CAMBRIDGE TRACTS IN MATHEMATICS General Editors
B. BOLLOBAS, P. SARNAK, C.T.C. WALL
118
Sets of Multiples
Richard R. Hall York University
Sets of Multiples
AMBRIDGE UNIVERSITY PRESS
CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi
Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521109925
© Cambridge University Press 1996
This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1996 This digitally printed version 2009
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication data Hall, R. R. (Richard Roxby) Sets of multiples / Richard R. Hall, p. cm. - (Cambridge tracts in mathematics : 118) Includes bibliographical references (p. -) and - index. ISBN 0 521 40424 X (hc) 1. Sequences. I. Title. QA246.5.H33 1996 512'.72 - dc20 95-39233 CIP ISBN 978-0-521-40424-2 hardback ISBN 978-0-521-10992-5 paperback
To the memory of my mother and father
Contents
page ix
Preface Introduction Notation
xi xv
0.2 0.3 0.4 0.5
First ideas Introduction Sets of multiples and primitive sequences Densities The Heilbronn-Rohrbach and Behrend inequalities Total decomposition sets
1
Besicovitch and Behrend sequences
1.1
1.3
Introduction Erdos' criterion Behrend sequences
1.4
Witnesses
2
Derived sequences and densities Introduction Upper bounds for tk(sad) Generalized Behrend inequalities Multilinear functions
0 0.1
1.2
2.1
2.2 2.3 2.4 2.5
Formulae for the densities tk(d)
3
Oscillation
3.1
Introduction A first lower bound for (sd, d)
3.2 3.3 3.4 3.5
Upper bounds for (sad, sad)
Primitive d Perfect sequences vii
1 1 1
3
13
22 26 26 26 36 60 66 66 68 82 88 91
96 96 105 112 117 122
Contents
viii
Probabilistic group theory Introduction The Erdos-Renyi theorem: first variant The Erdos-Renyi theorem: second variant The Behrend sequences Q*(t) A conjecture of Erdos and Renyi Divisor density Introduction Necessary and sufficient conditions Slowly switching sequences Slowly switching sequences: a reformulation
126 126 130 146 153 167
6.2 6.3 6.4 6.5
Divisor uniform distribution Introduction Applications of divisor density Weyl sums and additive functions Weyl sums and discrepancy Lower bounds for discrepancy
203 203 205 213 218 226
7
H(x, y, z)
239
7.1
Introduction Short intervals The asymptotic formula for E Rk The asymptotic formula for H(x, y, z)
239 240 244 250
4 4.1 4.2 4.3 4.4 4.5 5 5.1
5.2 5.3 5.4
6 6.1
7.2 7.3 7.4
170 170 179 192 198
Bibliography
258
Index
263
Preface
In this book I describe some of the developments which have taken place in the theory of sets of multiples since Halberstam and Roth's Sequences was published in 1966. My object is twofold: to give a coherent account of the general theory as it exists today, and to encourage others
to study an elegant and, perhaps to some persons, surprisingly subtle subject in which I believe much progress is possible. There are still many unsolved problems, some of them arising from the most recent work, and I have attempted to fit these necessarily loose ends into the text in such a way that the reader can see at which point a new idea is required. One of the attractions of the subject is the great variety of techniques which can be employed: thus one chapter (not the easiest) consists entirely of elementary inequalities, another involves Dirichlet series, contour integration and exponential sums, while a third
is probabilistic. Where probabilistic methods have been used, I have presented them in an accessible fashion as a non-probabilist writing for (perhaps mostly) non-probabilists. This tract is a companion volume to Cambridge Tract No. 90, Divisors,
written with Gerald Tenenbaum some years ago. Although there are references to Divisors (I refer to this book by its name throughout) at several points, Sets of Multiples is self-contained and can be read by persons unfamiliar with this area. I have quoted freely from joint papers with two collaborators: Paul
Erdos and Gerald Tenenbaum. It is almost automatic that in any long-standing collaboration some of the results will be due, on different occasions, entirely to one or the other author. Without embarking on
any sort of catalogue at this juncture I should like to make clear that many of the ideas in the book are due to my collaborators, particularly in Chapter 1 where I quote from Hall and Tenenbaum (1992) and Erdos, ix
x
Preface
Hall and Tenenbaum (1994). For example the fine, short proof of Erdos' criterion for Besicovitch sequences is Tenenbaum's. Where results were published individually by any author this will of course be clear from the references.
I should like to acknowledge my great debt to Paul Erdos who introduced me to this subject and from whom I have learned, or had the opportunity to learn, so much. I shall always be grateful to Gerald Tenenbaum for his collaboration - his results in this area speak for themselves. I wish to record my thanks to Heini Halberstam for his encouragement during 1990 to write this book, which was essentially planned during the British Mathematical Colloquium at the University of East Anglia that
year. Finally I am grateful to David Tranah for his patience and good spirits: I undertake not to write another book for Cambridge University Press in longhand. York
Introduction
The study of sets of multiples began in the thirties as an abstraction from one special problem. A number of mathematicians, including Behrend, Chowla, Davenport, Erdos and Schur, had been interested in abundant numbers, (the positive integers not greater than the sum of their proper
divisors), in particular whether the proportion of such integers < n converged to a limit with increasing n. This was proved by Davenport (1933), using an analytic method involving the Stieltjes moment problem
due to Schoenberg (1928), which Schoenberg had applied to a similar problem about the Euler 4-function. A few months later Erdos (1934) gave an elementary proof of this theorem; general ideas which developed into the subject now called sets of multiples can be discerned clearly in both these proofs. We shall not be concerned with abundant numbers in this book, nevertheless it may be helpful to use this historical example as
an illustration. We note the property that any multiple of an abundant number is abundant. This leads to the idea of a primitive abundant
number, which is minimal in the sense that its proper divisors are not abundant. The abundant numbers then comprise all the multiples of these primitives. This immediately raises general questions about the sequence, or set, of integers which are multiples of the elements of a given base sequence, for example (as above) whether the former, top, sequence possesses asymptotic density. In general the answer is no (Besicovitch (1934)). We call a sequence whose set of multiples does possess asymptotic
density a Besicovitch sequence (this terminology is due to the present writer and Tenenbaum). Erdos observed that a sufficient condition for this conclusion would be the convergence of the series of reciprocals of the base sequence (which he could establish in the case of abundant numbers).
