (3.43) and integrating from -T t o

I

(3.48)

l’l/logf(n) = 2 ( C ( t ) 6 x -T

To estimate C ( t )

-

O(T > 0)

, we

obtain

- 6/n2) d t t O ( T z 1 exp(-C l ~ g ~ ’ ~ c c ( llogzr1’5)) og

for t 5 0

6/r2

, let

where O
gt(2)f2t

(gt(3) - g t ( 2 ) ) p - 3

t

(gt(4) - g t ( 3 ) > p - 4 t . . .

gk(2) (p-2 - p - 3 ) tgC(3) (p-3 - p-4) t g 7 4 ) (p-4 - p -5)

5

and r

gt(r) (p-2- p - 3

t

p - 3 - p -4 -

is an integer such t h a t g(v)

2

i s not f i n i t e , then g ( t

m)

i s just

?;

which holds whenever z > 0 and y > 0

w

lim inf g(v) m

logzty/z

, we

v = 2,3

... .

It

l i m i n f g(v) > 1 ; moreover i f

Using the inequality

log(.ty)

.. .

t -2 = 9 (21) P

g(r) > 1 for

is c l e a r t h a t such an integer exists, since r

...I

t

obtain

.

87

LOGARITHMS OF MULTIPLICATIVE FUNCHONS

s

6 t exp(g (r)

.

t

s 6exp(g ( r ) ) / r 2

P

2 .

If

1 (p2- l ) - ' )

is s u f f i c i e n t l y small,

t < O

V

If n = p l

V.

...pi 2 , then

1

f ( n ) = g(vl)

...g(vi) .

If

f(n) > 1

(3.51)

Since 0

J

-

(C(t)

o -T

6/n2) d t =

-T

J -J -m

,

-m

we can derive from (3.48) (3.52)

1'

YlSX

0

l/logf(n)

= x

(C(t) -m

-

6/a2) d t t O(g-T(r) 2))

,

J . M . DE KONINCK AND A . IVId

88

by using (3.47) and (3.48).

, we obtain

Defining T

g-T (r)X = exp(-l'logg(r)

t

logr) = exp(-1 1ogX) = x1/2 , 2

which ends the proof of Theorem 3.4. For the first application, we look at a ( n ) isomorphic abelian groups of order n = P(w)

plicative, and that a@') where P ( w ) P(w) =

.

, the number of non-

We recall that a ( n ) is multi-

for every p

and for

v =

is the number of unrestricted partitions of

1 iff

w

1 and P ( w )

v

.

1,2...

,

Thus

is strictly increasing. The conditions

of Theorem 3.4 are satisfied, and (3.46) holds when f ( n ) = a ( n ) and g(k) = P ( k )

.

We note that lim inf g(w) =

t-

w

,r

and g(r) = 2

= 2

.

Other examples of multiplicative, prime-independent functions that satisfy the conditions of our theorem may be readily found among enumerative functions of certain algebraic structures. One such example is S(n)

, the nmber of non-isomorphic semisimple finite rings of order Finally let us consider d ( e ) ( n )

n

.

, the number of exponential di-

.

A divisor d = p bl , . . . p F is called an exponential divisor W 1 wi l Wi of n = p , ...pi if is called an exponential divisor of n = p , - - 'Pi visors of n W

b , Iv,,.

.,,bi Iwi .

It follows that

(n)

is a multiplicative, prime-

independent arithmetical function for which

(p')

, where

= d ( ~ )

is the ordinary nunber of divisors function. Since d(1) d ( v ) 2 2 whenever w 2 2

1

d ( ~ )

, and

, the conditions of Theorem 3.4 are satisfied,

and (3.41) holds with f ( n )

d e ) ( n ) and g ( k )

d(k)

.

In this example

LOGARI'I1IMs OF MULTIPLICATIVE FUNCTIONS

we have

l i m i n f g(v) = 2

and r = 2 , g ( r ) = 2

89

.

vtco

NOTES Early r e s u l t s concerning multiplicative functions f

1'

n5x

l/logf(n)

-

C

f

z/logz

f o r which

were obtained by De Koninck and Galambos Cl1.

Their paper gives t h e asymptotic formula (3.6) i n the special case when f(n)

a(n)

.

Theorem 3.1, which is a generalization of t h e i r method of

proof, is t o be found i n E. Brinitzer C11. Theorem 3 . 2 is from Ivid C21, where i t i s a l s o shown t h a t i f

is the number of integers n

f o r which f ( n )

N(I> = C z +

where d and

E

O(z.

5 I

exp(-dlog

and

3/8-c

N(z)

f.8,then z))

,

are a r b i t r a r y fixed positive numbers, and

The ccndition ak,k 2 - 1 / 2 i n Theorem 3 . 2 is necessary t o keep k f ( 2 ) being too close t o unity. If j @ e & and no condition of t h i s s o r t

90

J.M. DE KONINCK AND A. IVId

i s f u l f i l l e d , then Theorem 3.2 does not have t o hold. t h e multiplicative function f , ( n ) k

fob with a l , k =

= P

k

To see t h i s , define

by

al,kP k-l+

... t

ak - l , k P

+ ak,k

... = ak-1,k =-' , and

and we have

S l o g2 > e

= I c ,

so t h a t Theorem 3 . 2 does not hold f o r f = f, For the properties of

.

@*(n), o*(n) and t h e unitary convolution,

see Cohen C11 and C21. Theorem 3.3 could have been included i n Chapter 2, but we f e l t t h a t it i s more appropriate t o s t a t e it among r e s u l t s concerning recipro-

cals of logarithms of multiplicative functions. The p r o b a b i l i s t i c method sketched at t h e end of 1 2 has been outlined by De Koninck and Galambos C11.

For a f u l l e r account of probabilis-

LOGARITHMS OF MULTIPLICATIVE FUNCTIONS

91

t i c methods i n number theory the reader is referred t o t h e monograph of Kubilius C11, where a proof of the l a r g e deviation theorem may be found. This important result may be s t a t e d as follows:

Let A

f

be an additive function f o r which there e x i s t s a constant

# 0 such t h a t f(p) = A

f o r a l l primes

p

of a s e t Q of primes such t h a t the s e r i e s

with the possible exception

1 l/p

converges.

Also

PEQ o((log logs)1/2)

assume m

and s e t y = m/(log l o g x ) l / *

and

Then the number of positive i n t e g e r s , n not exceeding s

, for

which

f ( n ) < log l o g x

when m

5

0

t

m(1og l o g z ) 1 / 2

and t h e number of positive integers n

,

, not

f o r which f ( n ) > log l o g s

when m 2 0

where

, is

equal t o

t

m(1og l o g s ) 1 / 2

,

exceeding x

,

J.M. DE KONINCK AND A. IVId

92

From the expression f o r Qe(m) m4

rn3

6(10glogc)~/~ while as u + m

we infer as

1 2 log l o g e <

m+m

m3

1/2 Qdm) < 6 (log log c)

’

we have

(see e.g. J. Galambos

121, p. 66). To see that (3.37) holds we take

rn = (log log c)’, 0 < 6 < 1/2, obtaining as c

*AX

-t

(1t o ( 1 ) ) exp(Qe((log l o g z ) 6 ) ) G(-(loglogc)6))

= o(

1

(log log c)2

1 .

W e are indebted t o Professor J. Galambos f o r the above discussion concerning the p r o b a b i l i s t i c method. Theorem 3.4 was proved by De Koninck and I v i t C11. For a proof of the asymptotic formula

which was used i n ( 3 . 4 3 , see A. Walfisz [I], Chapter V, 56.

Though f o r

our purposes a s l i g h t l y weaker r e s u l t would s u f f i c e , we have s t a t e d

LOGARITHMS OF MULTIPLICATIVE FUNCTIONS

93

Walfisz's r e s u l t because it is the sharpest known r e s u l t about the number of square-free integers not exceeding x

.

Although the inequality (3.46) is a d i r e c t consequence of Lenuna 2 . 1 , it could have been proven by elementary methods, but the proof would have been much longer. A detailed account of enumerating functions of various algebraic

structures (including

u ( n ) and S ( n )

applied) i s given i n Knopfmacher

[ 11.

the number of exponential divisors of

, to

which Theorem 3.4 may be

For t h e properties of

n , see Subbarao

(n)

,

[I].

For generalizations of Theorem 3.1 and Theorem 3.2 we r e f e r the reader t o Chapter 8, Definition 8.2, 8.3 and Theorem 8.3.

This Page Intentionally Left Blank

CHAPTER 4

SUMS OF QUOTIENTS OF ADDITIVE FUNCTIONS

51. Introduction R.L. Duncan C11 proved that

Dmcan’s result was based on the elementary estimate

In Chapter 2 , we gave estimates of

1’ l / f ( n )

n 2

of additive functions f which improved estimate (4.2).

for a large class In this chapter,

we will improve the estimate given in (4.1) , by using an extension of the method employed to obtain an asymptotic expansion of the form (2.3) for sums of reciprocals of additive functions. Using the method developed in the proof of Theorem 4.1 we will deduce that, given any fixed positive integer M ,

where al

1 l / p ( p - 1) , and

the remaining ails

P

95

are computable constants

J . M . DE KONINCK AND A. IVIC

96

In 5.2 we investigate sums of quotients of "small" additive functions (where the "smallness" of f and

g

is defined in the formulation

of Theorem 4.1), and in 53 we turn our attention to sums of quotients of additive functions which behave "like c logn". This class of functions contains the functions log$(n) and logo(n) for the sum

5.2.

1' & 10 + ( n )

n2c

.

An asymptotic expansion

is stated in Corollary 4.1.

Sums of quotients of "sma1l"additive functions. Theorem 4.1. Let g and f be two additive functions such that

for all primes p

and all integers r 2 1

and 1 5 f(pP)

(4.5) where el and

c2

,

are two positive absolute constants. Then

(n) 1' h lnr;J:

(4.6)

, 0 s g ( p21) < c2r

< clr

+

3:

loglogx

(log log x)2

where

c c (1-p-l) (g(pP) -

m

'4 =

Proof. Let

r=1 p u

E

(0,ll and t

m

seen that

1

n=l

t g ( nuf ) (n) n-'

Re s > 1 , and therefore

E

ft

,

f(pP))/pP

It1

5

1

.

,

By (4.5) it is

converges uniformly and absolutely for

QUOTIENTS OF ADDITIVE FUNCTIONS

= (s(s))tU H(t,u;s)

where H(t,u;s)

97

,

is absolutely and uniformly convergent f o r Re s > 1 / 2

Using Lemma 2 . 1 we obtain uniformly f o r

It I

5

1

,

Iu

I

.

s 1

and the remaining powers of u have exponents not less than two, since 1 s f(pr)

from (4.5).

Differentiating both sides of (4.7) with respect t o

t

t

we obtain

uH( t,u; 1) logtu2 . l o g l o g x t O ( 1 ) ) , 7 r tu

98

J . M . DE KONINCK

AND

A. I V I C

where we have used Cauchy's inequality f o r derivatives of analytic functions t o estimate the e r r o r term. W e note t h a t f ( n ) = 0 i f and only i f

n = 1

.

i n (4.9), and dividing by u we obtain uniformly f o r u

Letting E

t= 1

(0,ll

where (4.11)

W e next integrate (4.10) with respect t o u to

l ( z t 3)

.

(4.13)

t

(logx)-2/2

The left-hand side of (4.10) becomes

=

since f ( n )

from E(X)

1'

nl?:

1 for

$# o(z(logs)-1'2 t

n >1

1

nsx

loglogx)

,

, and g ( n ) << s l o g l o g z

,

as can be proven using elementary methods. From (4.8) it i s seen t h a t F ( u )

E

C2C0,11

, and

G(u)

E

CICO,ll

.

99

QUOTIENTS OF ADDITIVE FUNCTIONS This implies

(4.14)

F(u) log'z

. loglogz.

F(1) l o g z

du

-

F ( E ( s ) ) logE(z)z

E (XI

Similarly, we obtain

(4.15)

-

G(1)logz

- loglogz

+

(

)

logz (log log 2)

since both F f r ( u ) and G r ( u ) a r e bounded on

C0,lI

J

.

From (4.10), (4.11),

(4.12), (4.13), (4.14), and (4.15) we obtain (4.6) since F(1)

1

.

A

straightforward computation shows t h a t m

A = G(1)

1 1 (1-p-')

- F'(1)

r=l

For the case q ( n ) = n ( n ) so that both F ( u ) C0,ll

(4.16)

.

and G ( u )

p

, f(n)

( q ( p r ) - f(pr))/pr

.

,s&

=

~ ( n )we have

uf(Pr>

a r e i n f i n i t e l y d i f f e r e n t i a b l e functions on

This means that we can obtain, by repeated integration by p a r t s , log log 3:

F ( u ) log'z. E (XI

du = l o g z

I

F.(1)

- log F 'log (1) + . .

J.M. E KONINCK AND A. IVI6

100

t

(

(-1>MF(M) (1)

(log log Xft1

(log log x)

11,

and 1

(4.17)

J

G ( u ) log'x.

G(-'l)

du = logx

(log log x)M

E (XI

-+ where M

2

(1)

1 i ) 9

O I (log log 3C)Mt1

1 is an arbitrary but fixed integer. This then proves (4.3)

with

Asymptotic formulae similar to (3.30) can easily be obtained for sums of quotients f ( n ) / g ( n ) by using the probabilistic approach out-

lined in Chapter 3 , whenever f ( n ) and g(n) are both "close" to loglogn in the sense of relation (3.33).

93. Sums of quotients of additive functions which behave "like clogn".

Theorem 4.1 is concerned with additive functions h for which h(p)

is independent of p

, and

h(pr)

of condition (4.5)) for each r >2 h where h(pr) varies with p

.

is not "too large" (in the sense

We now consider additive functions

(for r 2 1) "regularly", that is, we

require h ( p r ) to be "close" to logpr

as defined in the following theo-

rem. Theorem 4.2.

Let g and f be two additive functions for which

there exist two non-zero constants a and b such that, for each prime p and each integer r r l

,

QUoTIENTS OF ADDITIVE FUNCTIONS (4.18)

101

a r l o g p + R kr) g

gkr)

and

with IRh(pr) - R h ( p r - l ) and some c > 0

f(n)

*

I

, whenever

<

~ p - ~, ' uniformly in r h = g or h = f

0 for all integers n > 2

.

.

2

1

, for some x > 0

Assume, €or simplicity, that

Then, given any fixed positive integer

M ,

(4.20) m

where A1

=

1

r=l

( 1 - p - l ) ( ( g ( p r ) - f(pr))/pr

and the remaining

Ails

are

computable constants. Proof. Without loss of generality we assume that a = b = l . We first estimate Define g l ( n )

2'

rz-

(g(n))t (f(n))'

eg(n)

, f,(n)

for

= ef(n)

.

It1

<

1/4 and u

Then g1 and fl

tive, which implies that for Res > 1 we have

where

2 rl

E

1-3/4,01.

are multiplica-

J.M. DE KONINCK AND A. IVI6

102

From (4.21) we deduce

1 ( g l ( n ) n - l ) t ( f l ( n ) n-1)u = x 1 h(t,u;n)n-'t 1 h(t,u;n)(CEl

(4.23)

nsx

nsc

nsc

= x

1

nsz

h(t,u;n)n-l t o (

where the &constant is independent of

-E)

1 Ih(t,u;n) I) ,

nsx

t and u

.

Using (4.23) we next show that

(4.24)

, for

uniformly i n t and u

Using the inequalities ex

-

1s

3;c

,

(0 s x s 3/2)

some

E

'-

le

such that

O<

E

sl

.

11 s e l Z 1 - 1 and

we obtain

where C is a positive-constant, notnecessarilythesameoneateachoccurence. Suppose r

L

2

.

From the hypothesis on Rh we have

From the d e f i n i t i o n of

H(t,u;s)

we obtain

103

QWTIENI'S OF ADDITIVE FUNCTIONS

where, as i n the previous case, on p

and r

C

depends only on

E

, and

and

.

Thus, since h(t,u;n)

is a multiplicative function of n

I t can be shown that (4.26)

P a r t i a l summation then gives, f o r some

and

not

E

such t h a t

0

<

E

51

,

, we

have

J . M . DE KONINCK AND A. IVIi

104

1

(4.27)

n >x

h(t,u;n)n

-1

-€

<< 5

.

From (4.23), (4.26), and (4.27) we obtain (4.24) uniformly for It( s

and

rl

that if

u

€=€(A)

C-3/4,01

E

>

1

, wlth some

E

such that O < E < .~ (We note

, then the error term in (4.24) is

.)

O(1)

Using partial integration in (4.24), we obtain

Differentiating with respect t o t we deduce

o(xlt~tRettu

1 ,

where we have used Cauchy's inequality to estimate the error term. Defining Z(U)

by Z(U) =

-= ttutl

, we have, f o r

t=O

,

105

QWIEMS OF ADDITIVE FUNCTIONS

From (4.25) we have g ( p r ) - f(pr)

<<

p-A for a l l

r

2

1

, which

implies, by a d d i t i v i t y ,

(4.29)

for n t n

.

From (4.25), we have

Re(R (pr))

f

, so

2

that

(4.30)

where the last estimate follows by p a r t i a l summation and t h e f a c t that

We now integrate (4.28) with respect t o u

from -3/4

to

using the same procedure a s the one which was used i n Chapter 3, 5 1 . (4.30) we see that the left-hand s i d e of (4.28) yields

0

, With

106

J.M.

DE KONINCK AND A. IVId

Integration of the right-hand side of (4.28) yields

-

H(O,u,l) ' xu logx utl 1 log2x

(

1'

t

... t

0(x-€)

-3/4

(4.20) follows from (4.31) and (4.32).

I

We can now easily prove

that H(0,O;l) = 1 and thus obtain the desired expression for A1

.

Theorem 4 . 3 we have Corollary 4.1.. For any fixed integer

where all the constants Ai

A1 =

1 (1 - -)

P

M21

are computable, and in particular

1

21og(l

pl rr=l n {

- -)1 P

- log(1

-P

1

P-r

*

From

QWTIFNI'S OF ADDITIVE FUNCTIONS

107

NOTES

Although condition (4.5) in the statement of Theorem 4.1 can be somewhat relaxed, it can easily be verified, and in addition, many c m o n additive functions (such as n ( n ) and u ( n ) ) satisfy it. The generality of Theorem 4.1 lies in the fact that g and f are not necessarily integer-valued. By imposing a condition m r e stringent than f(pr) namely f(pr) G(u)

E

k

C

2

keN

, we could obtain

F(u)

E

2

1

,

CktlLO,ll and

[O,ll , which would give correspondingly more terms of the form

aiz/(log log3:)'

in (4.6).

To see that

we use (2.1).

since

p

-2

This gives

converges.

P2-

The original version of Theorem 4.1 can be found in De Koninck C21, where the asymptotic formula (4.3) is proved. In Chapter 5 we shall prove a formula which further sharpens (4.3). It is interesting to note that one can also use an elementary approach (thus avoiding deep analytical results like Selberg's Lemma 2.1)

J.M.

108

DE KONINCK AND A. IVI6

to estimate sums of quotients of additive functions. If g and f satisfy conditions similar to those required in Theorem 4 . 1 , then it is proved in De Koninck

C3l

and if g and f satisfy the conditions of Theorem 4.2, then his elementary method gives

2" h-7 ( n ) - a (1 + A/logz+O(log-2r.

nsx

.

loglogz))

In the proof of Theorem 4 . 2 , we note that from ( 4 . 2 2 ) it follows that

Z(u)

and H(O,u,l)

are infinitely differentiable in C0,ll

, so

that integration by parts employed in (4.32) is justified. Toprove (4.26), wenote that for

C>O

and R e s > l ,

m

where H(s) = gives

1 h ( n ) n-' n=l

is absolutely convergent for Re s

>

1/2 . This

QUOTIWS OF ADDITIVE FUNCTIONS

109

which is obtained from Lemma 1.1, by observing the condition can be replaced by

) zI

5

A

IzI s 1

.

Note i n this connection t h a t t h e analogous estimate

c- 1 nsx holds f o r

0
, but

not f o r

C> 2

.

The reason f o r this is t h a t the

corresponding Euler product f o r C > 2

does not converge absolutely f o r

1t

Re s > 1 only, since

t

c22-2s t

. ..

converges absolutely f o r R e s > l o g C / l o g 2 > 1 . In case C = 2 t h a t f o r some s u i t a b l e constants

1 2n(n)

nSx

one can show

D and E we have

DEC h g 2 x

t

Ex l o g x

t

O(x)

.

A proof of t h i s formula may be found i n the paper of E. Grosswald c21.