This led to much work on general primitive sequences (in which no element divides another), and to Erdos' criterion (1948a) for a sequence xi
xii
Introduction
to be Besicovitch. Meanwhile Davenport and Erdos (1937) proved that every set of multiples possesses logarithmic density, and Heilbronn (1937) and Rohrbach (1937) independently obtained the inequality which bears their names. This is a precursor of Behrend's inequality (1948). The complement of a set of multiples comprises the integers not divisible by any elements of the base sequence, and so there is a connection with the theory of sieves. Usually the emphasis in the two subjects is rather different: in sieve problems the sieving set consists simply of primes, but the sifted sequence will be the values of a polynomial, possibly at prime arguments. In sets of multiples the sifted sequence is nearly always Z+: the complications arise from the fact that the elements of the base sequence need not be coprime. Also, if this sequence is infinite, there is no analogue of a sieving limit. We regard the base sequence as fixed and we have to say what we can about the distribution of the set of multiples. One question which has arisen repeatedly in Erdos' work is whether the set of multiples has density 1, or equivalently whether almost all integers have at least one divisor of a particular type. To formalize
this I have called a sequence whose set of multiples has density 1 a Behrend sequence. Quite general necessary conditions for a sequence to be Behrend are now available; the sufficiency conditions are still rather specialized. Another leitmotif in Erdos' work has been to refine the base sequence to a threshold at which the set of multiples possesses positive density determined by a statistical law: examples occur in Chapters 4 and 7. More recent avenues of research include oscillation (we prove a-theorems, i.e. that the oscillations are large), and derived sequences, which relate to the following question. What can we say about the density of the integers which are multiples of two, or k, elements of the base sequence, given only the density of the set of multiples itself. This part of the subject is dominated by elementary but delicate inequalities and there are new problems which may be difficult.
Each chapter of the book has an introduction and so I shall not attempt a resume of the contents here. However I think it might be useful at this point to suggest possible orders in which the chapters could
be read. Chapter 0 is essential, and most readers would follow this with Chapter 1, which contains basic information about Besicovitch and Behrend sequences. Some choices are now possible. Chapters 2 and 3 go together in this order. They comprise the material on derived sequences and oscillations mentioned above and are elementary and combinatorial in nature, certainly presenting a new face to the subject. Chapters 5 and 6 also combine in order, dealing with divisor density (an alternative
Introduction
xiii
to asymptotic/logarithmic density which is particularly appropriate to this subject) and divisor uniform distribution. The techniques involved here are mathematically more sophisticated but also familiar: many analytic number theorists would find these chapters the easiest place to start after Chapter 1. A general problem about multiplicative functions, which seems to me fairly fundamental, arises and is explained at the end of §5.2. The discrepancy lower bounds and double variance lemma in Chapter 6 are the most recent work in this Tract. Chapter 4, on probabilistic group theory, concerns an idea of Erdos for constructing Behrend sequences (among many other applications), and is important for a full understanding of other parts of the book, but could be read at any time. I suggest that some of the technical details might be skimmed at first reading. Some of the hardest problems in the book, of importance outside number theory, occur here; a brief introduction to these is given in §4.5. Finally Chapter 7 presents the solution of a conjecture from Divisors about Tenenbaum's function H(x, y, z) which counts the set of multiples of an interval. This is a central example, but no other chapter depends directly upon it.
Notation
d a positive integer sequence, or a family of sets.
Jf(d) the set of multples of .. d. the asymptotic density of M. d, d upper, lower asymptotic density. 6 logarithmic density. T(n) the number of (positive) divisors of n. i(n, d) the number of divisors of n which belong to sd. i(n, y, z) as above, with d = (y, z]. co(n) the number of distinct prime factors of n. cw(n, t) the number of distinct prime factors p of n such that p < t. co(n, s, t) the number of distinct prime factors p of n such that s < p < t. S2(n) the number of prime factors of n counted according to multiplicity. 0(n, t), 0(n, s, t) similar to w, but the prime factors are counted according to multiplicity. P+(n), P-(n) the greatest, least prime factor of n. a I b a divides b. a' II b a' divides b but ar+i does not.
F(x, y) = card In : n < x, P-(n) > y}. 'V(x, y) = card In : n < x, P+(n) < y}. Exp expectation. The following is an index, by section numbers, of notation introduced in the text.
g(d)
0.2
A(d)
0.3
6 =.086071 ...
0.3
M(x) .T("V)
0.3
0.4 xv
xvi
t(d) d'(b) E(f;M) {d(S)} vp(a)
O(x, y, z) H(x, y, z)
d'(t)
Q(2)=2 log2-A+1
'cl(k)
tk(d) Irk
Pk
(Pk(6)
qk(6) tk,n
E (x, d), E (x, d) (SI, SI)
p(x) = [x] - x + Go(z)
Dd
0(n;f)
Notation
0
First ideas
0.1 Introduction
In this chapter we state and prove some of the classical results about sets of multiples and we make some definitions which have arisen in the more recent theory. This is presented in the later chapters.
0.2 Sets of multiples and primitive sequences
Let c = jai, a2,a3.... } be a non-decreasing sequence of positive integers. It may be finite or not, and we allow repetitions. We do not require that
al > 1 but this will usually be the case, when we shall refer to d as being non-trivial. The set of multiples of sad, which we denote by comprises all the distinct, positive multiples of elements of .sad. Thus if we define
i(n, si) = card {a : a I n, a E d}
(0.1)
.11(d) = In : n E Z+,r(n,Q) >_ 1}.
(0.2)
then we have
It is traditional to refer to ./#(d) as a set of multiples, but for the most part it will be better to regard it as an ordered sequence.
We may have 41 # d but t(sd1) = '/&(d2). We begin with the observation that in this circumstance we also have
(sail n d2) = l(d1).
(0.3)
Let us prove this by contradiction. Suppose (0.3) is false. Plainly nd2) c l(d1) so there exists an integer b e #(Vj)\,#(d1nA2): we may suppose it is the least such. Then b = meal = m2a2 where a, E ci for i = 1, 2 and mi E Z+. Since b . !(sa/1 nS12) we have a1 * a2. Suppose 1
0 First ideas
2
a1 < a2. Then m1 > 1 and a1 < b. Since a1 E . (d1) this implies, by the minimal property of b, that a1 E 4'(d1 n d2), i.e. a1 = m3a3 where a3 E d1 n d2, m3 E Z. We now have b = (mlm3)a3 E .#(d1 n d2), which is the required contradiction. This proves (0.3).
Let 9(d) denote the intersection of all the sequences d' for which M(d). Then
# (3 (d)) = ol(d),
(0.4)
moreover 9(sad) is a primitive sequence, that is a sequence of positive integers none of which divides any other. There is a substantial literature on primitive sequences, quite independent of the theory of sets of multiples,
and we record some of the properties of these sequences here. Let 9 be primitive, with counting function P (x) = card{9 n (0, x1 j.
(0.5)
We have
supP(x) _
[±!]
(0.6)
on the one hand because 9 = Z n (2 x, x] is an example and on the other because for each odd integer m < x, P(x) counts at most one integer n of the form 2km, (k = 0, 1, 2, ...). Erdos (1935b) showed that always 1
aEY
a log a
< 00,
(0.7)
and in the same volume of J. London Math. Soc. Behrend proved that
a1
:a<x} «
ll a
JJJJJJ
logx
(0.8)
90 g x
Neither of these results implies the other. Pillai (1939) gave an example to show that (0.8) is qualitatively best possible, and the question of the sharp constant on the right proved to be a resistant problem, still unsolved when
Halberstam and Roth (1966) appeared. Erdos, Sarkozy and Szemeredi (1967a,b) obtained essentially final results for this problem. We have sup
E{ aE.g/
111 a
:a<x} <(1+0(1)) J))JJJ
logx 0 90 g x)
(0.9)
0.3 Densities
3
but for each fixed 9, as x -> oo,
aV a
: a < x I =0 ( (log og x) ) .
(0.10)
Each of (0.9) and (0.10) is best possible. In view of (0.4) we may, for many purposes assume that Q( is primitive, indeed this will be essential in Chapter 3, §3.4. However, in both Chapters 2 and 3 when we deal with derived sequences, (these are defined in §2.1) ////
we shall find that such a reduction is inappropriate: to obtain the most general derived sequences we even allow 4 to contain repeated elements. A review of the literature concerning sets of multiples reveals that primitive sequences have not played as large a part as may have been expected. The results above demonstrate that a primitive sequence need
not be at all thin (certainly not enough to be useful), moreover the structural constraint ai%aj is quite awkward to handle.
0.3 Densities Various definitions of the density of an integer sequence are possible. We
shall be concerned with asymptotic, logarithmic and divisor densities, and we consider the first two of these here. divisor density is the subject of Chapter 5. Let A'' be a subsequence of Z+ with characteristic function X. We say that A' possesses asymptotic density dA' if X(n) = (dam + o(1)) x, (x -> oo).
(0.11)
n<x
In any event there exist
lim sup x X-00
X(n), lim inf x i
1
n<x
X--oo
X(n)
(0.12)
n<x
and we refer to these as the upper and lower asymptotic densities of A-, denoting them by dal'' and d''. We say that A' possesses logarithmic density Sir if x(n) n<x
n
= (S'' + 0(1)) log x, (x ->
00)
(0.13)
and there are definitions of upper and lower logarithmic density analogous to (0.12). We sometimes say, simply, `dir exists' instead of `A'
4
0 First ideas
possesses asymptotic density dl''. If dit'' exists then so does 6Y, moreover Sam' = d.. The converse is false: the most that we can say is that if S.'' exists then
< oir < dr.
d.
(0.14)
For example the sequence 00
U {(22m-1 22m1 n Z}
(0.15)
m=1
has upper and lower asymptotic densities 3 and 1 respectively, and logarithmic density 1. We prove the right-hand inequality in (0.14): if we apply this to the complement of .'' the left-hand inequality follows. Let the sum on the left of (0.11) be denoted by K(x). We have
1: n<x
x(n)
= x-1K(x) + I K(t)dz
(0.16)
.
By hypothesis K(x) < (dA' + E)x for x > x0(e) and, trivially K(x) < x. Hence for x > xo(E),
E &)n< 1 + log xo(E) + (a_* + E) log n<x
We divide through by log x and let x 1
log x n<x
X-00
(0.17)
oo to obtain X(n)
8r= lim sup
(_f).