This Page Intentionally Left Blank

CHAPTER 5 A SHARPENING OF ASYMPTOTIC FORMULAE

11. Introduction The purpose of this chapter is to sharpen some of the asymptotic formulae proven in the previous chapters. In particular, we improve the formulae

and

by introducing new leading terms. Our results will hold for certain classes of non-negative, integer-valued additive functions f for which we shall give sharp estimates of the sum

1'

nsz

.

l/f(n)

These estimates will be seen

to depend on two deep lemnas which give estimates for the sum

1zf(n) nsx

,

and which possess an intrinsic number-theoretic significance of their own. Both lemmas (due to H. Delanee) were originally derived with the purpose of estimating sums of the form

1

1

n % , f ( n ) =k

, where

k (21)

is a fixed

integer, and f belongs to a certain class of additive, non-negative and integer-valued functions. As corollaries to lemmas 5.1 and 5 . 2 we shall deduce sharpest known asymptotic formulae for the sums 111

J . M . DE KONINCK AND A. IVI6

112

1

1

nsx, w ( a )=k

and

c

1 .

a%, n ( n )-w ( n )=k

Because of the unifying principles underlying these lemas, namely, the convolution method and complex integration (carried out in detail in the proof of Lemma l.l), and because of our desire to keep the exposition as clear as possible, we consider it more appropriate to devote a chapter to the sharpening of the asymptotic formulae, rather than to have stated the best results in the earlier chapters. Before proceeding further we shall introduce the concept of a slowly oscillating function. A real-valued function f(x)

is called

slowly oscillating (or slowly varying) if it is positive and continuous and for every c > 0 ,

for x 1. xo

(5.1) Many functions that appear in error terms in asymptotic formulae for arithmetical functions such as logA J: , exp(C 10gl’~x)

, log log x etc.

are easily seen to be slowly oscillating. These functions possess a canonical representation of the form

where

and 6 ( x ) are continuous for x z x o

p(x)

lim 6 ( x )

= 0

every

>

so

x+-

.E

0

.

lim

p(r)

x+-

that the above representation yields L ( x )

A > 0 << xE

and

for

Slowly oscillating functions that will appear in the formu-

A SHARPENING OF AS-TIC

FORMULAE

113

l a t i o n of our theorems admit an asymptotic expansion i n t e r n of negative l o g l o g z ; that is, f o r every integer

powers of t a n t s A1,

. .. ,AM-l

such that

(loglogx)-l t A l ( h g 1 o g r ) - * t .

L(z)

where the 0-constant depends on M

g2.

there e x i s t cons-

M 2 1

(log log z) -M t 0( (log log z) -M-

.

The lemmas

Lemma 5.1.

Let f ( n ) be a non-negative, integer-valued additive

arithmetical function such t h a t f@) = 1 f o r every prime p p 2 0

3

, let

u,(p)

.

For every

denote the infimum of the s e t of r e a l numbers u

>

1/2

f o r which

(5.2)

if t h i s set is non-empty (and u0 (p) =

of a l l

p 2

0

the set E

f o r which

(finite or

e x i s t functions A, ( z ) Ao(0)

(5.3)

A1(0)

n-

...

t-).

1

<

o,(p)

otherwise).

Let E

be the s e t

l e t R > 1 be t h e supremum of

Then f o r every fixed integer N (z

I

t

0 there

< 1 such t h a t

, and

= AN(0) = 0

N

(-1 A3. ( z ) l o g - j z

t

J =O

where the 0-constant is uniform f o r zf ( n )

, and

m

, A1 ( z ) ,. .. ,A N ( z ) analytic on

z f ( n ) = z(logz)z-l

Proof.

t

Iz

I

< 1

O(l0g

-N-1

x))

,

.

is multiplicative since f ( n ) is additive,

therefore follows that f o r Re s

>

1

and f o r

12

I

p L E

(since

It

uo ( p ) < 1)

114

J . M . DE KONINCK

AND A. IVIC

where

W e now show t h a t the i n f i n i t e product i n (5.5) i s an analytic funct i o n of

and z

s

for

IzI 5 p

E

E

and Res > u l > o

(p)

.

To investigate

the convergence properties of i n f i n i t e products we note the following s i m ple result:

Suppose u,(x)

and v n ( x ) a r e two sequences of complex func-

tions defined on the same s e t A

, and suppose that f o r every

n

2

1 and

for x e B _ c A

where

un

and

(5.7)

vn

a r e positive constants such t h a t m

m

n=l

n=l

Then the i n f i n i t e product

is absolutely and uniformly convergent f o r for x

E

B

.

To see t h i s , we define w n ( x )

by

3: E

B

and i t s value is bounded

A

If

V > 0

115

SHARPENING OF ASYMPTOTIC FORMJLAE

is such a number that

then there e x i s t s a M

>

0

and

Vn 5 U

such t h a t f o r

Iz

I

2

Vn

5

U

for n

2

1

u

Since

we see t h a t f o r n

where

W,

V

= Me V;

2

1 and f o r x

+

MVn

, so

E

B

that (5.7) implies absolute and uniform con-

vergence of the product (5.8) f o r x

E

B

.

This product is bounded by

m

Numerating t h e sequence of primes

p1,p2,. . . , p , , .

..

we note that

the above r e s u l t holds f o r

provided t h a t (5.6) and (5.7) hold with n replaced by p Recalling t h e d e f i n i t i o n of

where

G

, we

have

.

,

116

J.M. DE KONINCK AND A. IVId

(5.10)

If

IzI 5 p

E

and Res 2 o1 > u

E

then, since zf@) = z

(p)

,

we have

and

since ul

>

1/2 and

1 ( 1 pf @ k ) p - k u l ) z

p

converges by ( 5 . 2 ) for u1 > 1/2.

k=2

Moreover

which means that

1 VP

P

<

+

-

for

121 5 p E

E

and Res 2 u > u 1

0

(p)

.

Therefore the product in (5.9) is absolutely and uniformly convergent in the region defined by

Iz

I

5 p E

E

and Re s > u1

the factors in (5.9) is an analytic function of

f is integer-valued and non-negative, tion of s

G(s,z)

s

.

Since each of

in this region, because

represents an analytic func-

in the same region. In particular, since R > 1 by hypothesis,

we find that for IzI s 1

where each g ( n , z )

is analytic on

IzI 5

1

.

117

A SHARPENING OF ASYMPTOTIC FOFMJIAE

From (5.4) we obtain

where

is t h e generalized d i v i s o r function defined by (1.10). Lemma

dz(m)

1.1 y i e l d s uniformly f o r

1zf(n)

(5.13)

n-

For From (5.5)

G(s,z) =

Iz

, we

I

5

= z

n

JzI

5

1g(n,z)

1

N

1 c i ( z ) logz-ix/n

n-'I

1 and f o r every n a t u r a l number

I

we have

k

(z) I

s1

.

have

n(1-zp-'t (i)pm2'

whence g ( p , z ) = 0 and

t

Ig(pk,z)I

implies that uniformly f o r

1 g(n,z)n-llogAn

nsx

Re z-N-1 O(1og

i=1

P

(5.14)

t

IzI

5

. . .) (1t zfS

k d(p ) = k

t

t

.fb2)p-2s

1 for

s 1 and every A

k

0 and

2

.

2

2 E

>

m

=

1g(n,z)n-llogAn n=l

t O(z

since

From (5.14) we obtain uniformly f o r

IzI s

1

logA z)

t

. . .)

This

0

,

,

118

J . M . DE KONINCK AND A. IVId

as follows: Writing

1

<<

since

1

nsx 1 / 2

1

1

t

nsx 1 / 2

x1/2<nsx

we have

Re z-N-1 log x s

Ig(n,z) I n - l

converges by (5.14) f o r

E

<

1/2

.

(5.14) simi-

larly yields

<<

x E-1/2

which proves (5.15) f o r

<<

E

logRe z-N-1 x a

.

< 1/2

Using

in (5.13), expanding

l o n z-i ( 1 - &) and using (5.14) and (5.15), we then

derive (5.3) with N

replaced by N-1

Every function A . ( z )

1 Di(z) c i ( z ) , where

i

3

t h e Di(z)

.

is thus seen t o be an expression of the form Is

are analytic functions, and the ci(z)’s

a r e analytic functions, as defined i n Lemma 1.1, which s a t i s f y c i ( 0 ) = 0 for i = 1 , 2

,... .

Thus every A . ( z ) 3

is an a n a l y t i c function on

IzI 5 1

119

A SHARF'FNING OF ASYMPTOTIC FORMULAE

satisfying A . ( O ) = 0 d

.

This completes t h e proof of Lemma 5.1.

Although the study of

n*,

1

1 is outside t h e scope of t h i s

w (n)=q

chapter, we wish t o point out t h a t Lemma 5.1 can be used t o obtain a sharp estimate of t h i s sum. I t is clear t h a t the lemma may be applied t o

1 z @ ( ~ ) is

nsc

a polynomial i n z

with the c o e f f i c i e n t of

the number of positive integers not exceeding r where q

is a fixed non-negative integer.

1

(logzy

f ( n ) = ~ ( n ). Thus

n=O

zq

equal t o

f o r which w(n) = q

,

Using

(log log .)n

2/n!

and Cauchy's c l a s s i c a l inequality f o r c o e f f i c i e n t s of a power series, we deduce, a f t e r equating coefficients of Corollary 5.1.

(5.16)

w

4

n*,w(n)=q

+ where each P . ( t ) 3

and N

i n (5.3), the following

For every fixed integer q z l

1

(r)

zq

O(a:

1=

N

1

j=o

rP.(log logr) 1 0 g - ~ - L

log-N-2 3:. (log logx)q-l)

,

is a polynomial i n t of degree not exceeding q - 1

,

is an a r b i t r a r y non-negative, fixed integer. Formula (5.16) is the sharpest known version of a c l a s s i c a l r e s u l t

of E . Landau. I t i s clear t h a t a similar formula holds when w(n) is

J.M.

120

DE KONINCK

AND A. IVId

.

replaced by n(n)

Lemma 5.2. Let f ( n ) be a non-negative, integer-valued additive arithmetical function such that for every prime p , p ( p ) f(p2)

1

For every

.

For every

p 2

0 let

p 2

0 and

0 define a multiplicative function h D ( n ) by

be the infimum of the set of real numbers

u,(p)

u > 1/3 for which

(5.18)

if this set is non-empty (and u be the set of

p2

0 for which

1

otherwise).

+m

, and let

< 1/2

+m)

that F ( 0 ) = 6/n2 , A o ( 0 )

(5.19)

u,(p)

=

Furthermore let I R>1

. Then for every.fixed integer A . (2) ,... , A N ( z ) which are analytic on

mum of I (finite or functions F ( z )

(p)

=

...

AN(0) = 0

z F ( z ) +z1’2 logz-2z(

ncz

be the supre-

N > 0 there exist

I

(2

, and

N

1 A j ( z ) log-jr t O(log-N-lz))

j=O

where the 0-constant is uniform for

1.~1 s

Proof. We begin by noting that

1

.

u,(p)

R

and the A . ( z ) ’ s that 3

appear in both lemmas are not necessarily the same functions. For Res > 1

s 1 such

and

IzI

5 p E

I (since

o,(p)

<

1/2)

A SHARPENING OF ASYMPTOTIC FORMULAE

where

so that

Thus gz(n)

z

-1.

For exp(Z(s))

d n

.

is a multiplicative function which s a t i s f i e s

and a l l

for a l l primes p g,(p2)

f gz(d)

zf(n)

(5.22)

k> 1

.

In p a r t i c u l a r g,@) = 0 and

W e thus have

(s

- 1)

, we

~ ( s )

define H(s,z)

by

in order t o obtain (5.26)

G(s,z)

= H(s,z) ( s - 1/2)l-'

W e now give a brief sketch of the proof.

. I t consists of

121

J.M. LIE KONINCK AND A. IVIi

122

(a) proving that the function V ( s , z ) .s

and z

for

Iz

I

for < 1

12

I

5 p

, since

E

R>

I

and for Res > u l

is an analytic function of (and, in particular,

> u,(p)

1 by hypothesis),

(b) using the inversion formula ctiT

X

with c > 1/2

, and

(5.28)

to obtain

for some suitable functions $ z and Qz

, where Gz(x)

is not G(s,z)

of (5.26),

1zf(n)

(c) recovering

nlz:

and

from Gz(x)

(d) showing that the functions F,A,,

by a convolution argument,

., .,A N

satisfy the conditions

specified in the lemma. We begin with (a).

From the definition of h

P

and from (5.23),

we have that

for

s

P

.

m

Thus

1 g,(P

k=2

) p -ks

converges absolutely and uniformly

A

for I z I s

p E

SHARPENING OF ASYMPTOTIC F O W

I and for Res 2 ul > u,(p)

123

, since, by hypothesis,

m

converges. Therefore all factors of the infinite product

k=3

(5.31)

P

k=2

are well-defined for these values of s and

, and the general term may

z

be written as

where

Furthermore,

Since the sum in (5.18) is bounded we have

Thus

1 U2P

<

+ m

P

, because

Furthermore, for

so that

1V

P P

<

+ m

.

ul > 1/4 and

Iz1

s

p

1 ((1

P

and Res

2

+ p)p

-201 )2 <

+m

.

u1

By the observation made in the proof of Lemma 5.1

J.M. DE KONINCK AND A. IVId

124

t h i s implies t h a t the product i n (5.31) i s uniformly and absolutely converk Noting that g z ( p ) is analygent for I z I 5 P E I and Res > u (p)

.

t i c because f

is integer-valued and non-negative, we see t h a t V ( s , z )

is a l s o analytic f o r these values of IzI

1

5

, since

and z

s

, and,

in particular, for

R > 1 by hypothesis.

In applying the inversion formula (5.27) it should be noted that which appears i n (5.25) is analytic i n any open neighborhood of

Z(s)

, that

1 which is f r e e of zeros of

s

~ ( s )

Z(1)

= 0

and t h a t

.

exp(Z(s)) = (s - 1) ~ ( s ) To obtain (b) we replace the contour of integration

Cc

- iT , c + i T 1 by the contour used i n the proof of Lemma 1 . 2 ,

with the point

1 replaced by

s

s =1/2

.

(This is done because t h e

generating function now has a singularity a t s = 1 / 2 i n Lemma 1.1.)

From (5.25), from t h e product representation given i n (5.24)

, and

for

V(s,z)

for

IzI 5 1

,

and not a t s = 1 as

It1

from (1.14), it is e a s i l y seen t h a t

2

tl and u

2

1/2 - a / l o g ( I t l ) , a > O

.

The evaluation of the integral appearing on the right-hand side of (5.27) i s analogous t o the evaluation performed i n the proof of Lemma 1.1, and therefore we omit the d e t a i l s . Gz (t)d t

, the

To recover Gz(z)

same simple Tauberian arguments used i n Lemma 1.1 are again

used t o yield (5.29), where, f o r some c > 0 IzI s 1

(5.33)

from

, we

have uniformly f o r

A SHARPENING OF ASYMPTOTIC FORMULAE

125

and

(5.34)

t

For

Iz

I

s

1

1 2ni

)

stl

~(s,z)(s-1/2)~-~. .-. 5

ds

.

, @z(x) is an infinitely differentiable function of

x whose derivatives are obtained by differentiating under the integral

sign, and

is the circle

y,

q

Is

- 1/21

= P

minus the point s = 1/2 - r

,

is as defined i n Lemma 1.1.

We now prove (c) by using (5.22) and the convolution method. If y

y(x)

is a function satisfying 1

where F ( z )

<

y < x

, then

is defined by

(5.36)

Using (5.29) we have m

m

m

126

J . M . UE KONINCK

AND A. I V I d

m

m

Y

Y

and therefore m

From (5.29) it follows that

and we thus have

The substitution t

xr

X/V

$i(X/t)

1

and thus

yields X

q v )

t-l dt =

Y

0-l

dv

,

A SHARPENING OF

127

ASYMPTOTIC FORMULAE

Using (5.38) and (5.37) i n (5.35) and r e c a l l i n g (5.29),

1z

(5.39)

n-

Y("

m

~ ( =~ x) ~ ( z-) x j + ; ( t ) t - ' d t - x

I +;(x/t)(t- C t l ) t - 2 d t

J

1

5

+;(t).(t - C t l ) t - 2d t .

The main d i f f i c u l t y c o n s i s t s of evaluating 1

W e first d i f f e r e n t i a t e and then i n t e g r a t e (5.34) t o obtain

(5.40)

z

1$ L ( t )

t

-1

1-z 1 / 2 - u H(l/Z-u,z)u x du 1/2 + u

dt

For y > 0 and R e s > 0

, we

define S(y,s)

by

(t - Ctl) t-s-ld t

.

m

(5.41)

S(y,s)

Y Noting that

€or Res

>

1

, we

obtain

we obtain

J . M . DE KONINCK

128

5

(5.42)

AND

A.

IVI6

(t - Ctl) t--l d t = ( s - 1 ) - l - c ( s ) s-l - S(y,s)

.

1

Using analytic continuation, we can show that (5.42) holds f o r y Re s > 0

.

Furthermore define TZ (x,y) sinr z -

T2(z,y) =

(5.43)

H(1/2

>

0 and

by

- u,z)

s(y ,1/2

- u ) u 1 -2 z 1/2-24

du

r

- 22ai

J ~ ( s , z ~) ( y s, ) ( s - 1/211-'

,

xs ds

yr

for

121 5

1

, z > 0 , y > 0 , and

independent of

r

0 < r < r,

, where

.

Noting that

is bounded and t h a t S(y,s)

H

i 1

H(1/2 - u,z) S(y

r

<<

<< y

-Re s

is

, we

have

B

e-'

, 1 / 2 - u ) u 1-zx1/2-u dU rllo x/y

(x/y)1/2-udu<< ( ~ / y ) l / ~(logx/y)-'

r log X/Y

r

where u = v(logr/y)-',

the integral on yr

and

r

(logx/y)-'.

'r

which implies that uniformly i n

Iz( s 1

We a l s o have that

dv

A

SHARPENING OF ASYMPTOTIC FORMULAJi

129

From (5.34) we have rl

$p)-

(5.45)

1-2

-1/2-u d u

r

+

&J

~ ( s , z ) ( s - 1/21 1 - 2 xs-1 ds

,

Yr

xr

which, combined with (5.42) and (5.43), gives f o r

z

t > 0

$ g ( x / t ) (t- Ctl) t - 2d t

1

1

+1 2ni

H(s,z) ((el)-'- 3(s) s-l) (s

- 1/2)1-2zsds + T 2 ( z J z / g ) .

yr Combining this with (5.40) we obtain

(5.47)

H(1/2-u,2)

$;(x) = -

5(1/2 - u)(1/2

- u ) -1 u 1-2 21/2-u du

r

+1 2ai

J

~ ( s , z )~ ( s s ) - l ( s - 1/21 1-2 xs ds

From (5.39) and (5.47) it f o l l o w s that

.

J . M . DE KONINCK AND A. IVI6

130

Using (5.33) and (5.43), we have uniformly i n

\ nz(t) m

(5.51)

x

IzI s 1

m

t - 2 d t << zexp(-c log1/2y)

Y

J t-3/2dt

Y

and

From (5.30) and the f a c t t h a t k f o r which h ( p ) 0 and h (p ) 5 2 P

P

Therefore from (5.48),

we obtain, uniformly f o r

121

(5.49),

s 1

h

P

for

is a m u l t i p l i c a t i v e function k 2 2

and

(5.50), (5.51),

IzI s 1 we deduce

(5.52),

and (5.53)

131

A SHARPENING OF ASYMPTOTIC FORMULAE

Defining y(x)

f o r some constant

D

>

0

.

Finally we evaluate tion

we obtain

by y = zexp(-c log1/2x)

$,*(x) from which (d) w i l l follow.

$;(x)

is evaluated exactly a s the function

$:(x)

The func-

(defined by (1.20))

was evaluated i n Lemma 1.1. Thus we obtain

In the preceding formula the IzI

5

1 so that the A . ( z ) ’ s 3

and furthermore A . ( O ) 3

at

z

- n (n = 0,l

y...)

3

( A . ( z ) = D.(.z)/i-(z-j-l)) 3 3

for j = 1,2,

0

are a n a l y t i c functions on

O.(z)’s

...

since

r(z)

1 Igz(pk)Ip-ka

PJk values of

s

Re s =

(I

> 1/2

and z

(5.55)

From (5.36) we then see that

(5.56)

has simple poles

.

W e note t h a t by (5.18) and (5.30) each sum uniformly bounded f o r

are also analytic,

and

Iz

I

5

1

, and

is

that f o r these

132

J.M. LE KONINCK AND A. IVI6

is an analytic function for

1 , and in particular that

IzI 5

which ends the proof of Lemma 5.2. An important application of Lennna 5 . 2 is to the so-called “R6nyi’s

problem”. This consists of sharpening the formula

c

(5.57)

7 2 3 , n (n)-w

-

1

(n)=q

dqx

,

as x-f- , where q is a fixed non-negative integer and the left-hand side represents the number of integers not exceeding x for which n(n)

- w(n)

q for a fixed non-negative integer q

, and

d

9

is the

density of these integers. It is known that d is always positive and 4 that

(5.58)

We note that do is the density of the squarefree integers, and thus the above formula gives the well-known value do that Lemma 5.2 can be applied to f ( n )

n(n)

-

w(n)

m

l0gZx

1

n=O

(loglogx)nzn/n !

6/n2

.

.

It is clear

Recalling

,

using Cauchy’s inequality, and then equating coefficients of zq in (5.19), we obtain the following sharpest known formula (due t o H. Delange) for R6nyi’s problem:

A SHARPENING OF ASYMPTOTIC FORMUM

Corollary 5.2.

(5.59)

n a , fi

For every fixed integer q

2

133

1

N

c (n)

-w

1 = d z +z1'2 1 P .(log log z) log-j-I5 9 (n)=q j=I 3

t o [ x 1/2 . (log log 2p-l logN+23:

I

is a polynomial in t of degree not exceeding q - 1

where each P . ( t ) J

,

and N is an arbitrary positive fixed integer.

The theorems

13.

Theorem 5.1.

Let f ( n ) be a non-negative integer-valued additive

arithmetical function such that for every prime p , f(p) k

f(p )

for every k

< Ck

fixed integer N

2

2

2

and for some fixed

C >

0

1

.

, and

Then for every

1 there exist computable constants el, ...,eN such

that

1'

(5.60)

n-

l/f(n)

elzL1(z)+

...+eN zLN(z)log

1-NxtO(zlog-Nx)

,

where each L .(5)( j = 1,...,N) is a slowly oscillating function asymptotic 3

to l/loglogx

.

Theorem 5.2. Let f ( n ) be a non-negative integer-valued additive arithmetical function such that for every prime p , f(p) = 0 , f ( p 2 ) = 1 and 0

<

f(pk)

<

Ck for every k

every fixed integer N such that

2

2

3

and some fixed C

>

0

.

Then for

1 there exist computable constants e o , e l , ...,eN

DE KONINCK AND A. IVId

J.M.

134

. ,N )

where each L .(x) ( j 1 y . . 3

to l/loglogz

.

,

o(x1/210g-N-1x)

t

is a slowly oscillating function asymptotic

Theorem 5.3. Let f(a) and g ( n ) be two non-negative integervalued additive arithmetical functions such that for every prime p

, f(p)

= g(p) = 1

for some

.

C > 0

, and , b3.

Ck

, g ( pk )

< Ck

for every k 2

2

2

and

1 there exist compu-

ly...,N) such that

(j

1,. . .yN). is a slowly oscillating function asymptotic

where each L .(x) ( j to l/loglogx

<

Then for every fixed integer N

table constants a j

3

f(pk)

.

Proofs. We note that L . ( z ) may represent a different slowly 3

oscillating function in each theorem. If f

satisfies the hypothesis of

Theorem 5.1, then Lennna 5.1 may be applied and so ( 5 . 3 ) holds uniformly in

z

for

121 5

1

c Pf@ p,kt2 for every u If u

t

2/3

>

1/2

.

In order to see this we note that if

k )P-ku

5

1

p , k22

c -u k

(P p

provided that p c p -u

1

<

=

c

-0

< P

c2-2/3

1

1 (P c p -0) 2 /(l-p c p -u ) < t -

P

1 - B for some fixed 0 5 B < 1

, then P p

p 2

5/6

.

A

for

p <

(g

22/3)1/c

135

SHARPENING OF ASYMPTOTIC FOFWULAE

.

Since

C

is fixed t h i s last number is greater

than unity, and therefore Lemma 5.1 applies, since R

>

1

.

W e note that f ( 1 ) = 0 ; the only other possible values of f o r which f ( n ) = 0 a r e t h e "square-full" numbers not exceeding z

nsx

, the

.

number of which is O ( Z ~ / ~ ) (A natural number is square-full if it is of al ai t h e form n = p1 ...pi where al 2 2 , . . .,ai 2 2 .) If f ( n ) * 0 , then f(n)

2

1 since f

is integer-valued and non-negative.

sides of the equality (5.3) by z that for

1

since

.

z

A .(z)/z

3

I z I 5 1 we have uniformly i n

Let to

and s e t t i n g B . ( z )

Dividing both 3

, we.see

z

be real and integrate (5.63) over z

from

E (z)

z

-2/3

From t h e left-hand s i d e of (5.63) we obtain

( E ( x ) ) ~ ( << ~ )~ ( z ) i f

f(n)

*

0

.

When we integrate t h e right-

hand s i d e of (5.63) we see t h a t t h e i n t e g r a l of the e r r o r term i s bounded bY

The main terns on the right-hand s i d e of (5.63) w i l l y i e l d integ r a l s of the form

(5.66)

log'z.

zlog.-j-'z. E (XI

1

d z = z l o g-j-1 z . B . ( z ) logZz.dztO(z1'3) 0

3

136

AND

J . M . DE KONINCK

where we have used t h e f a c t that analytic on

IzI

1

5

A

.(O)

0

3

.

is

Hence B . ( z ) = A . ( z ) / z 3

3

and thus, f o r x s u f f i c i e n t l y large,

For every fixed integer M z 1

(5.67)

A. IVId

log’x. dz =

H.(x) = 3

B

, integration by

p a r t s gives

.(2) logZ x log log w

0

(-l)M

t

(log log x)Mtl

B.(1) l o g r - 3

log l o g x

z

1

, e x i s t s since

B.(z) 3

L.(x) = H.(x)/(B.(l)logz) 3 3 3

l/loglogx loglogx

.

O[

4

(l0gZx

(log log Z y + l

(log log x ) M

logx (log log x)Mtl

t h e k-th derivative

j

IB!M+l)(Z)

. . . + (-1) M-1 Bj(M-l) (I)l o g x

t

+o[ where each B ( k ) (1)

I:+ 1

B Y ’ ( z ) logZ 32

( k = 1y . .

is analytic on

IzI

5

),

.,M) 1

.

of

B .(z) 3

at

Thus

is a slowly o s c i l l a t i n g function asymptotic t o

which admits an expansion i n terms of negative powers of From (5.64), (5.65), (5.66), and (5.67), Theorem 5.1 follows

with ci = Bi-l(l)

.

functions

which appear i n (5.3) can be e x p l i c i t l y calculated, and

A .(z) 3

From the proof of Lemma 5 . 1 it is seen that the

thus a l l the constants ci’s a r e computable.

A .SHARPENING OF A S W O T I C F

O

137

W

To prove Theorem 5.2 we use L e m 5.2 and the same method of proof as was used in that lemma. We recall that the hypothesis f ( P k ) < Ck

assures that R

Y

1 in Lemma 5.2 so that (5.19) holds uniformly on

1 whenever f satisfies the hypothesis of Theorem 5.2. Since

IzI s

fbk)>

0 for k

t

2 we have that f ( n )

0

if and only if n is square-

free, so that

where c

>

0 and 6 ( r ) = (10g~’~x)(loglogx)-1/5 (see Walfisz C11).

note that the weaker error term O(z1/210g-Az)

for any A

>

0

We

in the

above formula would have been sufficient €or our needs in what follows. Therefore dividing both sides of the equality (5.19) by z we obtain uniformly for

IzI s

1

where once again B3.(2)

A .(z) / z 3

.

In (5.69) we integrate over z E(Z)

=

2- 2 / 3

to 1

where

z

is real, from

as was done in the proof of Theorem 5.1. The left-

hand side of (5.69) becomes

As was seen in (5.66) ,

J . M . DE KONINCK AND A. IVIC

138

]

(5.71)

B . ( z ) log’x 3

. dz

=

l o g z x . dz

t O(E(S).)

,

E

where L . (x) is a slowly o s c i l l a t i n g function asymptotic t o 3

l / l o g log x

The integral of the e r r o r terms on t h e right-hand side of (5.69) from E(X)

x-*l3 t o 1 is

since the f a c t that F ( 0 ) = 6/71’ continuous on C 0 , l l

.

Combining

implies that

e

j

= Bj-l(l)

for j

2

1

eo =

J

(p1,p2)

6/71’) z - l

-

( ~ ( z ) 6/71’) z-l dz

is

,

and

0

.

W e begin the proof of Theorem 5.3 by defining every p a i r

-

(5.70), (5.71), (5.72), (5.73), and I

(5.74), we obtain Theorem 5 . 2 with

(F(z)

ao(p1,p2)

, for

of non-negative r e a l numbers, t o be the infimum of

the s e t of r e a l numbers a

>

1/2

f o r which

.

139

A SHARPENING OF ASYMPTOTIC FORMULAE

(5.75)

i f t h i s set is non-empty (and u

(p

0

p ) =

+ m

1’ 2

set of p a i r s of non-negative r e a l numbers We now assume that

1 and

p1 2

otherwise).

f o r which

(p,,p,) p2 2

Let

.

1

be the

E

uo(p1,p2)

<

Then, from t h e hypo-

theses of Theorem 5.3, we have

L75

c p,

Ck C k -ka 2c 2c P 1 P * P = c p 1 P2

P

k22

f o r every u If

(I

t

>

2/3,

1/2 p1 <

P

pc 1 p 2cp

whenever

9 21 / 3 ) ?/C (m

,

C C p1p2

100

Since m 2 ‘I3 > 1

81

<

, there

-u

p,

5

-20

1- B

(A

<

s -100 p

exists a

p

1<+m

f o r some fixed

.

u

p >

O< B< 1.

and therefore

Z1’3)1/c

81

2/3

c c -u

/(l-PlP2

1 such that

(p,p)

E

E

.

By using the same methods a s were used i n the proof of Lemma 5.1, we have

that

P

is a well-defined function for which is analytic i n

s

.

k=1 121 5

3

,...,N )

(j = 1

IuI

P

,u

Res

>U~(P,P)

Proceeding as i n Lemma 5 . 1 (or Delange

obtain t h a t f o r every fixed integer

where A . ( z , u )

,

p

N

2

0

is an a n a l y t i c function

,

C21),

we

1.

J . M . DE KONINCK AND A. IVId

140 )zJ< p

,

JuI < p

continuous for J z l 2

lul

p

such that A . ( z , 0 ) = 0

5 p

3

and (5.77)

R (z, z, u)

0 (z (log z)

where the 0-constant is uniform for

IzI

Rezu-N-2 )

and

5 p

Y

IuI

We now differentiate (5.76) with respect to z

5 p

.

use Cauchy's

inequality for the derivative of an analytic function to estimate the error term in ( 5 . 7 7 ) , and then set z = 1 Dividing by

u

which is possible since

we obtain uniformly for Iu I

p >

1

.

5 p

where

Thus both B . ( u ) and 3

since

p z

C.(u) 3

are analytic functions of u on

1 and A.(z,O) = 0 3

We recall that f(1) of n for which f(n)

IuI

L

1 ,

. 0 and that the only other possible values

0 are the square-full numbers. This yields

Combining this estimate with (5.78) we obtain

A SHARPENING OF ASYMPTOTIC FORMULAE

141

We now proceed as in the proof of Theorem 5.1, taking integrating (5.79) over

from E(x)

u

x-*l3 to 1

.

u

real and

The integral of

the left-hand side of (5.79) is

since g ( n )

<<

n(n) , f ( n )

2

1 if f ( n )

z

0

and

The integrals of the leading terms on the right-hand side of (5.79) are handled in exactly the same way as in the proof of Theorem 5.1, and the integral of the error term is O(x log-N x)

, so that after collecting

terms we obtain the conclusion of Theorem 5.3.

54.

Applications and remarks All three theorems of this chapter seem particularly well suited

for application to the well-known additive functions w(n) and n ( n ) k

k

Since w(P ) = 1 and n(p ) = k

, we see that both

the hypotheses of Theorem 5.1, that n - W Theorem 5.2 and

w

and

w

.

and n satisfy

satisfies the hypotheses of

satisfy the hypotheses of Theorem 5.3. As

immediate corollaries of our theorems we obtain the following improvements of Theorems 2.5, 2.6, 4.2, and the new result (5.83) (we note that the ails and the .L.(x)'s J

functions).

do not necessarily denote the same numbers or

DE KONINCK AND A. IVId

J.M.

142 Corollary 5.3.

For every fixed integer N

1 there exist compu-

L

table constants al = l,a2,.. . ,aN , and slowly oscillating functions

.. ., L ~ ( Z )

L~ (2) ,

(asymptotic to l/log log I) which admit an expansion in

terms of negative powers of loglogx

1'

(5.81)

nza

l/w(n)

, such that

... taBLN(x)

alxLl(z) t

logl-NxtO(xlog-Nx)

Corollary 5.4. For every fixed integer N table constants al

l,a2,

...,aN

...,

~ ~ ( z ) , ~ ~ ( x(asymptotic ) to

1'

nsx

l/a(iz)

alzL1(z)

l/loglogx) which admit an expansion in

t

, such that

... taBLN(z)

logl-NstO(zlog-Nz)

Corollary 5.5. For every fixed integer N table constants ao,al,...,aN

. . .,LN(x)

L1 (z) ,

c'

1 there exist compu-

2

(asymptotic to l/log log x) which admit an expansion in

-

l/(Q(n)

n-

Corollary 5.6. table constants al functions L~ (z),

.

and slowly oscillating functions

terms of negative powers of loglogx

(5.83)

1 there exist compu-

and slowly oscillating functions

terms of negative powers of loglogx

(5.82)

L

.

=

w(n))

, such that

= a x t a Z " ~ L ~ ( h~g )- l x t 0 1

For every fixed integer N

L

...

1 there exist compu-

1,a2,...,a ,b ,b *,..., bN , and slowly oscillating

.. .,~~(x) (asymptotic to

l/log log z) which admit an

expansion in terms of negative powers of loglogx

, such that

143

A SHARPENING OF ASYMPMTIC FOFMILAE

The constant a.

of Corollary 5.5 is not difficult to evaluate.

Inversion of (5.23) gives

f ~ ( d z) ~

g,(n)

so

d n

( ~ ' ~, )

that

which, by using (5.56) for f

Expanding F ( z )

= s2-w

, yields

in a power series we see, by the same reasoning m

that led to Corollary 5.2, that F ( z ) = (5.57).

This also proves (5.58).

1d

q=o

zq

, where

d

9

is given by

, we have

Noting that do = 6/n2

In concluding this chapter we mention that Theorem 5.1 can be generalized further by obtaining sharp estimates for

1'

n*

l/(f(n)f

(r E N

fixed) by using the method developed in Chapter 2. Furthermore, Theorem 5.2 can be modified for non-negative integer valued functions f such that f(p) =

.. .

f(pr-')

= 0

, f(pr)

which then leads to (5.61) with

= 1 and

0 < f ( P k ) < Ck

replaced by xl"

for k r r t 1

.

,

J . M . DE KONINCK AND A. I V I C

144

NOTES

The r e s u l t s of t h i s chapter a r e obtained from Lemmas 5.1 and 5 . 2 , which were proven by H. Delange 121 and C31. asymptotic expansions of the sums

1zf(n)

H i s deep r e s u l t s concerning

a r e the sharpest known, and

n<x

are presented i n somewhat greater generality than our lemmas. pose the range

Iz

to

1

from

E(Z)

I

.

For our pur-

l s u f f i c e s , since we have subsequently integrated

5

Lemma 5.1 is dependent on Lemma 1.1 (whose proof i s

again based on Delange "21), while the proof of Lemma 5.2 follows more closely that of Delange C3l. coefficients of

In Lemma 5.1 we have t o estimate the sun of

(<(s))'U(s,z)

sum of coefficients of

, and

i n Lemma 5.2 we have t o estimate the

3(s) ( 3 ( 2 ~ ) ) ~ - ~ v ( s, , zwhere )

a r e absolutely and uniformly convergent f o r Res > 1 / 3 +

i n which z 121 6

1

E

respectively.

I

1

, Re s 2

1/2

and

t E

We a r e thus faced with a convolution problem

i s not an integer, but an a r b i t r a r y complex number s a t i s f y i n g

, thereby

and s = 1 / 2

/z

and v ( s, z)

U(s,z)

producing an algebraic singularity a t the points

respectively.

s

1

This i s why a complicated contour of integra-

t i o n (introduced i n Lemma 1.1) w a s used. The above mentioned papers of H. Delange contain many results, re-

ferences and generalizations concerned with sums of the form where k

is fixed, and f is an additive function.

ginal paper concerning the problem of estimating R6nyi [I]; f o r the special case where k

1

,

For A. R6nyi's o r i -

c

~SZ,R (n)--w

, see

1 1 n
Schwarz C21.

Slowly o s c i l l a t i n g functions represent a class of functions t h a t naturally arises i n many branches of mathematics.

They were introduced by

J. Karamata C I J , who discovered the canonical representation f o r these

145

A SHARPENING OF ASYMPTOTIC FORMUM3

functions given i n 5 1 .

A comprehensive account concerning slowly o s c i l l a -

t i n g functions is t o be found i n E. Seneta's recent monograph C11.

A proof of Landau's formula f o r

w

4

(x) =

1 1 is t o be found nsx, w ( n )=q

i n E. Landau ClJ; f o r t h i s problem i n a more general s e t t i n g see Knopfmacher [ I ] , Chapter 6, 5 2 . The theorems of t h i s chapters a r e o r i g i n a l , and may be found i n the forthcoming paper of De Koninck and I v i c [21.

For the sake of brevity

we did not give a l l possible generalizations and applications of our k

results.

is s u f f i c i e n t t o ensure t h a t

The condition f(p ) < Ck

R > 1

when one applies Lema 5.1 o r 5.2, and, though t h i s condition could doubtl e s s l y be somewhat relaxed, it is an easy one t o verify and furthermore it covers t h e i n t e r e s t i n g cases of "small" additive functions such as w ( n ) and n(n)

.

"Square-full" numbers are a special case of the more general 'powerful" numbers.

For a fixed k

E

N

one may define the set of powerful

numbers G(k) as G(k)

k { n c N I (pln) => (p I n ) )

Square-full numbers a r e then j u s t

G(2)

.

.

I f we s e t

and

then A k ( x )

i s the number of powerful numbers n

i n G(k) t h a t do not

146

J.M.

exceed x

. m

Fk(s)

where G

Since for Re s > l / k

1 f,(n) n-' n=l k

(6)

DE KONINCK

1 (1 + p - k s

AND

A. IVId

we have

tp-(ktl)s

P

t

... )

= c ( k s ) G,(s)

has the abscissa of convergence equal t o l / ( k t l )

, the

con-

volution method inunediately yields

and

Powerful numbers were first investigated by P. E r d k and G. Szekeres CIJs where the above formula w a s proven.

Much sharper asymptotic for-

mulae f o r A (x) can be obtained by more i n t r i c a t e methods; f o r these k methods see P.T. Bateman and E. Grosswald C11, and Ivi; C3l.

CHAPTER 6 RECIPROCALS OF "LARGE" ADDITIVE FUNCTIONS

11. Introduction In Chapter 2 we studied asymptotic formulae for the sum

1'

n-

where f ( n ) was a "small" additive function belonging to the class

l/f(n), Su

of

Definition 2.2. Broadly speaking, one could say that a general "small" non-negative arithmetical function f ( n ) is a function for which

as

z

-+

, where

L(z)

is a slowly oscillating function. This implies

that one can think of L ( z ) as the "average order" of f ( n ) in a certain sense. For every

E

>

0 we have

which means that the average order of f

is small. On the other hand,

there exist non-negative additive functions f for which (6.1) does not hold, and furthermore for every D

>

0 one can easily find a non-negative

additive function f satisfying

(6.3)

147

1 48

J . M . DE KONINCK AND A. I V l 6

which means t h a t f must be i n some way very large. t o be the average order of

More precisely, one can define g ( n ) f(n)

if

(6.4)

as z

-f

m

, where

t i a l etc.).

is a "well-behaved" function (polynomial, exponen-

Thus we can say that f ( n ) i s "large" i f there is an

such t h a t f o r n

where g ( n )

g(n)

2

E

> 0

no(€)

s a t i s f i e s (6.4).

For our purposes we w i l l formalize the concept of "large" additive functions by considering a class h'

, which

contains the most interesting

"large" additive functions t h a t we have i n mind. Definition 6.1. let (6.5)

(6.7)

and

For K

fixed,

y >

0 and

6

a fixed r e a l number,

149

LARGE ADDITIVE FUN(TI0N.S

The class H of large additive functions is defined to be the class of all possible functions f , F and F1 obtainable by varying K, y

>

0 and 6

.

Our definition of class H

includes for K

= y =

1,6

0 the

functions

(6.10)

and

(6.11)

The functions f3

, B and

B1

are of great intrinsic interest.