< d,*' +
(0.18)
n
This holds for every e > 0 whence 8-*' < d.', and this implies (0.14). Our first two theorems which follow show that logarithmic density will play a central role in the theory of sets of multiples. Theorem 0.1 Let d be finite, or such that 00
1 < oo. E i-1
(0.19)
a1
Then d./#(d) exists. This is best possible in the sense that if fi(x) -> 00 as x -> oo, there exists a sequenced such that
E{I 1111 a,
but d.t(d) does not exist.
a; < x } < 1(x) J))JJJ
(0.20)
0.3 Densities
5
The essential part of this theorem, and the part which is not straightforward, is that there exist sequences d for which the set of multiples do not possess asymptotic density. This is due to Besicovitch (1934). The next theorem is due to Davenport and Erdos (1937), (1951). Theorem 0.2 For every Q,
exists. Moreover the logarithmic density
and lower asymptotic density are equal, that is &#(d) = Proof of Theorem 0.1 When sl is finite #(d) is a finite union of arithmetic progressions and so possesses asymptotic density. Moreover we may derive a formula for the density from the inclusion-exclusion principle : if d = {al, a2, ... , an } then
ddf(d) _
1
ai
[ai aj i<j
i
+
1 i<j
-.
.
.+
[a,,(-1)n_1 a2,..., an]
(0.21)
in which we have used the standard square brackets notation for least common multiple.
Next, let s./ be infinite, and (0.19) hold. Let .,(sl) comprise the positive integers n for which ailn but for j < i, aj%n. Let Xi(n) be the characteristic function of and X(n) be the characteristic function of ms(s). Then 00
(0.22)
X(n) = I Xi(n). i=1
Every .I&i(d) is a finite union of arithmetic progressions and possesses asymptotic density d.Ri(si), for which there is a formula similar to (0.21). We deduce from (0.22) that for all N, N
d. # (d) < dl #(,,V)
(0.23)
i=1
whence 00
dA 1(sal) < d
(sal).
(0.24)
i=1
On the other hand we have for all N N
dM(sd) <
00
dM1(s./) + i=1
(0.25) i=N+1 ai
0 First ideas
6
and we deduce from (0.19) and (0.24) that fore > 0,
d. #(d) < A(d) + e.
(0.26)
Thus, subject to (0.19), dt(d) exists and
dt(d) = A(sal).
(0.27)
Finally we drop condition (0.19). We require the following result of oo, we have Erdos (1936): as T
d(T) := dN(Z n (T,2T]) - 0.
(0.28)
For completeness we shall prove this in Lemma 0.3 below; let us assume (0.28) for the moment. By the definition of asymptotic density, for each T there exists xo(T) such that for all x > xo(T) we have card{n : n < x, n E(Z n (T, 2T])} < 2d(T)x.
(0.29)
Let {Tk : k E Z+} be a sequence of integers satisfying the three conditions 00
Ed(Tk) < E,
(0.30)
k=1
card{k :
Tk < x} < fi(x) for all x,
Tk+1 > xo(Tk) for all k.
(0.31)
(0.32)
These are consistent, all requiring that Tk should be large. Let 00
d = U{Zn (Tk,2Tk]}
(0.33)
k=1
and consider the counting function M(x) of Tk whence
For all k, M(2Tk) >
2. Next,
M(Tk) <
2d(Th)Tk < 2sTk
(0.34)
h
by (0.29) and (0.30). Therefore dA1(d) < 2e, and we choose c < 4, does not exist. We notice that for positive integers T we whence have
< log 2
(0.35)
T
so that (0.31) implies (0.20). This proves Theorem 0.1.
Notice that for this proof it is sufficient to have lim inf d(T) = 0,
0.3 Densities
7
the result proved and employed by Besicovitch (1934). The following quantitive version of (0.28) is Lemma 1 of Erdos and Hall (1980). Lemma 0.3 Let d(T) be defined by (0.28). Then T)-b
d(T) << (log
(0.36)
where
6=1-
1 - log
= .086071...
(0.37)
Tenenbaum (1984) obtained a more precise result in which the righthand side of (0.36) is divided by (log log T), and gave a lower bound for d(T) which establishes that the exponent 6 is sharp. We discuss the matter further in Chapter 7.
Proof Let S2(n, T) denote the number of prime factors of n which do not exceed T, counted according to multiplicity. We begin by showing that for each fixed yo < 2 we have E yO(n,T) << x(log T)Y-1,
(0.38)
n<x
uniformly for 0 < y < yo, 2 < T < x. This may be proved by the inequality of Halberstam and Richert (1979) (Theorem 00 of Divisors) which gives the upper bound yn(n,T)
x
logx>n<x
(0.39)
n
for the sum in (0.38). We have a(n T)
y n
1
< <<
f \1-y/ P
p
71
p
I )-Y log X tog T P
log x(log T)Y-1
\1n<x
p
(0.40)
by Mertens' theorem. We combine (0.39) and (0.40) to obtain (0.38). Now consider the integers n < x which have a divisor d E (T, 2T]. We split these into two classes according to whether S2(n, T) < K log log T or not, (with K at our disposal). For arbitrary y, (0 < y < 1) the cardinality
0 First ideas
8
of the first class does not exceed y-K log log T
T
yc2(n,T)
nSx
1
dIn
T
(log
T)-Klogy
T^ yri(dT) E yO(m,T)
(0.41)
m5(x/d)
T
on inverting summations and writing n = md. We estimate the inner sum by (0.38), replacing x/d by x/T, to obtain <:
T (log T
)v-1-K logy
yn(dT) T
« x(log T
)2y-2-K log y
(0.42)
using (0.38) again. We choose y = x/2 (which is in order provided x < 2
as we need y < 1) to minimize the exponent of log T. This becomes K - 2 - K log(K/2) = -6 if K = 1/ log 2. We estimate the cardinality of the second class similarly, but more crudely since we ignore the divisibility condition. For arbitrary z, (1 z < yo) this does not exceed Z-K loglogT
E
ZS2(n,T)
T)Z-1-KIogz
(0.43)
n<x
by (0.38). We choose z = K = 1/ log 2 which again makes the exponent of log T equal to -S. We assemble these estimates to complete the proof. We have written out a detailed proof of Lemma 0.3 with two objectives, first that the reader should have a complete proof of Besicovitch's important theorem and second that he should understand the role of the variables y and z, since this method will appear repeatedly throughout the book. We require a lemma for the proof of Theorem 0.2. This is the Tauberian theorem due to Hardy and Littlewood (see Hardy (1949)) and Karamata (1931). We state this with sufficient generality for applications later in the book. Lemma 0.4 Let f (t) be non-decreasing and such that for some fixed a > 0 we have
Jf00° e °`d f (t)
F(a + 1) a
U
as
-4 0+.