For example, for a fixed integer rn the number of solutions of B ( n )

=

m

is the number of partitions of rn into primes (not necessarily distinct), the number of solutions of f3(n) = rn , p 2 ( n )

=

1 is the number of parti-

tions of rn into distinct primes, while the number of solutions of Bl(n)

rn

is the number of partitions of m into powers of distinct pri-

mes. The function ~ ( n )is the additive analogue of the multiplicative function a ( n )

fl

p

, whose sum of reciprocals was investigated in

P In

Chapter 1. The average order of

~ ( n )and B(n)

(in the sense of (6.4))

is a2n/(610gn) ; this can be seen from the asymptotic formulae

150

J.M. DE KONINCK

AND A. IVId

(6.12)

and (6.13)

Our goal i s t o obtain estimates f o r

where f

,F

i f and only i f

and F1 belong t o H n

1

, and

(note t h a t

the same f o r F

f(n) 2 0

and f ( n ) = 0

and F1 ) . The techniques

based on the properties of the generating series

or

which successfully worked i n previous chapters, seem t o be of no use here. The d i f f i c u l t y l i e s i n the f a c t that there is no obvious way ( i f any a t a l l ) t o factor out a power of the zeta function from the generating s e r i e s .

W e therefore abandon t h e approach v i a Dirichlet s e r i e s and proceed instead with investigations of a more elementary nature.

W e shall not be able t o

obtain asymptotic formulae f o r sums of reciprocals of functions belonging

t o H , but only good lower and upper bounds f o r these sums. Estimates furnished by Theorem 6.1 extend, of course, t o

B

,B

and B1

, and

53 we give estimates f o r sums of quotients of these functions, and an

in

LARGE ADDITIVE FUNCITONS

151

asymptotic formula for

as well.

Bounds for sums of reciprocals

12.

Theorem 6.1. Let f , F and F 1 be functions belonging to class H

of Definition 6.1. Then there exist two positive constants 0

< el < c2

such that if

(6.14)

(6.15) then (6.16)

(6.17) and (6.18) where c1 , c2 and the <<-constantsdepend only on h of Definition 6.1. Proof. We begin by first establishing the lower bounds. Let

(6.19)

Ak

( n I ( n < z )A

(p2(n)

1)

A

(p(n)

5

z”k)l

,

152

J.M. DE KONINCK AND A. IVId

where p ( n )

denotes t h e largest prime factor of

large and k = k ( x ) of

k

<

logx

n

w i l l be suitably chosen.

-

, as

y

+

there are at l e a s t

u

(6.20)

3k~”~/(4logx)

primes not exceeding xlIk , which means t h a t

(6.21)

1

nEAk

provided t h a t

k

1

U U (U - 1) (k) =

.. .(U - k t 1)

2

k!

satisfies

(6.22)

U

- k

f o r x s u f f i c i e n t l y large.

(6.23)

t 1.2

If

k

(k/2)k

>

2

2U/3

6

k!

, then ,

which cmbined with (6.20) and (6.21) gives

11

(6.24)

neAk

For n

(6.25)

E

Pk

we have

2

x l o g -k x

.

n n

E

i s a product Ak

, since

is square-free.

u 2 ( n ) = 1 is equivalent t o the condition that n ~ ( y ) yllogy

is s u f f i c i e n t l y

If

, then

different primes each not exceeding xlIk

the prime number theorem

,x

03

,

From

it follows t h a t

LARGE ADDITIVE FUNCTIONS

153

since n i s square-free, and therefore

Now h ( z ) of Definition 6.1 i s monotonically increasing f o r z > z

0 ’

so t h a t w e can write

Noting t h a t t r i v i a l l y

f o r every

E

>0

, we

u(n) << log 3:

f o r n < z and t h a t

obtain from (6.24), (6.25), (6.26) and (6.27)

2 O(zE) t

rexp(-(k+l) loglogz

-

logh(zl/k))

.

The exponential term has a maximum whenever ( k t 1) log l o g z t log h(zllk) = (k

has a minimum.

(6.29)

This

k = k ( z ) = (C

OCCUTS

t

+ 1) l o g l o g z t Kk-’ log’z

. (log l o g rl / k .j6

for

~ ( l ) logY/(Ytl)z )

. ( l o g l o g z ) (6-1) /(Y+l)

154 where

J.M. DE KONINCK AND A IVIC denotes an absolute constant. It is obvious that (6.22) holds

C>O

then for x sufficiently large, and the lower bounds of Theorem 6.1 follow then from (6.28) and (6.29). To establish the upper bounds it is enough to establish the upper

bound in (6.16), since the other two proofs are identical. We now have for y y(x)

>

that

x

because

in the second sum, since, as previously noted, h ( x ) is increasing for x>x

.

For the function

$(X,Y)

1

1

n=, P (n)sy

we use the following estimate due to N.G. de Bruijn Cnl:

(6.31)

$(x,y)

<

c3xlog2y. exp(-a(1oga

t

logloga-ch))

,

where c3 and c4 are some positive absolute constants, lim y w

m

,

155

LARGE ADDITIVE FUNCTIONS

and (6.32)

Suppose t h a t y

as x

log

loga

(6.33)

-f

-.

-

is chosen i n order t o s a t i s f y (6.32) and t h a t

1%Y

= (C

t

o(1)) log l o g x

Then from (6.31) and (6.32) we conclude that

W e now choose y = y(x)

so t h a t

(6.35)

Recalling t h e d e f i n i t i o n of

(6.36)

y

exp((C

t

(see (6.5)), we obtain

h(x)

o(1)) logl'(Yt1) z * (log log s)(1-6) /(Y+l))

I t is c l e a r that (6.32) and (6.33) are s a t i s f i e d f o r t h i s choice

of

y

, and

t h a t t h e desired estimate follows from (6.34) and (6.36).

Theorem 6.1 suggests that, a s x

(6.37)

log

l'l/f(n)

nCc

- .logx -

+

-,

Clogy/(ytl)x.

( l o g l o g x ) (Y+6)/(Y+l)

There i s , however, a big difference between

l'l/f(n)

nza

and

J . M . DE KONINCK AND A. IVI6

156 log l ' l / f ( n ) n-

, in

the sense t h a t an asymptotic formula l i k e (6.37) does

not necessarily lead t o an asymptotic formula f o r

l'l/f(n) n%

.

Conjectures

concerning asymptotic formulae l i k e (6.37) may be found i n the last chapter.

53.

The functions

B

, B and

B1

W e turn now our attention t o the functions

B(n)

, B(n)

and B1(n)

which are defined by (6.9), (6.10) and (6.11) and belong t o the class H The proof of Theorem 6.1 gives i n t h i s special case

( t h a t is, f o r

k = y = l , 6 = 0 ) . Corollary 6.1.

(6.39)

For any

E

and

> 0

zexp(-(Z+E)(logz. loglogIt:)1/2)

3: 2 z ~ ( E )

1 1/B(n)

*:<

292%~

and

<<

Proof. k

exp(- (1/2

~t:

- E ) (log ~t:. log log x)1/2)

In t h i s establishment of the lower bounds, we take

(logIt:/log l o g z ) 'I2 (giving c 2

2

t

E)

, while

i n the establishment

.

LARGE ADDITIVE FUNCI'IONS of the upper bounds we take c

1

1/2 -

E)

y

157

exp( (logz . log log3c) 'I2) (giving

.

In Chapters 4 and 5 we have obtained sharp asymptotic formulae f o r

(6.41)

Since functions B and B are related to each other in the same way as

w

and

R

are related, it seems natural to investigate the asymp-

totic behaviour of

(6.42)

expecting it to behave like x

as

3: + m

, as does the sum (6.41).

That

this is indeed true is shown by

Theorem 6.2.

There exist positive constants C1 ,

C2

such that

and

Proof. We first show that it is sufficient to prove (6.43). Since

and from the Cauchy-Schwarz inequality we have

J . M . DE KONINCK AND A. IVId

158

where i n S1 the sum is over those n for which B(n) < k B ( n ) the sum is over those n for which B(n)

S2

a large number which w i l l be chosen l a t e r .

f o r some integer r p

p

.

2

, then

2

Therefore the number of

is <<

(6.48)

1xp-r

<< x2-r

P

S2 =

for some C3

>

, where

We note that i f

n must be d i v i s i b l e by pr

n s

3:

f o r which pr

k = k(x)

is

B(n) t rB(n)

f o r some prime

divides n f o r some

Thus we have

c

0

.

To estimate S1 let

s1

S' t 1

s; ,

i s the sum over those n

divisible by p 2

f o r which B(n) < kB(n) and n

for some prime p > L

t o be chosen l a t e r ; obtain

kB(n)

in

1 B(n)/B (n) r>k 25n%c,rsB(n)/ B ( n )
(6.49)

where S;

.

t

, and

S;

, where

L = L(z)

is

is a large number

is the sum over the remaining 12's. Thus we

159

LARGE ADDITIVE FUNCTIONS

a a If n = p1 1.. . p l then each a

i

(6.51)

is one of the integers over which S;

= 1 whenever p

B(n) = ( a l - l ) p l t

i

.

> L

This + l i e s

...+ ( a i - 1 ) p i t B ( n )

5

i s summed,

that

L(alt

...t ai - i ) t @ ( n )

Therefore we have

where we have used (6.38). From (6.47), (6.48), (6.49), (6.50), (6.51), and (6.52) we theref o r e have (6.53)

s

s

O(ktc/L)

3: t

+

O(x exp(-C3k))

.

.

O ( ~ log L 3: exp (-c4 (log x log l o g s ) '/*)I

Noting t h a t t r i v i a l l y

(6.54)

t

S 2

x

t

O(1)

, and

. log l o g r ) 1 / 2

k

(logx

L

exp(C,-(logx

choosing

Y

and (6.55)

. log log x)'I2

,

.

J.M. DE KONINCK AND A. IVId

160

where

C5 = C,/2

, we

obtain (6.43) from (6.53).

Theorem 6.2 can be e a s i l y generalized a s follows: Theorem 6.3.

If

t i v e constants C1 , C2

r > 0

is a fixed number, then t h e r e e x i s t posi-

such t h a t

(6.56) and (6.57)

Proof.

The proof of (6.56) is almost i d e n t i c a l t o t h e correspond-

ing proof of (6.43)

except t h a t i n S1 t h e sum i s over those

which B(n) < k l I P p(n)

and i n S2 t h e sum is over those

n

for

n f o r which

B(n) 2 klIr B(n) . This leads t o an estimate similar t o t h a t i n (6.53),

where

k

i s replaced by k1Ir

. The choice

k = (logx

. log l o g z ) r/2

completes the proof of (6.56). To prove (6.57) we again have t r i v i a l l y

(6.58)

Using t h e Cauchy-Schwarz inequality i n the form

161

LARGE ADDITIVE FUN(TTI0N.S where

we obtain

so t h a t (6.57) follows from (6.56), (6.58) and (6.60).

I t seems i n t e r e s t i n g t o a l s o i n v e s t i g a t e

(6.61) and (6.62)

and t o compare these suns with (6.43).

-

on t h e average considerably l a r g e r than greater than cz as

3:

+.

One feels t h a t B l ( n ) B(n)

, so

should be

t h a t (6.61) should be

f o r any constant c > 0

.

This follows e a s i l y

from Theorem 6.4.

As x

,

+.-

(6.63)

Proof.

x

.

Let

Let p1 < p 2

<

. .. < p k

be those primes which do not exceed

Zi be defined by pili

S

x

li+l

<

pi

(i 5 k )

,

162 and ti

J.M. DE KONINCK AND A. IVI6 by ti

1 and

2

t i p i li s

(6.64)

so that ti

pi

.

3:

< (ti

,

+ l ) P ili

We then have

I t is clear t h a t

and

Thus

(6.66)

since

1upi

isk

c l/p

=

loglogr

O(1)

t

P*

e

li-l

i s kPi

o(xl0g log r )

From the elementary inequality

23C-l

2

,

. z (r

2

1) we have p a

2

ap

163

LARGE ADDITIVE FUNCTIONS

for a

bl

, so

that

This yields

On t h e other hand, it i s obvious that

B(n)

2

as before denotes the l a r g e s t prime f a c t o r of n

B(n)

2

p(n)

, where

.

I f qa (q prime, n n a 2 1 an integer) i s the l a r g e s t prime power dividing n , we then have

p(n)

where we used w(n)

(6.69)

1

I

u(m) t 1 and

B l ( n ) / B ( n ) <<

2SnSx

1 w(n) <<

nsy

y log l o g y

.

This gives

1 B l ( n ) / B ( n ) << 1 B l ( n ) / p ( n ) < < x l o g x .(loglogz)2. 2InIx 2Sn5x

These last bounds can c e r t a i n l y be further improved. We conclude this chapter by giving an asymptotic formula f o r t h e

J.M.

164

DE KONIKK

AND A. IVId

(6.70)

which may be compared with (6.71)

A sharp asymptotic formula f o r (6.71) is furnished by Corollary 5.5, and the formula f o r (6.70) w i l l have a similar leading term.

However

the sharpness of Corollary 5.5 w i l l not be attained, since B(n) - @ ( n ) has much larger oscillations than Q ( n ) - w(n)

, and consequently

it is

much more d i f f i c u l t t o handle.

Theorem 6.5. (6.72)

I'l/(B(n) rsx

-

B(n)) = cx

t

O(x1'2 logx)

,

where 1

c =

(6.73)

J ( F ( t ) - 6/7r2) t - l d t

,

0

and

Proof.

We use the same analytical method which w a s used i n the

proof of Theorem 3.4 and 5.2, noting that

I'

is now the sum over those

integers n which are not square-free, since ~ ( n )= B(n) i f and only

.

n i s square-free. In what follows we assume that 0 5 t s 1 Observk k we have f o r Res > 1 ing that B ( p ) - B ( p ) = p ( k - 1) f o r k = 1 , 2 , if

...

LARGE ADDITIVE FUNCTIONS

where for Res > 1 / 2

and 0

t

2

1

5

m

(6.75)

1 g ( n , t ) n-’

G(s,t)

.

n=l

Here g(n, t ) is a multiplicative function of n and

1 g ( p k , t ) / 5 1 for 0

5

t

5

165

1 and k

2,3,

for which g(p, t ) = 0

... .

This implies uni-

formly in t

(6.76)

By partial summation we obtain from (6.74) and (6.76)

= xG(1,t)

t

O(Z~’~)

,

where

(6.78)

and therefore (6.79)

G(1,t)

=

n (1t k=2( g P k -- l$ )( k - 2 ) ) p - k )

P

F(t)

,

J . M . DE KONINCK

166

Recalling that B(n) = B(n)

AND

A.

IVI6

if and only if n is square-free, we

infer that uniformly for 0 s t s 1

= xF(t) t - l t

t-')

- 1p2(n) nsx

t-l = s ( F ( t )

- 6/7r2) t-1tO(r1/2t-1).

Integrating (6.80) over t from ~(r)= z-2/3 to 1

, we obtain

the conclusion of the theorem, since

(F(t)

-

6/7r2) t - l d t

<<

m(r)

s1 / 3

,

and

The second of these three estimates holds because of the continuity of (F(t)

-

6/7r2)t-l

on C0,ll

, which follows from

(6.79).

LARGE ADDITIVE FUNCTIONS

167

NOTES The results of this chapter are original. Some have been obtained jointly by the authors and Professor Paul Erdos, and will appear in De Koninck, Erdos and Ivie C11. We would like to express our gratitude to Professor Erdos for his collaboration and for his kind permission to inclu-

de these results here. Although we were at first interested only in the functions B(n) and B(n)

(as '*largett analogues of ~ ( n )and Q(n))

, the class H arose

naturally in the course of our investigations. It may be shown that the "small" additive functions of Chapter 2 satisfy (6.1).

If f c S a

, (note that

S,

was defined in Definition 2 . 2 ) ,

then, by (2.7),

where R(x,t)

is analytic for

(6.82)

R(z,t)

It1

<<

5

1

, and uniformly in

z

and t

zlogRe t-2

Differentiating (6.82) as a function of t

, using Cauchy's ine-

quality for the derivative of an analytic function, and then setting t=l

, we obtain

so that (6.1) holds with L ( z )

g(1,l) log logz

.

J.M. DE KONINCK AND A. IVI6

168

The inequality (6.2) follows from the canonical representation of L(x)

(Chapter 5, 5 1 ) ; see also the monograph of E. Seneta [I1 for a proof. To see that (6.3) holds with an arbitrary D > 0 define

Then f

is additive and we have

since there is at least one prime p

between x/2

and x for x 2 3

.

The average order of an arithmetical function (as given by (6.4)) in m s t cases differs from the maximal order F(n)

, which may be defined

as the function F(n) satisfying

For example, from ( 2 . 2 ) and ( 2 . 3 ) , it is seen that loglogn is the average order of both w(n) and o(n)

(6.84)

which was used in (6.51).

n(n)

5

, while

logn/log 2

,

Furthermore for some C

>

0

,n

t

3

,

LARGE ADDITIVE FUNCTIONS

(6.85)

w(n)

Clogn/loglogn

s

and C l o g n / l o g l o g n

for n=nk

,

,

n(n) and w(n) a r e t h e functions

which implies t h a t t h e maximal order of logn/log2

169

respectively ( a t t a i n e d f o r n = Zk

th e product of t h e first k

md

(6.84) follows tri-

primes).

v i a l l y from a

n = p1

.

-I-. .+a

1

k

2"n)

while (6.85) follows from

u2(d) s d ( n ) s e x p ( C l o g n / l o g l o g n )

(6.86)

d ( n ) i s exp(Clogn/loglogn)

which a l s o shows t h a t t h e maximal order of

bince the maximal order i s a t t a i n e d again f o r n = n k first that if

primes).

k

n

p:'.

..p:

for all p

is fixed.

and a

t h e product of t h e

is t h e canonical decomposition of

n 6>0

,

To see t h a t t h e l a s t inequality i n (6.86) holds note

d o =k l az. t l where

,

6

We now have

, and

thus

do 6 n

i=l

a.6

pi

z

n

, then

'

( a t 1)/pa6 s 1 f o r p z 2"'

and

J.M. DE KONINCK AND A. IVIe

170

The desired inequality in the form

(6.87)

then follows for 6 = ((1

t

~ / 2 )log 2) /log log n

.

A more detailed account of

this subject may be found in Knopfmacher C11, Chapter 5. The inequality (6.23) is obvious from Stirling's formula for the gannna function if k is large, but it is also easily obtained by mathematical induction. For k = 6 we have 36 = 729

>

720

6!

, and the induc-

tion hypothesis yields

2;

(k+l)!(l+l/k)k

>

,

(k+l)!

since

In Definition 6.1, h ( x ) is a slowly oscillating function for 0< y < 1

.

It would be of interest to investigate

l'l/f(n) n%

(as well as F and F1 as given by (6.7) and (6.8))

, where

where

h(x)

is a

general non-decreasing slowly oscillating function, or even if h ( s ) = x%(x) where u > 0 and L ( x ) is slowly oscillating.

,

LAFGE ADDITIVE FUNCTIONS

171

The formulae (6.12) and (6.13) a r e not d i f f i c u l t t o prove.

From

the prime number theorem we obtain by p a r t i a l summation

(6.88)

P“Y c p

-

y2+ 0 2logy

1

- 5

Ilog2,

’

which gives

(6.89)

since

1 Urn2

rnsx

=

~ ( 2 )t 0(1/x) = n 2 / 6

t

0(1/x)

.

The proof of (6.13) is similar, and one can also prove

(6.90)

One of the few papers i n which B

and B

a r e investigated i s t h a t

of K. Alladi and P. Erdos C11, which contains many interesting r e s u l t s including proofs of (6.12) and (6.13).

For instance they prove t h a t f o r a

J.M. JE KONINCK AND A. IVId

172

fixed integer rnrl

,

as x-tm

, which

of n

Here Pi(n) (i = 1,.

n

.

, and

ple of

k,>O

reveals the connection between B

. . ,rn)

denotes the i-th largest prime factor of

is a constant depending on rn

c ( 1 t Urn)

.

, which is a rational m u l t i -

They also prove

1 (B(n) -

(6.92)

and large prime factars

nsx

B(n)) = x l o g l o g x t + ( x )

,

by noting t h a t

=

1 pcx/p21 P2”

t

c

pCx/p31 p3sx

t

...=x l o g l o g x t o ( z )

,

since

and

Further r e s u l t s concerning

B(n) and B(n) may be found i n the

paper of K. Alladi C11. There is an extensive l i t e r a t u r e about

LARGE ADDITIVE

One of t h e well-known estimates of

173

FUNCTIONS

Y(x,y) (somewhat weaker than (6.31))

can be found i n Prachar C11, Chapter 5. An estimate sharper than (6.31) can be found i n de Bruijn C3l. In (6.28) we used the estimate

since

because xo i s fixed. Note from (6.12) t h a t

B(n) = o ( n ) f o r almost a l l n

a

so t h a t

following the proof of Theorem 6.4 we may replace (6.63) with the sharper result

This Page Intentionally Left Blank

CHAPTER 7 RECIPROCALS IN SHORT INTERVALS

91. Introduction This chapter is concerned with the study of the sums

(7.1) where f belongs to a certain class of non-negative, integer-valued arithmetical functions. In '(7.1),

n belongs to the "short" interval ( x , t~ h l

where (as is customary in such problems) "short" means that h = O ( X ) z+-

.