(0.44)
0.3 Densities
9
Then as t -> co, we have f (t) - t". In particular, if we set bn, (bn >_ Ofor n E Z+),
f (t) =
(0.45)
n<et
then the hypothesis (0.44) becomes
-
b1+n
00
F(OC + 1) -01
as a --> 0+
(0.46)
n=1
and the conclusion is that
E
bn
- (log x)"as x --+ oo.
(0.47)
n<x
For a proof see for example Hardy (1949), Chap. VII. Proof of Theorem 0.2 Let JI (d) be as defined in the proof of Theorem 0.1, that is the sequence of positive integers such that ai n, aj%n for j < i. has asymptotic density d.fli(d) and characteristic function I
Xi(n), and we have
0 00
X(n)
d` #i(d) as s -> 1+.
(0.48)
n=1
We denote the function of s on the left of (0.48) by F1(s)t'(s), where c(s) is the Riemann zeta function, and we have from (0.48),
0< Fi(s) < 1, (s > 1); lim Fi(s) = d.#i(d).
s1+
(0.49)
Let GN(s) = Fl (s) + F2(s) +
. + FN(s).
(0.50)
By (0.22), Xl (n)+X2(n)+ +XN(n) < X(n) < 1 (where X is the characteristic
function of .t(d)), whence GN(s)?(s) < c(s) and GN(s) < 1. Hence we may define 00
G(s) _
Fi(s)
(0.51)
t=1
and we notice that N
lim inf G(s) >_ lim GN(s) _ s-.1+
s-.1+
d.iLli(d)
(0.52)
i=1
by (0.49), whence
lim inf G(s) > 0(d) S-.1+
where A(d) is defined in (0.24).
(0.53)
0 First ideas
10
For any large positive integer no there exists N sufficiently large that 1 < n < no.
+ XN(n),
X(n) = Xl(n) + X2(n) +
(0.54)
Hence for fixed s > 1 and e > 0 there exists No(s, e) such that for N > No(s, e) we have 00
Xnn)
- e <
X(n)
n=1
(0.55)
n=1
Therefore 00
X(n) ns
n=1
=
s > 1.
(0.56)
We claim that for each fixed N, we have
S>1.
dsGN(S)<_0,
(0.57)
To see this, let us denote the sum on the right-hand side of (0.54) by X(N)(n) so that we have 00
E
(N) n
X
=
n
n=1
(0.58)
GN(sX(s)
We may differentiate the Dirichlet series term by term and (employing dashes in the usual way) we obtain
0 00
X
(N)
n log n (n g
= -G'N(s)C(s) - GN(s)Ns).
(0.59)
n=1
Since X(N) is the characteristic function of a set of multiples we have X(N)(n) > XM(d) for every n and divisor d of n. Therefore X(N)(n) log n =
(n
)
X(N)(n) E A dIn
(0.60)
X(N)(d)A (d) din
(where A is von Mangoldt's function), whence for s > 1 we have 00
E n=1
00
n
X(N) (ri
log
>-
d=1
A(m)
(d)
X(
>
d
(GN(S)S(S))
m=1
-V(s) C(s)
(0.61)
0.3 Densities
11
It follows from this and (0.59) that -G'N(s) >- 0 as required. We deduce from (0.49), (0.50) and (0.57) that for all s > 1 we have
+ d. # (d)
GN(s) < d,# 1(d) + &W2 (d) +
(0.62)
whence
G(s) < A(d),
s > 1,
(0.63)
and so in view of (0.53) we have
lim G(s) = A(d). s1+
(0.64)
We substitute s = 1+ a, (a > 0) in (0.56) and we deduce from (0.64) that 00
x(n)
0
(0.65)
x -+ co.
(0.66)
n=1
whence Lemma 0.3, with a = 1, yields x(n) ti A(d) log x, n<x
Therefore
n
exists and is equal to 0(d). We have 4A1(d) <
S H(am) by (0.14) and A(sd) < dmo(d) by (0.24) whence dmo(d) _ &#(d). This proves Theorem 0.2. Let d(N) comprise the first N elements of a. We notice that the proof also gives, by (0.24)
8#(4) = lim d l(d("')). N-oo
(0.67)
We have followed the original proof of Davenport and Erdos (1937). Perhaps influenced by the recent elementary proof of the Prime Number Theorem, in 1951 Davenport and Erdos gave a new proof of Theorem 0.2 which eliminated the Hardy-Littlewood-Karamata theorem and which is in this sense more elementary. The later proof is presented in Halberstam and Roth (1966). We believe the pre-war proof is the natural way to tackle this theorem; moreover we shall need the Hardy-Littlewood-Karamata theorem later in the book. We conclude this section with two definitions. These are each quite recent, but they simply isolate and formalize two central problems in this subject which have been present for many years. Definition 0.5 We say that a sequence 4 c Z+ is a Besicovitch sequence if its set of multiples .mo(d) possesses asymptotic density.
0 First ideas
12
For example, every finite sequence is a Besicovitch sequence. The formal definition appears in Erdos, Hall and Tenenbaum (1994). It is a little paradoxical that the sequence which Besicovitch constructed should be non-Besicovitch, nevertheless it seemed clear to these authors that his name should be attached to the sequences with a positive property. Definition 0.6 We say that a sequence d c Z+\{1} is a Behrend sequence if its set of multiples mo(d) has logarithmic density 1. This definition appeared in Hall (1990b). By the Davenport-Erdos theorem (Theorem 0.2) we may deduce that if d is Behrend then d-&(d) = 1 and so an equivalent, but logically more complicated, definition would be to require that .1(d) possesses asymptotic density 1. A further consequence of (0.67) is that .sad is Behrend if and only if it is non-trivial and
lim d.A'(,(N)) = 1,
_QI(N)
N-+oo
= {al, a2, ... , aN }.
(0.68)
Let d be a (non-trivial) sequence of pairwise coprime elements. Then the familiar inclusion-exclusion formula gives
d,#(d")=1-
N C1
I
a;
whence ./ is Behrend, in this special case, if and only if a
: aEs =00.
(0.69)
It will emerge that in the general case this condition remains necessary, but can be rather weak. One thread of the book will be to find applicable, realistic criteria for a given sequence to be Besicovitch or Behrend and we begin this study in Chapter 1. We regard the notion of a Behrend sequence as the more fundamental: it would be possible to construct a coherent theory of sets of multiples without any reference to asymptotic
density (an idea with which the author of this work, perhaps rather briefly, toyed).
Erdos and Tenenbaum have noticed recently the following interesting fact about Behrend sequences. Theorem 0.7 Let sd be Behrend and M(x) denote the counting function of ol(d). Then we have, uniformly in x,
M(x + y) - M(x)
Y, (y - 00).
(0.70)
0.4 The Heilbronn-Rohrbach and Behrend inequalities
13
Proof Let e > 0 be fixed. By (0.68) there exists N = N(s) such that 1 - (s/2), where as before d(N) comprises the first N elements of q. Since .i#(d(N)) is a finite union of arithmetic progressions its intersection with (x, x+y] has cardinality d.1&(d(N))y+0(1), uniformly
in x. Hence there exists yo = yo(s) such that for y > yo(s) we have M(x + y) - M(x) > (1 - s)y, and (0.70) follows.
0.4 The Heilbronn-Rohrbach and Behrend inequalities
Clearly for all sequences d and . we have ol(d) c .iGZ(d U 9). We consider the quantitative effect of adjoining . to 4 and we begin with the case where .
has just one element b. Notice that the conditions a%n, b I n
(0.71)
are equivalent to
n=bm, a'%mwhere a'=
() a,ab
.
(0.72)
Let us write
.% (d) = Z+ \ J((/)
(0.73)
and
,,Y'(b) _
at
=
(a ab) : a E A
}
.
(0 . 74)
We employ the letter .9 for the complement because of the now almost standard notation
t(d) =1-
(0.75)
We deduce from (0.72) that
.l(d u {b}) = '(d) u b (d'(b)) .
(0.76)
If b .#(d) then a' > 1 always, that is Ql'(b) is non-trivial. For example if c is finite then .9 (d'(b)) has positive asymptotic density and .4(d U {b}) has strictly greater density than #(d). Theorem 0.8 Let b 0 .%((d ). A necessary and sufficient condition that
sd/(d U {b}) > &#(d)
0 First ideas
14
is that the sequence d'(b) defined in (0.74) should not be a Behrend sequence. In any case we have s
(.d U {b}) <
(1-
b) sa(d) +
(0.77)
Proof The union on the right-hand side of (0.76) is disjoint and so
U {b}) = SJI(d) + bt (,q/'(b)) ,
(0.78)
and t (d'(b)) = 0 if and only if sd'(b) is Behrend. This proves the first part. For any b we have .i (d' (b)) s 9 (d) whence t (.,5('(b)) < t(d),
which we insert in (0.78) to obtain (0.77). This completes the proof. A corollary is the next theorem, proved independently by Heilbronn (1937) and Rohrbach (1937).
Theorem 0.9 (Heilbronn-Rohrbach) Let d = {al, a2, ... , aN }. Then
t(d)
ri N
(1-a;1)
(0.79)
=
Proof We can rewrite (0.77) in the form
t(d U {b}) > (1 - b) t(d)
(0.80)
and (0.79) follows by induction. In fact (0.80) is the most useful form of the inequality and is sometimes loosely referred to as the Heilbronn-Rohrbach inequality. Strictly this is (0.79). An immediate consequence is
Corollary 0.10 A necessary condition for d to be a Behrend sequence is that (0.69) holds.
The example q = 2Z+ shows that (0.69) is insufficient for c to be Behrend. We construct a rather more interesting example after the next theorem. Before proceeding to this we give another definition, suggested by Theorem 0.8.
Definition 0.11 We say that the sequence W is taut if, for every a e W, we have
t(d \ {a}) > t(d).
(0.81)
0.4 The Heilbronn-Rohrbach and Behrend inequalities
15
A taut sequence is necessarily primitive, and every finite primitive sequence is taut. A Behrend sequence cannot be taut, by (0.80), and it may well be primitive: the sequence of primes is an example. We shall be able to classify the taut sequences later. (Corollary 0.19.) The following result generalizes (0.80) and is due to Behrend (1948).