,

, as

A n extensive literature concerning

(7.2) for various arithmetical functions f (or classes of functions) already exists. In addition to possessing an intrinsic interest, asymptotic formulae for (7.1) and ( 7 . 2 ) often lead to inequalities of the type

(7.3) where an is an increasing sequence of positive integers with interesting number-theoretic properties (such as a sequence of primes, a sequence of integers representable as a sum of two squares, etc.). approach to the estimate 175

Thus the classical

176

J.M. DE KONINCK AND A. IVIC

(7.4) where p ,

is the n-th prime, is based on establishing the asymptotic for-

mula

+ ( r th) - +(r)=

(7.5)

1

A(,)

x<nsxth

= (1 t o ( 1 ) ) h

,

as h-+m and c

(7.6)

h z x ,

whence (7.4) follows by taking x = pn

.

Our approach to the estimation of the sum ( 7 . 1 ) is based on the general method developed in Chapter 2 . We first estimate

(7.7)

.

We

where the error term we shall obtain will be uniform for IzI

5

shall then integrate this estimate over

, where

E(Z)

z

from

~ ( r to )

1

1

will be a suitably chosen function satisfying

lim

m

E(Z)

= 0

.

This procedure is particularly well-adapted to functions f for which z f ( n )

is generated by Dirichlet series of the form

RECIPROCALS IN SHORT INTERVALS where IzI 5

177

, considered as a function of s , is absolutely convergent on 1 , and Res = u 2 uo , for some uo < 1 . By an heuristic approach G

one may give an estimate for (7.1) for those functions f for which an asymptotic estimate of one would expect, for h

l’l/f(n)

n a

is known.

~ ( x )as

I+-

From Theorem 2.5, for example,

,

(7.9) but we are unable t o give a lower bound for h = h ( x ) such that (7.9) holds

82.

A n asymptotic formula for z f ( n )

in short intervals

We now prove Theorem 7.1, which will enable us to derive an asymptotic formula for the sum (7.1) whenever f belongs to a certain class of multiplicative or additive functions satisfying (7.8). Theorem 7.1

Let f ( n ) be a non-negative integer-valued arithme-

tical function such that (a) f ( n ) is multiplicative and f ( p )

(b) f ( n )

is additive and f(p)

If h = o ( z ) as

I+-

0

1 for all primes p for all primes p

, then uniformly on

IzI 5

,or

.

1

(7.11)

and a

(S

346/1067) is a constant for which the asymptotic formula

178

J . M . DE KONINCK AND A. I V I d

holds. Proof. We first assume that (a) holds. Let g,(n) by (7.11).

Then, for

If n

= pm

IzI 2

1 and for all n

, where (p,m)

f and the fact that f ( p )

1

be defined

,

, then by the multiplicativity of

1 we obtain

This implies

where c(n) is a multiplicative function such that 0

(7.16)

c(n)

if there is a p

such that p I In

,

= 2w (n)

otherwise.

If (b) holds, then (7.15) and (7.16) are still true, since g,(n)

179

RECIPROCALS IN SHORT INTERVALS

is multiplicative whenever f is additive. Moreover

p(1) zf(P)

gz(p)

For Res

>

1/2

t p(p)

= z o - zo

0

.

,

P

n=l

where (7.18)

is a Dirichlet series which is absolutely convergent for Re s > 1/3 the rough estimate

1 d(n)

n a

<<

xlogz ,

we obtain from (7.17)

The Mcibius inversion formula and (3.11) yield

(7.20)

and so we obtain

.

Using

J.M. DE KONINCK AND A. IVId

180

where we have used (7.15), (7.16), (7.19), and p a r t i a l summation. We note that a l l the 0-constants are uniform i n z

for

IzI

s 1

.

The main problem now is t o estimate the l a s t sun i n (7.21). t h i s we define

(7.22)

f

u(n) = d nc ( d )

Then we have, as i n (7.21)

(7.23)

,

-

To do

181

RECIPROCALS IN SHORT 1WERVAL.S m

h

1 c(n) n-’ n=1

t

t O ( ~ Z - ~ logx) ’ ~ t O(Z”~

1

c(n) xli2
lxth C

~

-

[-InI

logz)

.

On the other hand (7.17) and (7.22) yield

Thus we need t o find an estimation of

which is a ‘pure” divisor problem since u(n) does not depend on z

.

The simplest procedure i s t o derive an asymptotic formula f o r t(x) =

1 u(n) , and then

n a

t o use

To do this w e note t h a t from (7.24) we have

where m

(7.26)

V(s) =

1 v ( n ) YJ-’ n=l

.

= ~ ( s ~ ) ~ ( 2 s )

We wish t o show t h a t , f o r suitable constants A1 , A g , A g values, though computable, are unimportant here)

(their

J.M.

182

1v ( n )

(7.27)

nSx

where a

A1x

t

DE KONINCK AND A. IVId

A2x1'*logcc

i s defined by (7.12).

t

A3x1l2

t O(cc 1/(3-a)10g(4-a)/(3-a) X I

,

To prove ( 7 . 2 7 ) we use the convolution

method and we note t h a t (7.26) implies

(7.28)

Setting yz2 = x , y

,z

(where z w i l l be suitably chosen

> 1

l a t e r ) , we have, from (7.28),

=

s1 t s2 - s3

.

Using (7.12) we obtain

(7.30)

s3 =

Cyl

1d(n)

nsz

= (y t O(1)) ( z log z

yz log z t (2y

XZ

-1

logz

t

t

(2y

- 1) z

t O(za

log2 2))

- 1)yz t O ( z log 2 ) + (2y

- 1)Xa-l

t O(2

0 (z"y log2 2 )

logx)

t

O(xza-210g2x) ,

183

RECIPFOCALS I N S H O E INTERVALS

and (7.31)

S2 =

1 d ( n ) C ~ n - ~ =l

nsz

+

= p(2)

n
z - l l o g z + (2y - 1) 2-1 +o(za-210g2z)

m

-

2

j

( F 2 l o g t + (2y - 1) t - 2+ O ( t a - 3 log2 t)) at

2

= s2(2)

-

2

-1

l o g z - (2y

+ 1) z - l + o ( z a - 2 log2 2 ) ,

since integration by p a r t s y i e l d s m

j

P l o g t

. at =

2

-1

l o g z + 2 -1

.

2

Therefore

s2 = 252(2)

(7.33)

- 22 -1 l o g z -

(2y

+ 1)22-1 +

o(3cza-2log22)

+ O(zlog2)

.

Using t h e Euler-MacLaurin summation formula and defining Y(Z)

=

2

-

C21

-

1/2

, we

have, f o r O < s + l

,

"(2)

by

184

J . M . DE KONINCK AND A. IVI6

= s(s) t

1-s

5 - - Y(x) x-s 1-s

t

,

o(x-s-l)

and therefore by p a r t i a l sumnation

1rn-ll2logm

2y1I2 logy - 4y1I2

t

c

t

logy)

m*Y

,

where C i s a computable constant. Furthermore

=

1 7 5I;(

t O(z

X1l2

log x t c 1 X1l2

logx)

t

x1/2 y1/2 log x/y

o(xza-2 log2x)

where we have used (7.12) and (7.34), and Recalling that yz2

t

t

4yx

1/2

y

1/2

,

c1 i s

a computable constant.

x and using (7.29), (7.30), (7.33), and

(7.35), we obtain- (7.27), since we have

RECIPROCALS IN SHORT INIERVALS

x

1/2

Y

1/2

logx/y - 2xz-110gz

185

.

0

Finally, to obtain the error term of (7.27), we choose

z = (xlogx)1/ ( 3 - a )

(7.36)

so that the error terms z log x and XZ'-~

Noting that 1/3

<

1/(3- a)

5

log2x are equal.

.

0.37373029..

we easily obtain,

from (7.24) and (7.27),

(7.37)

where

and B2

, B3 are also computable constants. This means that for

h = o(x)

which, combined with (7.21) and (7.23), proves (7.10).

We now give an application of Theorem 7.1. For

Iz

I

5

R < 1

J . M . IIE KONINCK AND A . IVId

186

is certainly an analytic function of mial in

z

z

, and

1

zf(lz)

x<nsxth

is a polyno-

, so that (7.10) implies

1

(7.39)

zf(n)

C ( z ) h t R(x,z)

x
where R ( x , z )

is an analytic function of z on

thermore uniformly in

, 12

I

5 R <

1

, and fur-

z

We now proceed as in the proof of Corollary 5.2. The coefficient of zq

in

c

is

zf(n) x
c

A (x,h) 4

1 ,

x
and so equating coefficients in (7.39), and using Cauchy’s inequality for derivatives of analytic functions, we obtain Corollary 7.1.

(7.41)

A (x,h)

4

, where

If h z x 1’(3-a)log2a:

c

, then , as

1 = (d t o(1)) h 4 x
x+-

,

,

0 is a fixed integer, d G(q) (O)/q! 4 satisfies the hypothesis of Theorem 7.1, and a is given by (7.12).

uniformly in q

q

2

,f(n)

Among the multiplicative functions satisfying the hypothesis of

, the number of non-isomorphic abelian groups with elements, and S ( n ) , the number of semi-simple non-isomorphic rings

Theorem 7.1 are a ( n ) n

with n elements. The most important additive function satisfying the

RECIPROCALS IN SHORT INTERVALS hypothesis of Theorem 7.1 is n ( n ) - w(n)

187

, but in this case we are able

to obtain a better result than Corollary 7.1, due to the fact that z f ( n ) is multiplicative when f

is additive. By multiplicativity we have, in

case (b) of Theorem 7.1, that

which implies that, for

IzI

s 1/2 0

(7.43)

Ig,(n)

Thus, for Res

>

I

b(n)

1/2

,

m

(7.44)

B(s)

, if there is a p

such that pI In ,

otherwise.

1 b(n) n=l

r[ (1 + p - 2 s

+p-3s

P

+p + . ..)

which means that b ( n ) is the characteristic function of the "square-full" integers or of G ( 2 ) (see Notes, Chapter 5).

Following the proof of Theo-

rem 7.1 it is seen that the error term in the estimate of depends on (7.45)

where in this case

which gives

1 zf(n) x<nsx+h

J . M . DE KONINCK AND A. IVIC

188

1 v(n)

(7.47)

1

nSx

km2n3

1

.

3

Using e i t h e r (7.34) o r the calculus of residues i t is seen t h a t

where the error term R(x) =

0(2”~)

as x - + m

.

This error term has been

estimated by various authors, and the best-known r e s u l t (Srinivasan C11)

is (7.49)

R(x) << x 105/40710g2x

Proceeding as before, we obtain (7.39) with R(x,z) << hx-1’4 uniformly i n z

for

JzI5 1/2

t

x 105/407

10g2

J

.

For the special case f ( n ) = Q ( n ) - ~ ( n ), we obtain Corollary 7.2.

(7.50)

If

t

1 0 5 / 4 0 7 log3

h>x

1 m n ~ + h , n) -u(n) =q

uniformly i n q

, where

,

( d t o(1)) h 4

then

, as

x+m

,

,

q 2 0 i s a fixed integer and

To obtain a number-theoretic corollary similar t o (7.3) from (7.50),

RECIPROCALS I N SHORT INTERVALS

n(n) = u(n)

we observe t h a t

, x = qn , where

q=0

i f and only i f

n

189

is square-free.

Taking

qn denotes the n-th square-free number, we obtain

(7.51) where t h e l a s t inequality i s a consequence of

n

qn << n

<<

, which

follows from the elementary estimate

11

qn3

=

-6x

O(x1/2)

t

2.

Sharper estimates f o r the difference between consecutive squaref r e e numbers can be obtained by special methods.

13.

Reciprocals i n short intervals Having proven Theorem 7 . 1 we a r e now ready t o obtain estimates f o r

the sum (7.1), which is the main aim of t h i s chapter.

We w i l l use both

Theorem 7.1 and the general method developed i n Chapter 2. Theorem 7 . 2 .

Let f ( n ) be a positive integer-valued m u l t i p l i c a t i f(p) = 1 f o r a l l primes

ve function such that

1 l/f(n) x
(7.52)

as

z + m

, where

mined by (7.12) Proof. G(0) = 0

,

W e first state

(

1

G ( z ) z - l dz t o(1) 0

p

.

Then, f o r

I

h

,

i s defined i n (7.11) and t h e value of

G(z)

is deter-

CL

. We may use (7.10) and take

G ( z ) 2-l

and integrating over

is continuous f o r z

from E(X)

real, 0 < z s 1

z

Osz

log-1’2x

5

1

.

to

.

Since

Dividing (7.10) by z 1

, we

obtain

J.M. DE KONINCK AND A. I V I C

190

Similarly

h

1

3

h

~ ( z )

E (XI

as x + m

.

J ~ ( zz) -

~ t~ o(h) z

,

0

The integrals of the e r r o r terms are o ( h ) for h t z11(3-a)logzX,

and ( 7 . 5 2 ) follows.

Now l e t f ( n ) be an additive non-negative function such that f(p)

0 f o r every prime p

.

For

z

r e a l such t h a t

0 < aI 1

,

(7.43)

follows from ( 7 . 4 2 ) , and proceeding as i n Chapter 2 we have uniformly i n 2

(0 < a s 1)

In the most interesting case, t h a t is when f ( n ) = n(n) - w(n) have from t h e proof of ( 7 . 5 0 )

(7.55)

, we

RECIPROCALS IN SHORT IFRERVALS

191

where

Dividing by z E(X) = log-l'*x

and integrating as before over

If h > x'05/lro7 log3x , then

1' l / ( Q ( n ) - w ( n ) ) = x
as x + -

from

to 1 we obtain

Theorem 7 . 3 .

(7.56)

z

(F(z)

- 6 / ~ ~ ) z - ~t d~(l) z

, where

NOTES

The problem of estimating

-

pn

, the difference between con-

secutive primes, is one of the classical problems of analytic number theory. The first significant result is due to Hoheisel [ 1 1

, where he proved

J.M.

192

DE KONINCK AND A. IVI6

(7.57) for e to e

-

1

>

1/33000

.

1 - 1/250

>

.

This result was quickly improved by Heilbronn C11

Subsequent research brought many improvements. For

example Ingham C21 showed that

gives

8>

of zeros -T

5

y

5

1 - 1/A

6 t iy

p

T

in (7.57)

.

(6,y

, where

real) of

U(a,T) ~ ( s ) for

denotes the number which 6

a

2

1/2

and

His approach to (7.57) was via (7.5) , and in this way he

proved, in Ingham C31, that (7.57) holds with e close to e

2

>

1/2

>

5/8

.

This value is

, which is the best possible value attainable by this

method.

To see how (7.5) yields an estimate for the difference between consecutive primes note that

where (7.60)

S =

1 logp x<pa ~xth,ar2

<<

log2x(l

t

<<

- 21’2)

(x t h) 1’2

Suppose that the interval (x,x

log2x. 1 1 x
t

<<

log2x t hx-1’2 log2x .

hl contains no primes. From

(7.5), (7.59) and (7.60) it follows that, as x+m

,

RECIPROCALS IN SHORT INTERVALS h/2

5

(lto(1)) h =

1

x
A(,)

<<

which is a contradiction whenever h > C log2 x Therefore (x,x any c > 0

.

thl

log2x

193 hx-1/210g2x ,

t

for some suitable

C > 0

.

contains a prime if (7.5) holds whenever h >xc for

For h = x c

, x = pn '

implies (7.57) with e c

.

it follows that pntlE (p,,p,

c

t p,l

. This

Improving estimate (7.58) , Montgomery [ 2 1 obtained (7.5) for hzx3/5tE and (7.57) with e 3/5 t E . Huxley C11 refined Montgomery's method to obtain h > x7/12tE in (7.5) and e

7/12 t

E

in (7.57). That

the E-factor in exponents is redundant was shown by Ivic [41, who showed that (7.5) holds for h 2 x7/1210&~x . This gives (7.61) The inequality (7.57) can be derived for several values of e without using asymptotic formulae such as that in (7.5). an asymptotic formula for $(x t h )

-

$(z)

Without using

, Iwaniec and Jutila

C11 recently

proved (13/23 has been now decreased to 11/20 by Iwaniec and Heath-Brown)

(7.62)

by employing a successful combination of sieve and analytic methods. Their estimate (7.62) is the best one hown, and suggests that e be attained without sharp zero-density estimates for

uniformly in u for u

2

1/2

I

~ ( s ),

1/2

t

E

such as

may

J . M . DE KONINCK AND A. IVId

194

The above estimate is known i n the l i t e r a t u r e as "the density hypothesis", and the range of

(I

for which (7.63) holds has been extensively

The best-known r e s u l t , due t o J u t i l a C11, i s t h a t (7.63) holds

studied. for

CI

2

11/14

.

For some recent density r e s u l t s , see I v i t C5l.

Theorems 7 . 2 and 7.3 a r e o r i g i n a l and unpublished, while Theorem 7 . 1 i s similar t o those found i n I v i e C61 and C71, where a sharper r e s u l t

concerning

is proven.

An a l t e r n a t i v e approach i s used i n these papers.

I t consists

i n estimating

(7.64)

-n

5

t

5

n

, where

f

s a t i s f i e s t h e hypothesis of Theorem 7.1.

Observing

that 0

m

*

0

,m

an integer,

(7.65) -n

2n

m = O

,

we obtain

This approach has a s l i g h t advantage over t h e one used i n t h i s The condition t h a t f be non-negative and integer-valued (so 1 z f ( n ) i s a polynomial) can be somewhat relaxed by only r e that x<nsxth

chapter.

RECIPROCALS IN SHORT INTERVALS

195

quiring f to be integer-vdlued.Howeverit does not seem likely that a result such as Corollary 7.2 (based on multiplicativity of z f ( n ) when f is additive) is obtainable by this approach. It was proven by Ivii C6l that

uniformly for integers k > l

, where

The best-known value of a for which (7.12) holds is a s 346/1067. An intricate proof, which is certainly beyond the scope of these notes,

can be bound in Kolesnik

C21.

The yet unpublished result a 5 35/216 of

Kolesnik C11 leads to a slight improvement. For a s 346/1067 we have U(3- a )

5

0.37373029..

..

The conjectured value is a

1/4

t

E

, but

this seems to be unapproachable by present-day methods. The problem of estimating the '%richlet

1d(n)

nsz

is known in the literature as

divisor problem", after Dirichlet, who proved in the 19th

century

(7.69)

1 d ( n ) = xlogz n 3

t

(2y - 1)zt ~ ( z ) ,

where (7.70)

R(z)

<<

z

1/2

.

196

J.M. DE KONINCK AND A. I V I 6 A

comprehensive account of this problem may be found in Chandra-

sekharan C11, Chapter 8, where it is proven, using Bessel functions, that

For the sake of completeness we shall give an outline of a proof of the above result. (All the known proofs of R(s) a < l/3

are difficult).

<<

xa logA x with

For the rest of this chapter let $(2)

2

-

,

Czl - 1/2

which is standard notation, although in some cases it may be somewhat confusing, as the symbol J, is usually reserved for

$(z) =

1 A(n) , in our

nsx

case no ambiguity will arise. By the convolution principle, introduced by Dirichlet himself,

2cc(log21/2

- (2 +

ty

O(1)

- $(z1’2)

O ( & )

-

-

2$(z1/2)21/2

-2

$(2/?2)

nsx1/2

p)

-

[21/2]

197

RECIPIUXALS I N SHOHT INTERVALS

where we have used the elementary estimate

To see that (7.72) holds it is sufficient to assume that s is a positive integer. Define an by

an

(7.73)

logn

t

1/2n - 1

-

1/2

-

1/3

- ... -

l/n

.

Using

for -1 t s l we have an+l - an

(7.74)

=

f(n)

Sunnning (7.74) we obtain (7.72) where

<<

y =

n- 3

lim an n*

. (Euler's constant).

We

have thus shown that the error term in Dirichlet's divisor problem is

(7.75)

Expanding $(y)

into a Fourier series (for y not an integer),

we obtain

(7.76)

$(y)

-

03

1

n=l

-1 . (m) sin(2sng)

.

Using the fact that partial sums of the above series are uniformly bounded, we obtain the following lemma which is essentially due to van der

J.M. DE KONINCK AND A. IVIe

198

Corput (see Walfisz C11, p. 90): Let F( y ) be twice differentiable for X S Y [ ~ " ( y I)

2 E

for x s y

> 0

.

SY

We split the range n

i

SY

such that

Then

. x1/2 in (7.75) into O(1ogx)

intervals

of the form (IV,ZiVI and then use the above lemna with F ( t ) = zt-' X = IV

,Y

2117

, and

E

Z X I V - ~ to obtain (7.71).

,

A entirely elementary

proof of the same result (with log2x instead of logx)

can be found

in Vinogradov C11. One could also obtain (7.27) from a general convolution theorem of Tull 111, but our derivation, though longer, is self-complete.

and may be established by integration by parts (for more details concerning (7.34) see Chapter 8). To obtain the leading term in (7.48) one may use the Perron inver-

sion formula

where the prime

'

indicates that u ( n ) is replaced by u ( n ) / 2 whenever

x is an integer. When one moves the line of integration, the leading term in (7.48) is then simply the sum of residues at and

s

s=

1

, s = 1/2 ,

1/3 respectively. The best approach to the estimation of the

RECIPROCALS IN SHORT INTERVALS

199

error term in (7.48) consists again in transforming it into sums involving the function $(t) = t

-

Ctl - 1/2

is due to Srinivasan C11.

formula for

1a(n)

n-

.

The best result

This is also the error term in the asymptotic

(see Notes, Chapter 1).

The more general divisor

problems, such as the estimation of

1 1 ,

a b mn%

(1 5 a < b

, a , b integers) can also make use of estimates involving

(see Richert C 1 I) .

$(t)

For qntl - qn , the difference between consecutive square-free numbers, see Richert

C21,

where he proves

Further slight improvements were obtained by Rankin [ T I and Schmidt C11. Their basic idea is to investigate the range of h for which

1 p2(n) x
1

p(d)

( 6 / d t o(1))

h

, and reducing the problems to the es-

d 2 In

timation of sums involving $(t)

.

All of these divisor problems may be

reduced to the problem of estimating

J.M. DE KONINCK AND A. IVId

20 0

(7.78)

where b 5 2a and f

is differentiable a large number of times (and may de-

pend on several parameters).

To estimate (7.78) van der Corput developed,

i n the 19201s, the i n t r i c a t e theory of ''pairs of exponents", which is one of the deepest methods i n analytic number theory.

A presentation of the

basic r e s u l t s of h i s theory (which was l a t e r simplified by P h i l l i p s and Rankin) can be found i n Rankin C11 o r Richert C11. The two-dimensional

van der Corput method is the basic tool of Kolesnik C11 and C21 and S r i nivasan C11.

Thus i n addition t o proving a

also proves an estimate f o r

5(1/2

+

it)

5

346/1067 i n (7.12) Kolesnik

, which

rBle i n many branches of analytic number theory.

plays a very important H i s l a t e s t r e s u l t (Ko-

lesnik Cil) is 5(1/2

+ i t ) << It1 3 5 / 2 1 6 + ~

CHAPTER 8 RECIPROCALS OF ADDITIVE FUNCTIONS RESTRICTED TO PARTICULAR SEQUENCES OF INTEGERS

81. Introduction Our attention so far has been focused on sums of reciprocals of functions f taken over all integers n s x for which llf is defined. It seems natural to consider the possibility of evaluating the same sums

when n is restricted to a subsequence of the natural numbers N

.

In

other words, we can consider the sum

where f is additive and

is a subset of N

.

The purpose of this chapter is to estimate (8.1) for those functions f such that f E Sa of Chapter 2 or f is the logarithm of a multiplicative function as in Chapter 3. For f with

R

case, that is, when f E take

EL

and finally

12.

, where

= { n E N :k In}

A

EN .- k l n 1 ,

R

k

2

E

Sa

we shall consider (8.1)

1 is a fixed integer. In the other

(see Definition 8.2), we shall successively = { n E N . -n3(modk)l

= { n E N : (n,k)

11

(where (k,Z)

1

,

.

"Small" additive functions and quotients of additive functions We first consider the class of arithmetical functions defined in

Chapter 2, more precisely, the set of functions Sa , a E R

mi

, of Definition

20 2

DE KONINCK AND A. IVId

J.M.

2.2. Our goal is to estimate

1'

, where

l/f(n)

nsz, ns O(mod k)

and

feSa

k'>l is a fixed integer. In order to do this, we state the following definition. Definition 8.1. Let

and let k E J . If D ( t )

feSa

responding function of Definition 2 . 2 , then for t

E

[O,ll

is the cor-

, Bi(k,t)

is

defined by

x

and A i ( t )

n

(tf(P

U

p-a + t f ( p

at1

+

. . .) D ( t ) t-5

(i-1)

is defined by =

Bi(t)

We shall also write B i ( t )

for B i ( k , t )

Ai(t)

for each i 1,2,...,a t 2

.

The desired estimate for

1'

and Ai

l/f(n)

n%, n-0 (mod k)

for Ai(l)

.

is closely related,

as we shall see, to the behaviour of

where we write n O(k)

instead of n z O(mod k)

.

Therefore, we shall

seek an appropriate factorization of the Dirichlet series (8.2).

This is

REST~ICTICN TO PARTICULARSEQUENCES OF INTEGERS

203

provided by the following r e s u l t . Lemma 8.1.

Let

k

be a multiplicative function such t h a t

m

1 k ( n ) n-' n=l

converges absolutely f o r Re s = o > o

integer, then we have f o r Res

.

If

k

2

1 is a fixed

a > uo

n=l ns

P

The proof is by induction on ~ ( k ,) the number O f d i s t i n c t a prime factors of k If w(k) 1 , then k = p , 1 , and since h is Proof.

.

multiplicative, we have f o r Res

(8.4)

-1

h(n) = 7

n=l n a1 n:O(P1 1

1

m

n=l (n,pl) =1

>

uo

a 1 h(V1 1 -+ a (nPl1Is

c

m

n=l (n,pl) =1

...

a . +I

a.

which proves (8.3) f o r w ( k ) 1 . Assume t h a t ( 8 . 3 ) holds f o r Now l e t k = pal l . . . p ,ar As i n (8.4) we have

.

w ( k ) = r - 1.

204

J.M.

DE KONINCK AND A. I V I e

Defining the multiplicative function 6(n) by a 1 if (n,p, r) 1 6(n) otherwise ,

,

we have, using the induction hypothesis,

c

m

n=l

m = n8

a

nzO(pll . . .p,-a,P- 1)

n

l

m

h(n)

n=l 1

t...)

n8

a

l (wl ...PraP )=I

Substituting this in (8.5),

we obtain the desired result.

The representation of the Dirichlet series ( 8 . 2 ) provided by Lem-

ma 8.1 (with h(n) = tf(n)) for

enables us to derive an asymptotic expansion

1' l/f(n) , which we state as na,nE0(k) Theorem 8.1. Let f e S a

and let k be a fixed positive integer.

Then

(8.6)

1'

nsx n:O(k)

l/f(n)

a Ai a x 1 i=1 (log log x)z

.( (loglogx)atl X

RESTRICTION TO PAFCKULAR SEQUENCES OF INTEGERS

where the Ails Proof.

20 5

a r e defined i n Definition 8.1. Applying Lemma 8.1 with h ( n ) = t f ( n )we obtain f o r

O < t S l

-1

(8.7)

tf(4= -

n = l ns n=O(k)

>

0

, it

(I+-tf (PI t . . . ) -1

PS

plk

Since f E S a Res

n

, and

the products appearing i n (8.7) a r e regular f o r

follows that

t . . . ) -1 t-t-

P

nsx n=O(k)

where D ( t )

P2

i s defined i n Definition 2 . 2 , and

R(z, t) = uniformly f o r

0
5

1

.

O(Z

x)

Proceeding a s i n the proof of Theorem 2.4 we

obtain Theorem 8.1.

Applying Theorem 8.1 t o the functions observing t h a t

w

and n of

Su

, and

J.M. DE KONINCK AND A. IVId

206

n[lt$C.+E P Ik

t . . . )-1

n -,[

$Pa)

t...)

1

Patl

Pal Ik

P

t

,

F1 t=l

we obtain the following c o r o l l a r i e s . Corollary 8.1.

1’

n a n=O(k) where a;

k

and M are fixed positive integers, then

M

a: i=l (log log x)

1

l/w(n)

1

If

, and

X

O( (loglogx)Mtl

the remaining constants, which depend on k

, are

com-

putable. Corollary 8.2.

where b;

1

, and

If

k

and M are fixed positive integers, then

t h e remaining constants, which depend i n k

, are

com-

putable.

Following the method of proof i n Chapter 4 we s h a l l now estimate sums (over

nzO(modk)) of quotients of additive functions.

Theorem 8.2. f o r a l l primes p

Let g and f be two additive functions such that

and a l l integers r t 1

20 7

RESTRICTION M PAFWCULAR SEQUENCES OF INTEGERS

where C1 and

C2

a r e two positive absolute constants.

Then, f o r

k>l

a fixed integer, (8.8)

1'

$$- =

nn=O(k)

k

Bx loglogz

t

t

o

[

X

(log log x)2

where B is a computable constant depending on F,g

, we

special case g = R , f = u

(8.9)

m-r;

1'

nlz:

Q(n)

+

, and

k

.

In the

have M

1

X

n=O(k)

where bf21

is an a r b i t r a r y but fixed integer, and the

t a b l e constants depending on k Proof. It( s 1

, and

hi's

a r e compu-

.

Applying Lemma 8 . 1 t o

h ( n ) = t g ( nu)f ( n )

where

tE E

u ~ ( 0 , l I we obtain

Using (4.7) and the f a c t t h a t the products i n (8.10) a r e regular

,

208

J.M. DE KONINCK AND A. IVId

functions of

s

for Res > 1 - E

, we obtain uniformly for

It I < 1

, Iu I

5

1

n:O(k)

is absolutely and uniformly convergent for Res > 1/2 .

where K(t,u;s)

Following the proof of Theorem 4.1 we obtain (8.8) and (8.9),

by observing

that

The reader might wonder why we did not try to obtain estimates for the more complex sums

for ( k , Z ) = 1

.

What we can obtain is a "reasonable" representation for

the series

(8.13)

where f is additive and O < t < 1 X1' X2'".'

x9(k)

.

Indeed, let (k,Z)

be the characters modk

, where x1

=

1 and let

is the principal

RESTRICTION TO PARTICULAR SEQUENCES OF INTEGERS character (xl(n)

=

1 if (n,k) = 1 and zero otherwise).

If

209 6o

is the

m

abscissa of absolute convergence for ve, then for Re s

n-2 ( k )

1 h ( n ) n-’ n=l

and h is multiplicati-

=6> 6

II

.

But

hence

A

close analysis of relation (8.14) with h ( n ) replaced by

tf(n)

shows that it is not possible to factor out from (8.14) an expression of the form ( ~ ( s ) ,) which ~ would allow us to use the powerful Lemma 2.1. Nevertheless a careful study of the function

where x

f

x1 , might lead to an estimate for the intricate expression

210

J.M.

DE KONINCK AND

A. IVId

where for Re s > 1

is the L-series associated with x ( n )

, and

G(t,s;X)

is absolutely and

uniformly convergent for Re s > 1/2 . The problem is then transformed into the problem of estimating

which is very difficult.

93. Reciprocals of logarithms of multiplicative functions

The arithmetical functions defined by logarithms of positive multiplicative functions, which have been already studied in Chapter 3 , are more easily handled than other additive functions. Indeed, the fact that the estimation of sums of reciprocals of these functions, even when n is restricted to certain special sequences of integers, does not require the use of the powerful Lemma 2.1 allows us to somewhat enlarge the scope of our results. The first attempt to study sums of reciprocals of logarithms of a multiplicative function restricted to a congruence class of integers was made by J . M . Tourigny. He proved that, given a fixed prime p , preassigned positive integer M ,

and any

RESTRICTION TO PARTICULAR SEQUENCES OF INTEGERS

where al

l/po

ding on p ,

.

, and

the remaining ails

Here, of course,

a

211

are computable constants depen-

denotes the sum of divisors function.

In this section, our ultimate goal is to provide a general estimate for

(8.17)

, and

1 or k

where ( k , Z )

f belongs to a certain large class of po-

sitive multiplicative functions. We shall proceed in two steps. First we extend estimate (8.16) to the family

5

of Definition 3.1, giving way rather easily to a precise

estimation of (8.17) for n-O(modpo)

Then, considering a larger class

, we prove several identities concerning generating functions of functions f E & restric-

of functions

& and taking

.

(k,Z) = 1 or

(k,Z) = k

ted to certain congruence classes. Finally we deduce asymptotic expansions for (8.17). Our first result is as follows: Theorem 8 . 3 . integer. If f.

where, for -l/v

Let p ,

$ , then

5

t

5

0

,

be a fixed prime and M a fixed positive

J.M. DE KONINCK AND A. IVI6

212

where ht (n) is defined by (3.8). Proof.

In order t o prove (8.18), we define the function f o ( n )

a s follows: nzO(modpo)

f(n)

if

0

otherwise

fo(n)

,

.

Then

and

(8.19)

To see that (8.19) holds we note that the above sum is zero whenever p o l n by the definition of 9(n) by 9(n) = ( f ( n )

a

a

dlmpoO,PoOI Id

f,

.

-a t

1

Assume n = *

poem a

with

Then

a

a -1

d lmpO0'Po0

I Id

(m,po)

1

, and

define

213

RESTRI(JTI0N TO PARTICULAR SEQUENCES OF INTEGERS

from the multiplicativity of h,

.

Using the method of proof of Theorem 3.1, we have from (8.19) that

which is similar to the crucial relation (3.14) obtained in the proof. Now proceeding with (8.20) as was done with (3.14), we finally obtain (8.18).

We now enlarge the scope of Theorem 8.3 by considering a class of

.

functions somewhat larger than the class belonging to this new class

& ,

defined below, our sums will be over

those integers n s x which satisfy n-Z(modk) We begin by defining

k

t

5

ht(n)

np

P In

>

0 that for every

(8.21)

where a(n)

.

be the set of all positive mltiplicati-

ve functions f such that for some a 5

when ( k , Z ) = 1 or k

& .

Definition 8.2. Let

formly for -l/a

Then for certain functions

, and

ht(n)

<<

0 and some fl E

>

<

0

we have uni-

0

(a(n))BnE

is the multiplicative function defined by

J.M.

214

DE KONINCK AND A. IVId

2

I t i s obvious t h a t

& , where

i s the class of multi-

plicative functions defined i n Definition 3.1, since from (3.11) it can be seen t h a t (8.21) is s a t i s f i e d .

8

I t may also be shown t h a t

is the class of functions defined i n Definition 3.2.

8 If

2

fE

, where B ,

then for a = 1 , B -1 we have

ht@

3

-

1-

-1

( ~ + Q ~ + , .~. . P + aj J j p

= O ( l t ) p-l)

for -l/a

2

t h a t for

-j t

and

since

-j+llt

s cp-1

t s 0 and f o r some suitable C > 0

It1 s l / a

-1

- ( l + a l , j - l ~ +...+uj-l,j-lp

.

Here we used the f a c t

1x1 s 1/3

..

t t ( t - 1 ) . (t-ntl) (n) = n!

and

for p

2po

, where X

>0

is the constant appearing i n Definition 3.2.

thus obtain for some suitable C > 0

since , t r i v i a l l y

We

21 5

R E S T R I f l I O N TO PARTICULAR SEQUENCES OF INTEGERS

Before proceeding t o estimate (8.17) f o r c e r t a i n functions f E %,! we give a general i d e n t i t y f o r the generating function of any multiplica-

t i v e function f r e s t r i c t e d t o a congruence class f o r which t h e abscissa m

of convergence

u

1 f ( n ) n-' n=l

of

0

is f i n i t e .

Let f be a multiplicative function such that

Lemna 8.2. m

1 f(n)

converges absolutely f o r Res > u

n-'

then

n=l

m

n=l

, k'

where d = (k,Z) modk'

k/d

, I'

, and

= Z/d

the x's a r e t h e characters

. Proof.

If

(k,Z) = k (k, I )

becomes (8.3), while if

, then

n i l @ ) means n-O(k)

1 we have

d=1

, and

, and

(8.22)

(8.22) follows

from (8.14). Lemna 8.3. Res>l

and - l / a

Let f E 5

t

5

0

and l e t

( k , 2) = 1 o r k

.

Then f o r

,

216

J.M. DE KONINCK

AND A. IVId

where (8.24)

F(s,k,t)

p

=

t...)

-2a

2

t Y L - d mx t 1 ( P k'

cf(p)p

t

t . . .)

-1

P 2s

PS

-(atl)a t

at1 t

where d = ( k , l ) mod k

-1

k ' = k/d

, Z'

= Z/d

(f@

P(

and the

t...)

a) tP1 ) s

,

x's are t h e characters

. Since f i s positive and -l/a

Proof. 0 s ( f ( n )n-a)t

I,

1

.

t

5

0

we have t h a t

Thus the proof follows immediately from Lemna 8.2

with f ( n ) replaced by

(.f(n) n-")

I,

Cf(n)n-a)t

,

since the non-negativity of

implies the non-vanishing of t h e s e r i e s

217

RESTRICTION TO PARTICULAR SEQUENCES OF INTEGERS f o r Re s

2

1 , and assures absolute and uniform convergence f o r

of t h e Dirichlet s e r i e s generated by

Re s > 1

( f ( n ) n-")*, - l / a I t 2 0

.

We a r e now ready t o employ our basic method which was outlined i n Chapter 2 and was used i n Chapter 3 f o r summing reciprocals of logarithms of multiplicative functions.

1

mula f o r

We shall f i r s t e s t a b l i s h an asymptotic for-

, where f

belongs t o a subclass of

Let

&

ft(n)

&x, n Z ( k )

&,

, defi-

ned by Definition 8.3.

(8.25)

df(p)p-a)t

E

, where

[-A,O]

We c l e a r l y have

, then

and some 6

>

0

1 t O(p-&)

i n a r b i t r a r y but fixed, and

A >0

except possibly f o r O(xE)

fE

be t h e class of functions be-

such t h a t f o r every prime p

longing t o

for t

f

integers

nsr

5

4.

u

5

f o r which f ( n ) = 1 , In order t o see t h i s , l e t

from (3.4) we have

1- p - 8 s (f(p) p - a p s (1 - p - B ) - 1 = 1 O ( f 8 )

,

which implies (f(pj p T t =

{

(fb)P -"1l l q t / y

=

(l+O@-B))t/y = l t O ( p - 8 )

since t belongs t o a fixed i n t e r v a l .

From ( 3 . 4 ) we have f ( n )

, >>

n

a/2

.

21 8

J . M . DE KONINCK AND A. IVIi

Thus, f e

h1.

Similarly, using (3.20),

L e m 8.4. k

Let f~

.&,

and - l / a

&

5

A1 ,

5

t

5

0

.

If (k,Z) = 1 o r

, then

(8.27)

1

ft(n) = G(l,k,t) zattl attl nsx n-Z(k)

atG(l,k,t) + R ( 0 ) attl

where G is defined by

where d

(8.29)

(k,Z)

,k

k'ld

, and o(z l t a t - p 1

R(a,t)

uniformly in t for some fixed

p >

0

.

Proof. From the definition of ht(n) we have

(8.30)

which in view of (8.23) implies

Y

219

RESTRICTION M PARTICULAR SEQUENCES OF 1NTEC;ERS

m

(8.32)

B(s,k,t)

and f o r every

E >

1 b(n,k,t) n-' n=l

,

0

1 b(n,k,t)

(8.33)

=

n-

,

O(Z~-~'€) t ) '~'Z(O

since uniformly i n t

To see that (8.34) holds, we note t h a t using (8.25) we have f o r Res>l

n(1+X ( p ) f S

+0(f6) p - s + X*(p) ( p - 2 Q f ( p 2 ) ) t p - 2 s +.

P

=

n

P

(1 - x@) p - s ) -l

L(s,x)

t O(p-6)

P

n (l+O(p-6)p-St...)

P where

r[ ( 1 - x@) p-) ( 1 t x(p> p - s

.. 1

L(s,x)

C(s,x>

>

p - s t . . .)

J . M . DE KONINCK

2 20

AND

A. IVId

and

is regular f o r Re s > max (1- 6,1/2) for X = X

1

and k

, which

implies then (8.34), since

fixed,

A l l the products i n (8.24) of the type

are non-vanishing f o r Re s = 1 a regular function of

s

.

This means t h a t i n (8.24)

f o r Res > 1- p f o r some fixed

F(s,k,t) p >

0 , Using

a convolution argument we then i n f e r from (8.31) and (8.33) that

The above equation gives

(8.36)

is

RESTRICTION TO PARTICULAR SEQUENCES OF INTEGERS

I t i s now c l e a r why a condition such as (8.21) was needed. recalling t h a t from Theorem 1.4 we have f o r every

E >

221

Indeed

0

we obtain from (8.21)

f o r some

E~

> 0

.