Theorem 0.12 (Behrend's Inequality) For all 9 and -4 we have
t(d U q) >_ t(.d)t(a).
(0.82)
Equality holds if (a, b) = 1 for all a E a, b E 4. If V and R are finite, primitive sequences this condition is necessary for equality to hold.
We give some examples of equality at the end of the proof. Corollary 0.13 Let sat = -41 U d2 U ... be a Behrend sequence. Then 00
E 610Wk) = oo.
(0.83)
k=1
We notice that the sequence d as above with S1k = Z n
(ek12
2ek12]
(0.84)
is not Behrend, by Lemma 0.3 and (0.83). On the other hand it fulfils (0.69).
Corollary 0.14 The sequence d U 9 is a Behrend sequence if and only if at least one of sad and -4 is Behrend. In particular any tail of a Behrend sequence is Behrend.
The next corollary is due to Ruzsa and Tenenbaum (199x). I have not seen their proof of this interesting observation and give my own. Corollary 0.15 (Ruzsa-Tenenbaum) Any Behrend sequence may be split into an infinite disjoint union of Behrend sequences.
Proof Let a be the given Behrend sequence. We construct 41, d2, d3, ... by apportioning the successive elements of at to these subsequences in consecutive finite runs in the order at1, at2, d1, at2, at3, at1, ate, at3, at4, d1, ... etc.; at each stage the remaining tail of at is Behrend by Corollary 0.14, moreover at the nth stage we ensure that the particular subsequence being augmented has t < 1/n. We can do this by virtue of (0.68); this same result then establishes that every ai is Behrend.
0 First ideas
16
Corollary 0.16 If .sad is Behrend then
i(n, d)
oo p.p.
(0.85)
This follows from the previous corollary. See also Hall and Tenenbaum (1992).
Theorem 0.12 is important and we give two proofs, Behrend's own and
that of Ruzsa (1976). Another proof appears in Halberstam and Roth (1966). These proofs are quite distinct. Behrend considered the cases of equality in his paper but Ruzsa did not. For the sake of completeness, and furthermore because we require his interesting method later in the book, we supply the extra ingredient needed for this question in Ruzsa's proof here. We begin with Behrend's
proof: we note that originally Behrend restricted his result to finite sequences (the extension is easy) and both proofs begin with this case.
Proof (i) (Behrend) Lemma 0.17 Let d be any non-empty sequence and c be a positive integer. Let -4 be a (possibly empty) sequence all of whose elements are prime to c. Then
I 8d(cd U -4) = s.(.4 u -4) +
(1 - !)
(0.86)
Proof Let -4/- = .11(d) \ M(R). We consider the integers n belonging
to J1(cd U 4) but not to ^4). This requires that n = mca for some m E Z+, a E .sad such that b%mca for any b E -4. Since (b, c) = 1 by hypothesis, this last condition is equivalent to bjma for any b, that is n E c.K. Hence
!Z(cd U 9) = #(R) U c.N'
(0.87)
is a disjoint union and so
SJI(cd U -4) = 6-&(R) + 6X. c
(0.88)
We may substitute c = 1 in (0.88) and eliminate 6X between the result obtained and (0.88) itself. This proves the lemma.
Now assume d and 9 finite, and write a(d) for the sum of the elements of Q. We proceed by induction on N = o(d) + a(M). If d is empty we put a(d) = 0 and t(sar) = 1. We note that (0.82) holds if either s/ or R is empty or contains l's (is trivial). It also holds if (a, b) = 1
0.4 The Heilbronn-Rohrbach and Behrend inequalities
17
for every a E d, b E a. This is a consequence of the inclusion-exclusion formula
_
1
-
1
1
(0.89) + ... + i<j [a1, ai] i<j
1-
i
ai
out and the expression for t(d U -4) separates.
From the foregoing, (0.82) holds when N = 0. Let N > 0: the induction hypothesis is that (0.82) holds for every d and -4 such that a(.4) + 6(s) < N. We may assume that v and . are non-trivial and that there exist a and b such that (a, b) > 1. Let p be a prime which divides at least one element of each sequence. We write
d = p(i 1 U d2, -4 = Pal U -4 2
(0.90)
where Q2 and 22 but not 41 and 21 may be empty. By the lemma, we have
1
t(d1 U 4) + I 1 -
l/ t(d2))l (
t(d1 U 42)t(,41 U -42) +
-1
C1
- -1
)+
(!t(1 U
(i_ p) t(42) I
t(d2)t(,42)
(1 - p) {t(d1 U d2) - t(d2)}{t(A U 42) - t(-42)}.
(0.91)
We have t(d1 U -Q/2) < t(d2) etc. so that the last term on the right is
< 0. Also v(di) + 6(d2) = o-(d) - (p - 1)601) < v(d) whence 6(d1 U d2) + U01 U 42) < a(&/) + Q(s) = N.
(0.92)
By the induction hypothesis we have both t(d1 U 9/2)t(,41 U -42) < t((d1 U -41) U (d2 U -42)) and
t(d2)t(.2) < t(d2 U 142) We insert these inequalities into (0.91), striking out the non-positive final term. This yields
I t((d1 U -41) U (d2 U 42)) + t(p(d1 U A) U (d2 U -42))
C1
- p) t(S12 U 42)
0 First ideas
18
= t((Pd1 U S/2) U (P-41 U -42))
= t(d U -4)
(0.93)
employing the lemma again. This completes the induction and establishes the theorem for finite sl and M. The extension to the general case is common to the two proofs and is a straightforward application of (0.67).