Using p a r t i a l sumnation t o estimate

we obtain from (8.36)

which i n view of remarks made e a r l i e r gives

(8.40)

1

n ~ xn-2 , (k)

( f ( n ) n-’)*

= xG(lyk,t)

t

0(x1-’)

.

P a r t i a l sumnation f i n a l l y gives (8.29) f o r some fixed p > 0 which is not necessarily the s a m a t each stage of the proof.

J.M. DE KONINCK AND A. IVId

222

We are now ready to establish an asymptotic formula f o r the general

sum (8.17). Theorem 8.3.

Let f e

A1 , and l e t

1 or k

(k.,Z)

.

For an

arbitrary but fixed integer M z l we have

n-Z (k) where al = l / k

, and more

generally

a

(-1)

j

where E ( t ) = G(l,k,t)/(at

t

1)

j-1

E

(j-1)

(0)

, G(l,k,t)

,

being defined by the function

appearingin the statement of Lemma 8.4. Proof.

We make use of (8.26), which bounds f ( n ) away from unity,

except f o r O(xE) integers n s x nsx

satisfying f ( n ) 2 2

. Since

.

Let

c'

denote summation over those

f ( n ) >> naI2

2

for n z n

0

, we

for c > a

Since f ( n )

>>

na/'

for a l l n

, and

f(n) 2 2

for n z n o

,

have

RESTRICTION TO PARTICULAR SEQUENCES OF INTEGERS

1 f ( n ) dt= n~~:,nZ(k)

(8.44) -l/c

1

( E ( t ) xa t t l t a t E ( t ) +O(xl t a t - p

223

1) d t ,

-l/c

where E ( t ) = G(l,k,t)/(at t 1 )

.

Integration by p a r t s gives

Since E ( M ) ( t ) xat

O(1)

for t E Cl/c,Ol

, the

last above integral

is bounded, and, moreover, f o r O s i S M we have

X

di)(-l/c) x -a/c

<<

(a log xli

With uj

(8.45)

1

1-a/c

-

(l0gx)i

, it

defined by a = (-1) j-1 E (j-1) (0)

follows t h a t

j

E(t)x

attl

-l/c

dt

x

M

a

1

X

i = l ( a logx)

Furthermore

a t E ( t ) d t = O(1)

(8.46) -l/c

,

i

) dt

O(xl-')

,

-l/c

and Theorem 8 . 3 follows from (8.42)

,

( 8 . 4 3 ) , ( 8 . 4 4 ) , ( 8 . 4 5 ) , and ( 8 . 4 6 ) .

J . M . DE KONINCK

2 24

AND

A. IVId

Before finishing t h i s chapter, we would l i k e t o consider the sum

when f

E

k,

.

Using Lemna 8.4 and the f a c t t h a t

does not depend on 2

for

(k, 2 ) = 1

- $ ( k ) G(l,k,t) z a t t l

, we

of (8.28)

obtain uniformly f o r - l / a s t s 0

4 ( k ) a t G(l,k,t) at t 1

att 1

G(l,k,t)

O(xltat-p

1 -

Using the method developed i n the proof of Theorem 8.3 we obtain then the following Theorem 8.4.

Let f

E

A

and l e t k

2

1 be a fixed integer.

For an a r b i t r a r y but fixed integer M 2 1 we have

(8.49)

1'

nsx

where al = l / k

i

X

i=1 (alogz)'

(n, k) =1

, and,

more generally, a

= (-1)

,i

where E(t)

a

M l/logf(n) = @ ( k ) z1

G(l,k,t)/(at

t

1)

j-1

E

(j-1)

, where

appears i n the statement of L m a 8.4.

(0)

9

G(l,k,t)

is the function which

225

RESTRICI'ION TO PARTICULAR SEQUENCES OF INTEGERS NOTES

Only the most elementary facts concerning characters and L-series are used i n t h i s chapter.

Two of these are the orthogonality relations $(k)

c

(8.50)

x(n) = n(mod k )

and

(

$(k)

if

x=xl

if

x'xl

if

n-l(mdk)

Y

a proof of which can be found i n standard works such as Chandrasekharan C11 or Prachar C 11. Foxmula (8.9) was obtained by J . M . Tourigny in his Master's thesis (Universit6 Laval, W b e c , 1975).

Theorem 8.2 seems t o be new, while

Theorem 8.3 generalizes ThBorhme 4 of A. Mercier C11.

The identities in-

volving Dirichlet series that were used i n this chapter are also from Mercier's paper, which continues the work of De Koninck and Mercier C11. Identities involving Dirichlet series with multiplicative coefficients, characters, subseries etc., were also investigated by T.M. Apostol i n [21,

Cal and C4l. I t is possible, of course, t o investigate when

&

chapter.

is a subset of

l/f(nl

which differs from the ones used i n this

Such an example has already been given i n Chapter 2 .

an estimate for

(8.52)

W

1'

nsx, ne p

There,

J.M.

226

DE KONINCK AND A. IVI6

was obtained i n Theorem 2.9, where integers.

denotes t h e s e t of square-free

Although generalizations of (8.52), obtained by replacing

w

by other additive functions, are c l e a r l y possible, i n general t h e estima-

1'

t i o n of

6 x ,nE R

l/f(n)

fl

i s d i f f i c u l t , even i f

gers whose d i s t r i b u t i o n is well-known, such as

8

consists of inte-

= { p - 1 :p

i s prime)

.

= {n:n=k2t12,k,Z~~u{O)}

or

I n Lemnas 8.3 and 8.4 one can also take t a E

with

- l / a s Re t s 0

This would allow us t o d i f f e r e n t i a t e (8.27) with respect t o t take t 0

1

nsz

, thereby

logf(n)

.

providing an estimate f o r

1 logf(n) nsk n=l( k )

This was done by A. Mercier C11

, who

and then and

obtained f o r

(n, k) =1 example

n-l (k)

1 lOgo(n)= q z l o g x m (n, k ) =1

where p ,

tants:

t

Apx+O(log2z)

i s a fixed prime, and A1 , A 2

,

a r e two e x p l i c i t l y given cons-

.

RESTRICTION TO PARTICULAR SEQUENCES OF INTEGERS

227

Using the methods of p r o b a b i l i s t i c number theory outlined i n Chapt e r 3 and t h e results on d i s t r i b u t i o n problems of arithmetical functions obtained by Galambos (C3l

, C41)

i t i s possible t o give an estimate of t h e

type

where f(n)

is "close" t o

( 3 . 3 3 ) , and where

log l o g n

i n the sense implied by r e l a t i o n

is a set of positive integers.

However, i n t h i s

case, a much weaker e r r o r term than the one i n (3.39) is obtained.

This Page Intentionally Left Blank

CHAPTER 9 OTHER ESTIMATES AND SOME OPEN PROBLEMS

51. Introduction In this last chapter we shall give some estimates for sums which involve reciprocals of arithmetical functions, as well as a list of open

1'

problems for further research. Although the sums

1'

nlz:

n-

l/n(n)

and

l / A ( n ) are very easy to estimate, since they involve the well-horn

functions n(n)

and A(n)

, we consider them for the sake of completeness.

We also prove Theorem 9.3 which gives, in principle, a general method for estimating the sums

(9.1) and

where g is multiplicative and f is additive. In order to keep the exposition clear and concise, we do not give any generalizations of this result. The basic idea is to consider F ( s , z ) to factor out a power of

to

, using the property that

~ ( s )

multiplicative function of n result similar

1 g ( n ) z f ( n ) n-s n=l m

.

g(n)zf(n)

and is a

If we are then able to obtain (using a

Lemma 2.1) a uniform estimate of 229

J.M. DE KONINCK AM) A. IVI6

230

for

1

(21 5

, then,

broadly speaking, the estimate f o r (9.1) follows when

we d i f f e r e n t i a t e (9.3) with respect t o z

and then l e t z = 1 ; t h e e s t i -

mate f o r (9.2) follows when we divide (9.3) by z from lim

*

to 1

E(Z)

0

E(Z)

.

, where

and integrate over

z

is a suitable function satisfying

E(Z)

We would l i k e t o point out t h a t (9.9) was proved i n some-

what greater generality by De Koninck and Mercier [ I ] .

Their paper con-

t a i n s several other r e s u l t s , including estimates f o r

1

(9.4)

nsz, n: 0 (mod p , )

where p ,

w(n)

,

1

n-,nzO(modp

)

Q(n1

,

c

nsz,n=O(modp )

d(n)

Y

is a fixed prime.

Another problem which naturally arises is the estimation of the SUmS

(9.5) and 1

(9.6)

where A > 0

n 3

is a fixed constant, and f , g

a r e "well-behaved" non-negative

additive functions, such as functions belonging t o Su i n Chapter 2 .

, which

was defined

The analytic methods which we used i n e a r l i e r chapters a r e

e a s i l y adapted t o yield estimates f o r (9.5) and (9.6) i n a large number of interesting cases.

To keep the exposition c l e a r we have used these me-

thods (Theorems 9 . 4 and 9.5) only f o r the case f =

w

, g = Q ; it

should

OTHER ESTIMATES AND SOME OPEN PROBLEMS

2 31

be clear how to proceed in other cases as well.Section 3 contains a list of open problems. It is our modest hope that the problems presented here will induce further research. They occurred to us during the course of our investigations and we felt that they were of sufficient interest to be included here. While no one of these problems seems very easy, a number of them ought to be within reach. Indeed, an earlier version of this text has listed, as an open problem the proof of

as x+-

.

In collaboration with Professor P. Erdos we have proved mean-

while a much sharper result, namely Theorem 6.2.

92.

Miscellaneous estimates. Theorem 9.1

(9.7)

Proof.

Using the elementary estimate

and the prime number theorem in the form

232

J . M . DE KONINCK AND A. IVIC

which yields (9.7) a t once. Theorem 9 . 2 .

For every fixed integer

N c .x l'l/h(n) = 1 & t 0 2 nSz i = 2 log x

where c 2 = 1

, and

Proof.

the remaining

ti's

N 2

1

X

are computable constants.

W e now use the prime number theorem in the form (Prachar

Cl1, Chapter 3)

t o obtain

f o r some C1> 0

.

Since we have

p a r t i a l integration of

yields (9.8) 2

.

OTHER ESTIMATES AND SOME OPEN PROBLEMS

233

Theorem 9.3.

1d ( n ) w ( n ) = 2 z l o g z . l o g l o g z

(9.9)

n2c

+ Azlogz+O(z) ,

and

c

a

1' d (n)

(9.10)

xlogx

nsz

Bi

i=1 ( l o g l 0 g z ) i

(log log z) a+1

where A

~ ( {1i o g ( i - i / p ) t ( l / p + 3 / 2 p 2 + 4 / 2 p 2 +...I P

( 1 - -1) ~ 1

P

-

,

r'(2))

is an a r b i t r a r y fixed natural number, and t h e B i t s are a l l computable

a

constants. Proof. for

Noting t h a t

Res > 1 and

121

2

1

d ( n ) z u ( n ) is a multiplicative function of

, we

n,

have

where by Delange's L e m (Chapter 5, equations (5.6), ( 5 . 7 ) , and (5.8)) G(s,z)

(9.12)

=

1 ( 1 - p -s ) 22

(1+2~~-~+3zp-~~+...)

P

is absolutely and uniformly convergent f o r Re s > 1 / 2 any fixed A

>

0

the r e s t r i c t i o n m

.

(9.13)

Iz

I

5 A

for

Now using Lemma 2 . 1 (where i t may be e a s i l y shown that

It I

1 Ibt(n) I n - l log3 2n

n=l

and

2

1 may be replaced by

It I s B

is uniformly bounded f o r

It I

if 5 B)

we f i n d

J.M. DE KONINCK AND A. IVI6

234 where, uniformly for

Iz(

s 1

,

R ( t , z ) << tlog2Re 2-25

(9.14)

.

To obtain (9.9), we differentiate (9.13) with respect to z

Setting

G(1

2)

=

H(z)

and estimating

we obtain uniformly for

= H'

121 _<

(2) x

Setting z = 1

1

%?&

az

.

by Cauchy's inequality,

,

10g2=l t + 2H(z) 3: log2z-1t log logz +o(x)

, noting that H(1) = 1 , and then computing

we obtain (9.9). To prove (9.10) assume that z

,

gives, uniformly in z

(9.16)

.

is real and O < z ~ l Then (9.13)

C'

d ( n ) Zw(n)-l

nSx

K ( z ) x log2z-1z+ O(z2-l)

y

.

where K ( z ) G(l,z)/(zr(Zz)) is continuous on C0,ll Let E ( X ) = log- m x , and integrate (9.13) over z from ~ ( tto ) 1 to obtain

1

t E

K ( z ) log2z-1z.dz+O(tlogl/E(t))

.

OTHER ESTIMATES AND SOME OPEN PROBLEMS

235

Noting t h a t

writing

and subsequently i n t e g r a t i n g by p a r t s

K ( z ) log2'-l

2.

dz

, we obtain

(9.10), since 2

and

"

K ( z ) log2z-1

2.

log 1 / E

(2) << 2

dz << log 2 E (2)- 1

0

log l o g 2

,

1

.J

-1

l K ( z ) I dz << log

5

.

0

Theorem 9.4.

If

A> 0

is a fixed number and a r l

is a fixed

integer, then 1

(9.18)

, and t h e remaining

Ci's

.+ O I ( l o g l o g 2 ) a + 1

a r e computable constants depending

. W e start from equation ( 2 . 2 2 ) , viz

Proof.

(9.19)

2

i

where C 1 = l on A

a

I

,w(n) = ~$#rlogt-lx n 3

+ O(210gt-22) ,

J . M . DE KONINCK AND A. I V I C

236

where the 0-constant i s uniform f o r

0
5

1

.

Multiplying by tA-l we

obtain

(9.20)

lftW(n)tA-l n%r

% (1 t ) tA-lz l o g t - ' x

where the 0-constant is again uniform f o r

t

0
O(t-'x logt-2x)

2

1

.

,

Now we define

h(t)

bY

(9.21)

h(t) + g

and integrate (9.20) over t the proof of Theorem 2 . 4 .

t

Noting that

1 t ) tA-l

*= g

from E(X)

(1 t ) tA

(logx) -1/2(a+2)

to 1

, as

We then obtain

X

( t ) logt-l x . d t

.

in

OTHER ESTIMATES AND SOME OPEN PROBLEMS

237

and evaluating the right-hand side of (9.22) as in the proof of Theorem 2.4, we obtain (9.18) with

ci

Theorem 9.5. If

Ci(A)

a>

1'

n*

such that

a

1 = x 1 Di i n) fi n i = 2 (log log x)

J(fJJ

Y

2 is a fixed integer, then there exist com-

putable constants D2,...,D (9.23)

i-1 (i-1) h = (-1) (1)

X

Proof. We start from equation (4.7), viz.

where the 0-constant is uniform for 0 < t s l

(9.25)

H(t,u;s)

=

n (1 - p

P

-c L,

, 0 < u s 1 , and

tu (1 t tup-S t tu2p -2s

t

tu3p-3s +

. . .)

is absolutely and uniformly convergent f o r Res > 1 / 2 , inrplying that, for Jtl s 1

,

IuJ

5

1

,

(9.26)

where the a .

,-j

2,

E(2)

-

= (logz)

are suitable constants. From (9.24) we obtain, f o r

I/2 ( a t 2)

Y

238

(9.27)

J.M. DE KONINCK AND A. I V I 6

1]

E(X)

where K(t,u)

E(X)

1,t~( n )- 1 U n(n)- 1 dtdu nsx

is defined by

<<

1

X logx

E XI

Furthermore we have

1

? J ) dkY

<<

z(loglogz)2 log x

OTHER ESTIMATES AND SOME OPEN P R D B W

Integration by p a r t s gives

j

'(9.31) E

K ( t , U ) logtu-12 . d t

($1

= c

(-1)i-l

at1

i = 1 (log l o g 2 ) i ui

-

From (9.26) it i s seen t h a t

is a regular function of

(9.32)

j

K(t,u)

ata+'

E. (XI

<<

j

u

*

IuI < 1

for

tu-1 log

, which

implies

at

at1

U

1ogtu- l2.

at

l o g u - l x - log = u log l o g x

E

(x) u-1X

E (2)

Repeated i n t e g r a t i o n by p a r t s then y i e l d s

il

K ( t , u ) 1ogtu-12. d t d u

E 2) E 2)

a+1

c

(-1f-l

i = 1 (log l o g z )

i

j

M(l$,i)

E (2)

logu-12. &

2 39

J.M. DE KONINCK AND A. IVIe

240

(9.33)

+

*I

1+

1

(loglogx)

a

= i = I2

1

u ( l o g l o g x ) a + 2 d'

OIJ)

Di

OI

+

(log l o g x )

10gu-l x

1

1.

(loglogx)a+l

which, combined with (9.27) and (9.30), gives Theorem 9.5. I t i s clear t h a t Theorem 9.5

( h i t h e r t o unpublished) can be gene-

ralized t o give

(9.34)

C' f1(d* 1..f,(n) n*

where the Ei's functions

83.

= x

a

Ei 1 i =k (log log x ) i

X

+ OI

(loglogx)"+l

I'

a r e computable constants, f o r s u i t a b l e "small" additive

fi (such as w and Q), belonging t o

Open problems.

1. Define p , ( n )

% , p*(n)

=

Q (n)

Sa

, for

and p*(l) be undefined. In a c e r t a i n sense p , ( n ) sent the average prime f a c t o r of

n

.

.

n22

and l e t p,(l)

and p*(n) both repre-

Is it true t h a t , a s

x-tm

,

(9.35) and

1 l/p*(n) 2Sn3

(9.36)

2.

loglogx

.

1 l/B(n)

?

2SnSx

Find asymptotic formulae f o r t h e sums

1

23Kx

and

241

(TTHFiR ESTIMATES AND SOME OPEN P R O B W

1.

-1'S2(n)

,

which are related t o the sum

2SnsX

l'l/f3(n) n<x

.

Is it true t h a t , as x+- ,

3.

f o r some A > 0 ? This conjecture should be compared t o Theorem 6.2.

similar formula should hold when B(n) i s replaced by either B1(n)

(9.38)

B(n)

A

or

, and more generally we expect l1l/f(n) = x exp(-(B + o(1)) logy/(yt1) x

nsx

for f e H

, which w a s

. (log logx) (Y+W(Y+l))

defined i n Definition 6.1.

4. Estimate

(9.39)

5.

Estimate

(9.40)

(We believe t h a t t h i s sum is equal t o

6.

(9.41)

Estimate

(C + o (1)) x log log x , as x + m

)

.

J.M. DE KONINCK

24 2

(We believe that t h i s sum is equal t o

AND (C t

A. IVIf

o(1)) 2

, as

z - + m , f o r some

suitable c 2 0 ) .

7.

Estimate

(9.42)

where p ( n )

is the largest prime factor of

sunis equal to

(Dto(1))xloglogx

, as

n

.

x+m

(We believe that t h i s

, for

some D > l ) .

Estimate sums which are mre general than those appearing i n

8.

problems 3, 4 , 5, 6 , and 7 , such as

where

P>

0

9.

is a fixed number.

Let A be the s e t of non-negative arithmetical functions f

satisfying

(9.43)

Describe the set A

(for example, we have shown t h a t

Q(n) , d(n)

and the constant functions belong t o A .)

10.

Derive sharp asymptotic formulae f o r sums of the type

24 3

OTHER ESTIMATES AND SOME OPEN PROBLEMS

(9.44)

that is, f o r quotients of additive functions one of which behaves "like

, and

c logn"

t h e other is a "small" additive function. These problems

should be compared t o t h e r e s u l t s of Chapter 4 .

11. Investigate t h e sum

(9.45)

is an integer tending t o

where N

, and

an

is an increasing sequence

of positive integers such t h a t

(9.46)

where L ( z )

is a slowly o s c i l l a t i n g function such that

l i m L(z) F

While it is obvious t h a t (9.45) i s O(1ogN)

, the

0

.

problem appears

t o be very i d f f i c u l t even when one considers the special case of estimating

(9.47)

a(x1

7

I t is not hard t o see t h a t

Beginning with

one might argue h e u r i s t i c a l l y as follows :

2

When

Jim i n f S > 0 . N+-

N-tm

we have

J . M . DE KONINCK AND A. IVIC

244

S =

"j'

N-u

2-0

which implies that

,jl

dr(u)

2

du

=-'

2

N/2

N

2

N j2

l i m sup S <

w

t

m

N/2

.

A weaker, but more reasonable conjecture i s

S = o(log1ogN)

, as

N+m

.

(See Erd6s C11).

12.

One may also consider estimates f o r the sums

(9.48) and (9.49)

where f

is a non-negative additive function, and k

analogous multiplicative problems of estimating

1

f(n

k
terest.

- k)

when f

B

1 f(n nSx

1z f ( n ) n a

t

k)

The

and

is a multiplicative function a r e of a great in-

The basic method of Chapter 2 used t o estimate

f i r s t considering

i s fixed.

N

l'llf(n) nlx

by

f a i l s when one t r i e s to estimate (9.48) o r

245

OTHER ESTIWTES AND SOME OPEN PROBLEM3

(9.49).

This is because the function zf(ntk) is not multiplicative,

whereas z f(n)

was a multiplicative function of n

.

13. A problem analogous to 12 also seems interesting. This consists of estimating

(9.50) and (9.51)

where f is a non-negative additive function and k e N is fixed. This appears to be very difficult even in the case of simple additive functions

such as fi or

w

.

One may argue heuristically that, as x+m

since the average order of ~ ( n ) is loglogn

.

,

The corresponding mul-

tiplicative problem o f estimating

(9.53) and (9.54)

where f

(9.55)

is multiplicative, is also difficult. Thus, in order t o prove

1

k<+x

d@

- k)

x n (1t P ;Ik

xloglogx)

,

J . M . DE KONINCK AND A. IVI6

246

(the so-called Titchmarsch divisor problem) one has t o combine sieve re-

sults with Bombieri's theorem on primes i n arithmetic progressions (see Halberstam-Richert C11, pp. 110-112).

14.

denote the r - t h l a r g e s t prime f a c t o r of n

Let P,(n)

some results concerning C11).

Pr(n)

(for

see t h e paper of K . Alladi and P. Erdos

Find an estimate f o r

(9.56)

15.

i s an additive, non-negative arithmetical function

If f(n)

satisfying

(9.57)

as

, what

s + m

16.

is the set of primes p

f o r which f(p) = 1 ?

Estimate

(9.58)

T h i s sum seems t o be of i n t e r e s t , since u(n) visors of n

, while

is the sum of di-

B(n) i s the sum of d i s t i n c t prime divisors of n

,

so t h a t the quotient o(n)/B(n) represents, i n some sense, t h e degree of 'tompositeness" of n perty i s

.

A "smaller" function which enjoys a similar pro-

OTHER ESTIMATES

AND

SOME OPEN

24 7

PROBLEMS

and an estimate for the s m a t o r y function of this last function i s furnished by Theorem 9.3.

A similar problem would be t o estimate

(9.59)

17.

L e t f(n)

be a non-negative function f o r which, a s x + w

(9.60)

is a slowly o s c i l l a t i n g function such t h a t

where L ( x )

l i m L(x)

x+-a

=

.

Under what conditions does (9.60) imply t h a t , as x + m

,

(9.61)

(This is true f o r many well-known functions satisfying (9.60) such as

w(n)

, logo(n) , etc.) 18. Let f(n)

ture t h a t f o r

(z(5

1 ,h =

h2

xi'4tE

o(z)

1

(9.62)

as x + *

s a t i s f y the hypotheses of Theorem 7.1.

x<&th

, where )

G(z)

, h r x 1'4+E

zf(n) =

and

( G ( z ) t o(1))

is as i n Theorem 7.1.

E

h

>0

We conjec-

,

,

A similar conjecture (with

whould presumably hold throughout Chapter 7 .

24 8

J.M. DE KONINCK AND A. IVIE 19.

where f

The problem of estimating

is a non-negative additive function i s not e n t i r e l y solved by

the r e s u l t s of Chapter 8.

In p a r t i c u l a r , we again point out the problem

of estimating

where

It I s 1 and

x

i s a non-principal character modk

, which,

in

view of the discussion i n Chapter 8 , 12, i s closely r e l a t e d t o t h e behaviour of the sum (9.63).

20.

An i n t e r e s t i n g , but d i f f i c u l t problem which was not touched

i n any of the previous chapters is t o obtain the so-called I f g(z)

Q-estimates.

is an increasing real-valued function s a t i s f y i n g limg(z) x-)-

f(z) is a real-valued function, then the notation

as z- t-

, means

t h a t there is a s u i t a b l e constant C > 0 such t h a t

f(z) >

Cg(z)

o r respectively

holds f o r an i n f i n i t e number of values of z

.

f(z) <

-

Cg(z)

Furthermore

-,

OTHER ESTIMATES AND SOME OPEN PROBLFAE

means that both

Qt

and Q- hold, while f(x) = n(g(x))

249

means

A classical theorem (see Prachar C1l)in prime-number theory s t a t e s t h a t

1

=

(9.65)

nlx

0-type theorems provide

A(n) = x

t

~-+ ( x l / log ~ log logx)

.

an insight i n t o the true order of e r r o r

terms i n asymptotic formulae f o r arithmetical functions, although very often there is a disappointing discrepancy between the best-known and the best-known

0-result.

0-result

Thus, f o r instance, one can compare (9.65)

with the best-known 0-result (see Walfisz C11)

In many of the formulae we have obtained, it seems d i f f i c u l t t o formulate a possible

N

0-result, since we have asymptotic expansions where

i s a fixed arbitrary integer, such as i n Corollary 5.3, where we had

(9.67)

l'l/w(n) = alxL1(x) ?6X

t

. . . t a f l L N ( x ) logl-Nx

t

O(x log-'x)

,

although it would be interesting t o how the'true order of the left-hand side i n (9.67).

have

In many other cases, such a s i n Theorem 3.4, where we

250

J.M.

DE KONINCK AND A. IVId

an il-result would be highly desirable.

Among possible conjectures con-

cerning il-results we mention here

(9.69)

C'B(n)/B(n) = x

I. il+(x1'2)

nSx

w h i c h should be compared with Theorem 6 . 2 . enough t o be within reach.

-

,

This conjecture seems weak

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l11 On an additive arithmetic function, P a c i f i c J. Math. 71 (1977), 275-294. Apostol, T.M. c11 Mathematical Analysis (Addison-Wesley Publishing Co., Mass., 1957). L21 I d e n t i t i e s of t h e type 40 (1973), 341-345.

c f ( n ) ~ ( nn-')

C 31 Note on s e r i e s of t h e type 23 (1975), 49-50. c43

, Proc. h e r .

Zf(n) p ( n ) n-'

, Nordisk

Math. SOC.

Mat. Tidskrift

Arithmetical subseries of s e r i e s with m u l t i p l i c a t i v e terms, B u l l . Greek Math. SOC. 18 (1977), 106-120.

Ayoub, R.

C11 An introduction t o t h e a n a l y t i c theory of numbers (American Mathematical Society, Rhode Island, 1963). Bateman, P.T. and Grosswald, E. [I1 On a theorem of ErdOs and Szekeres, I l l i n o i s Journ. Math. 2 (1958), 88-98.

.

Brinitzer , E [I1 Eine asymptotische Formel ftlr Summen tiber d i e reziproken Werte a d d i t i v e r Funktionen, Acta Arith. 32 (1977), 387-391. 25 1

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Chandrasekharan, K. C11 Arithmetical functions (Springer-Verlag, Berlin-Heidelberg-NewYork, 1970). Cohen, E. 111 Arithmetical functions associated w i t h t h e unitary divisors of an integer, Math. Zeit. 74 (1960) , 66-80. C21 Unitary products of arithmetical functions, Acta Arith. 7 (1961), 29-38.

De Bruijn, N.G. C11 On the nunber of integers sx whose prime factors divide n I l l i n o i s J. Math. 6 (1962), 137-141.

,

C21 On the nunber of integers sx and f r e e of prime factors >y Nederl. Akad. Wetensch. Proc. Ser. A54 (=Indag. Math.), 13 (1951), 50-60. C3l

On the nunber of positive integers sx and f r e e of prime fact o r s >y 11, Nederl. Akad. Wetensch. Proc. Ser. A. 69 (=Indag. Math.), 28 (1966), 239-247.

.

De Koninck, J.M. C11 On a class of arithmetical functions, Duke Math. Journ. 39 (1972), 807-818. C2I S m of quotients of additive functions, Proc. h e r . Math. SOC. 44 (1974), 35-38. C3l

Some remarks on additive functions, Journ. Nat. Science and Math. 1 7 (1977), 31-41.

De Koninck, J.M., Erdus, P. and I v i z , A. C I A Reciprocals of c e r t a i n large additive functions, Can. Math. B u l l . ( t o appear)

.

De Koninck, J.M. and Galambos, J. C1J Suns of reciprocals of additive functions, Acta Arith. 25 (1974), 159-164. De Koninck, J . M . and I v i t , A. [I] An asymptotic foxmula f o r reciprocals of logarithms of c e r t a i n multiplicative functions, Can. Math. Bulletin 2 1 (1978), 409-413.

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C21

253

Sums of reciprocals of c e r t a i n a d d i t i v e functions, Manuscripta Mathematica, 30 (1980) , 3029-3041.

De Koninck, J.M.

and Mercier, A.

C11 Remarque sur un article de T.M. Apostol, Canad. Math. B u l l . 20 (1977), 77-78.

Delange, H.

C11 Sur des formules dues 3 A t l e Selberg, B u l l . S c i . Math. 2, 83 (1959), 101-111. C21

Sur des formules de Atle Selberg, Acta Arith. 19 (1977), 105-146.

C31 Sur un th6or;me de E n y i 111, Acta A r i t h . 23 (1973), 153-182. Dixon, R.D. C11 On a generalized d i v i s o r problem, Journal of Ind. Math. SOC. 28 (1964) , 187-196. Duncan, R.L.

C11 On the f a c t o r i z a t i o n of i n t e g e r s , Proc. Amer. Math. SOC. 25 (1970), 191-192. C21

Some applications of t h e Turk-Kubilius inequality, Proc. Amer. Math. SOC. 30 (1971), 69-72.

Erdos, P.

C11 Problems and results on combinatorial number theory 111, i n : Number Theory Day (ed. B.M. Nathansen), LNM 626, (Berlin-Heidelberg-New York, 1977). Erdos, P. and Szekeres, G . C11 Uber d i e Anzahl der Abelschen Gruppen gegebener Ordnung und uber e i n verwandtes zahlentheoretisches Problem, Acta S c i . Math. Szeged 7 (1935), 95-102.

Galambos, J . [ I ] On the d i s t r i b u t i o n of prime independent number t h e o r e t i c a l funct i o n s , Arch. Math. (Basel) 19 (1968), 296-299.

"21 The asymptotic theory of extreme order statistics (Wiley, New York, 1978).

[: 37

Distribution of arithmetical functions. A survey, Ann. I n s t . Henri Poincarb, Sect. B, 6 (1970), 281-305.

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2 54 C41

Distribution of additive and multiplicative functions, in: Theory of Arithmetic Functions (Lecture Notes Series, Vol. 251, Berlin, 1972).

Grosswald, E. C11 Topics from the theory of numbers (The Maanillan Co., New York, 1966). C2]

The average order of an arithmetic function, Duke Math. J . 23 (1956), 41-44.

Halberstam, H. and Richert, H.E. [: 11 Sieve Methods (Academic Press , London-New York-San Francisco , 1974). Hardy, G.H. and Ramanujan S. Cll Asymptotic formulae for t h e distribution of integers of various types, Proc. London Math. SOC. 2 (1977), 112-132; (S. Ramanujan, Collected Papers, Chelsea, 1962). Hardy, G.H. and Wright, E.M. C11 An introduction t o the theory of numbers (Third Edition, Oxford University Press, 1960). Heilbronn, H.

C11 Uber den Primzahlsatz von Herrn Hoheisel, Math. Zeit. 36 (1933), 394-423.

Herstein, I . N . C1-l Topics i n algebra (Glaisdell, Waltham, Mass .-Toronto-London, 1964).

Hoheisel, G. C11 Primzahlprobleme i n der Analysis, S i t z . Preuss. Akad. Wiss 33 (1930) , 580-588. Hwcley, M.N. C11 On the difference between consecutive primes, Invent. Math. 1 5 (1972) , 164-170. Ingham, A.E. C11 The distribution of prime numbers (Hafner, New York, 1971).

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c21

On the estimation of (1940) , 291-292.

C31

On the difference between consecutive primes, Quarterly J. Math. (Oxford) 8 (1937), 255-266.

N(u,T)

Math. (Oxford) 11

Ivi;, A. c11 On the asymptotic formulae for some functions connected with powers of the zeta function, Mat. Vesnik (Belgrade), 1 (14) (29) (1977), 79-90. C2I The distribution of values of some multiplicative functions,

Publs de 1'Inst. Math. (Belgrade) 22 (36), 1977, 87-94.

C3l

On the asymptotic formulas for powerfull numbers, Publs de 1'Ins. Math. (Belgrade) 23 (37), 1978, 85-94.

c41

On sums of large differences between consecutive primes , Math. Annalen 241,(1979) , 1-9.

C51

A note on the zero-density estimates for the zeta function,

C6l

The distribution of values of the enumeratine function of nonisomorphic abelian groups of finite order, AFchiv der Math. 30 (1978) , 374-379.

Archiv Math., 33 (1979), 155-164.

C71 On the number of non-isomorphic finite abelian groups in short

intervals Math. Nach. (to appear).

Iwaniec, H. and Jutila, M. C11 Primes in short intervals, Arkiv fur Matematik, 17 (1979), 167-176. Jutila, M.

C11 Zero-density estimates for L-functions, Acta Arith. 32 (1977) ,

55-62.

Karamata, J. [I] Sur un mode de croissance rggulikre des fonctions, Mathematica (Cluj) 4 (1930), 38-53. Knopfmacher, J. C11 Abstract analytic number theory, (North-Holland/American Elsevier, Amsterdam-Oxford-New York, 1975). Kolesnik, G .A. c'11 On the order of 5(1/2+it)

and A ( R )

, Pacific J. Math.

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( t o appear). C21 On t h e estimation of c e r t a i n trigonometrical sums (Russian), Acta Arith. 25 (1973) , 7-30. Kubilius, J. C11 Probabilistic methods i n the theory of numbers (Transl. Math. Monographs , h e r . Math. SOC , Providence , R. I. , 1964)

.

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Landau, E

.

[ I ] Handbuch der Lehre Von Der Verteilung Der Primzahlen I (Chelsea

Publishing Co.,

New York, 1953).

Mercier, A. C11 Somes de fonctions additives r e s t r e i n t e s 3 une classe de congruence, C a n . Math. B u l l . 22 (1979), 59-73.

..

Montgomery, H L Cl1 Primes i n arithmetic progressions, Mich. Math. J . , 1 7 (1970), 33-39. [23

Zeros of L-functions,Invent. Math. 8 (1969), 346-354.

Niven, I . and Zuckerman, H.S. Cl1 An introduction t o t h e theory of numbers (John Wiley & Sons, New York-London-Sydney, 1966)

.

Prachar, K. [11 Primzahlverteilung (Springer-Verlag, Berlin-Heidelberg-New York, 1957). Rankin, R.A. C I A V a n der Corput's method and t h e theory o f exponent p a i r s , Quart e r l y J . Math. (Oxford) (2) 6 (1955), 147-153. S n y i , A. [I] On the density of c e r t a i n sequences of integers, Publs. de 1 ' I n s . Math. (Belgrade) 8 (1955) , 157-162. Richert, H.E. C11 Lber d i e Anzahl Abelscher Gruppen gejebener Ordnmg, Math. Zeit. 56 (1952), 21-32.

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London Math. SOC. 29 (1954), 16-20.

Schmidt, P.G. C11 AbschPtzungen bei unsymmetrischen Gitterpunktproblemen,Dissertation, Gijttingen, 1964. Schwarz, W. [II Einige Anwandungen Tauberscher SPtze in der Zahlentheorie B, J. reine angew. Math., 219 (1965), 157-179. C21 Eine Bemerkung zu einer asymptotischen Formel von Herrn RGnyi,

Archiv der Math. 21 (1970), 157-166.

Selberg, A. C11 Note on a paper by L.G. Sathe, Journal of Ind. Math. SOC., 18 (1954), 83-87. Seneta, E. [li

Regularly varying functions (LNM 508, Springer-Verlag, Berlin, Heidelberg, New York 1976).

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Vinogradov, I.M. Cl1 Elements of number theory (Dover, New York, 1954). Walfisz, A. [I1 Weylsche Exponetialsummen in der neueren Zahlentheorie (VEB Verlag der Wissen., Berlin, 1963).

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D a s asymptotische Verhalten von Sumnen Uber multiplikative Funktionen I, Math. Annalen 143 (1961), 75-102.

SUBJECT INDEX

Additive function (definition) : 29.

, 230 , 246, 248. Arithmetic progression: 201f(*I Cauchy's inequality for coefficients of a power series: 10, 119, 132. Cauchy's inequality for the derivative of an analytic function: 98, 104, 140, 167, 186, 234. Cauchy-Schwarz inequality: 157, 160. Cauchy's theorem: 5. Convolution of arithmetical functions (definition): Dedekind's function

J, :

3.

71.

Density hypothesis: 194. Dirichlet divisor problem: 195f. Divisor function d(n) : xii, 21, 169f, 178f, 195f, 230, 233f, 242, 245f. Divisor function dk(n) : xii, 3f, 75f, 117. Euler's constant

y :

xiii, If, 21f, 30, 44f, 178f, 195f.

Euler $-function: xii, If, 21f, 65f, 71, 96, 106, 215f, 224f, 243f.

260

DE KONINCK AND A. IVIC

J.M.

Euler's t o t i e n t function 4 * : 71, 90. Exponential divisor of an integer:

81, 88, 93.

Finite abelian groups (properties) : 25. Gamma function:

x i i i , 4, l l f , 22, 25, 33f, 76, 84, 97f, 131, 167, 208,233f.

Generalized sum of divisors function ah : 66. Greatest prime divisor of an integer:

x i i i , 151f, 163, 173, 242.

Greatest squarefree divisor of an integer: 221. Hardy-Ramanuj an's theorem:

x i i , 16 , 28 , 68 , 149 , 213f ,

16.

Inversion formula f o r Dirichlet series (definition) : 26. Landau's formula:

145.

Large deviation theorem of Kubilius:

80, 91.

Levin-Feinleib method of integral equations : 25. Mabius inversion formula: Mobius function: 212f, 225.

68, 179,

x i i , I f , 22f, 48, 62f, 82, 92, 149, 151, 169, 177f, 199,

Multiplicative function (definition) : 1. Non-isomorphic f i n i t e abelian groups (number o f ) : 186, 199.

x i i , 13, 81, 88, 93,

Non-isomorphic semisimple f i n i t e rings of order n (number o f ) : 186. Partial summation technique:

22.

88, 93,

261

SUBJECX INDEX

P a r t i t i o n s of an i nt eger (number of unr es t ri ct ed): Perron's formula:

26, 198.

Powerful integers:

145f.

Prime-independent function:

x i i , 13, 88.

13, 78, 81, 88, 100.

Prime f a cto r s of n (number of d i s t i n c t ) : v i f , x i i , 30, 4 2 f , 61f, 78f, 95, 99, 103f, 107f, l l l f , 119, 132f, 140f, 152f, 157, 163f, 167f, 177f, 187f, 190f, 203, 205f, 214f, 225, 230, 233f, 245f. Prime f a cto r s o f n ( t o t a l number o f ) : x i i , 30, 45f, 62f, 78, 95, 99, 107f, l l l f , 120, 132f, 1 4 0 f , 157, 164, 167f, 187f, 190f, 205f, 230, 237f, 245f. 17, 152, 1 7 1 , 232, 243.

Prime number theorem:

P r o b a b i l i s t i c number theory: Renyi's problem:

7 7 f , 90f, 100, 227.

132, 144.

Slowly o s c i l l a t i n g (or varying):

v i i , 78, 1 1 2 , 134f, 170, 243, 247, 249.

Square-free in te ger s :

82, 132, 137, 152f, 164f, 189, 199, 226.

Square-full in te ger s :

135, 140, 145, 187.

S t i e l t j e s i n t e g r a l representation: S t i r l i n g ' s formula:

1 7 , 23.

170.

Strongly ad d itiv e function:

134.

Sun of a l l prime f act or s of

n : v i i i , x i i , 149f, 156f, 231, 240f, 247,

250.

Sum o f a l l d iv isor s of 243, 246f.

n : x i i , 2 1 f , 65f, 71, 89, 96, 106, 211, 226, 231,

262

J.M. DE KONINCK AND A. IVI6

Sum of distinct prime divisors of n : viii, xii, 149f, 156f, 240f, 246, 250. Sum of distinct prime powers that exactly divide n : xii, 149f, 156f, 241f. Tauberian theorem: 16, 26, 28, 124. Titchmarsh divisor problem: 22, 246. Turan-Kubilius inequality: 63. Uniqueness theorem for Dirichlet series (definition) : 27. Unitary analogue of the sum of divisors function u* : 71, 90. Unitary convolution of arithmetical functions (definition) : 24. Von Mangoldt function: xii, 176, 192f, 196, 232, 249. Zeta function: xiii, 1, 3f, 33f, 42f, 82f, 97f, 100, 108, 121, 124, 127f, 144, 150, 165, 171f, 175f, 192f, 209, 219f, 229, 233.

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