To establish the last assertion of the theorem we show that if d and M are finite primitive sequences and for some a, b we have (a, b) > 1 then
t(d u M) > t(d)t(R). Let p be a prime factor of (a, b) and write d and .4 as in (0.90). Our assertion will follow from (0.91) if we can show that
t(d1 U d2) < t(d2), t(A U 42) < t(,42).
(0.94)
We refer to Theorem 0.8. Let a"l> E c1 (which is non-empty). By hypothesis sI is primitive, whence a(2)%pa(') and so a(2)%a(l> for any a(e) E sd2. Hence
,2(aill) =
a(2)
(a(1) al2i)
:
a(2) E d2
(0.95)
is non-trivial and as it is finite, therefore not Behrend. We consider M similarly and apply Theorem 0.8 to obtain (0.94). This completes the proof. Before proceeding to Ruzsa's proof we give some examples of equality
in (0.82), in which (a, b) > 1 for some a E 4, b E A If W and . are finite but not primitive, Behrend's own example is c = {2, 4}, The elements 4 and 6 are simply redundant.
.
= {3, 6}.
Now let 4 and -4 be primitive but not finite. Let eP; denote the sequence of primes congruent to i (mod 8) and put
d = Si Yu {35}, 9 = 7Y3 U {35},
(0.96)
so that c and . are primitive and share an element. We have 3,
1 because Y1 and g 3 are Behrend. The element 35 is not
redundant but it does not affect the densities: we hardly need Theorem 0.8 in this case but it is useful to check that the sequences (5Y1)'(35) = Y1, (7°y3)'(35) = Y3
(0.97)
are each Behrend. Clearly &t(sl U -4) = 11/35. Finally we remark that if either d or -4 is Behrend then (0.82) holds with equality. Proof (ii) (Ruzsa*) We say that the arithmetic function f is multiplicatively non-increasing if f (md) < f (d) for all m, d E Z+. The proof involves * The purely number theoretic version of this is due to Ruzsa and Tenenbaum (199x), except for the equality condition, which is new.
0.4 The Heilbronn-Rohrbach and Behrend inequalities
19
sums of the form E(f ; M)
M
f (d)
(0.98)
dI M
where q is Euler's function and M E Z+. Lemma 0.18 Let f and g be multiplicatively non-increasing arithmetic functions. Then for all M > 1 we have
E(f g; M) >- E(f ; M)E(g; M).
(0.99)
There is equality in (0.99) if and only if f and g split M, that is there exist F and G such that M = FG, (F, G) = 1, and for every divisor d of M we have f (d) = f ((d, F)), g(d) = g ((d, G)).
Proof of lemma
(0.100)
We proceed by induction on k = co(M). If k = 1
then M = pv for some prime p, and the property that f and g are non-increasing implies
(f(pi) - f (ph)) (g(pi) - g(ph)) >- 0, (0 < h, j < v). We multiply by cp(p
(0.101)
h)cp(pv-i) and sum over all h and j to obtain
2E(fg;M)-2E(f;M)E(g;M)>-0.
(0.102)
Notice that f and g split pv if and only if at least one of them is constant on the divisors of p". Therefore if f and g do not split p° we have both
f(1) > f(p°), g(1) > g(p°) by the monotonicity, so that the left-hand side of (0.101) is positive when h = 0, j = v. We therefore have strict inequality in (0.102) and (0.99) as required. Hence the lemma holds when k = 1. Let co(M) = k > 1, and p% 11 M. We put M = Ml p° and define, for each j = 0,1, ... , v, the functions fj, gj by setting f3(d) = f (p'd), g3(d) = g(pid). These are obviously multiplicatively non-increasing. Now 1
E(fg;M) = M V iP
(Mi(P° ild)) fi(d)gi(d) J=o djM,
1
vp >
I pv
E(p(pv-i)E(figj;Mi) i=o
Eco(p° i)E(fi;Mi)e(g.i;Mi) 1=0
(0.103)
0 First ideas
20
where we have used, in the second step, the fact that Euler's (P-function
is multiplicative, and in the third step the induction hypothesis that (0.99) holds for w(Mi) = k - 1. We define f(pj) = E(fj;Mi), g(pj) _ E(gj; Mi), (0 < j < v), and we have shown that
E(fg;M) > E(fg;pv).
(0.104)
The functions f,g are multiplicatively non-increasing and we may apply (0.99) with k = 1 to obtain E(fS;pv) >- E(f;pv)E(8;pv).
(0.105)
Since
E(f;pv) =
1
p
y(p(pv-j)E(fj;Mi)
= E(f;M)
j=o
with a similar equation involving g, we may assemble (0.104) and (0.105) to obtain (0.99). It remains to consider the cases of equality, and we
leave it to the reader to check that the condition that f and g split M is sufficient for this. Suppose, conversely, that there is equality in (0.99). Then we must have equality in (0.105), moreover in (0.103) we must have
E(f jgj;Mi) = E(f j;Mi)E(gj;Mi) for every j, (0 < j < v). We apply the induction hypothesis in each case. Firstly, either f or say f, is constant. In view of the monotonicity of f, this implies that f j is independent of j, that is f (pjd) = f (d) for every divisor d of Ml. Secondly we require that for every j, f j and gj split Ml, that is f and g split Ml. This means that for every j there is a factorization Ml = FjGj in which (Fj, Gj) = 1 and f (d) = f ((d, Fj)) , gj(d) = gj ((d, Gj)) , dIMi. Let F = h.c.f.(Fo, Fl,..., Fv). We deduce that f (d) = f ((d, F)) for every d dividing Ml, and hence for every d dividing M. Also gj(d) = gj ((d,MI/F)) for every divisor of Ml and hence g(d) = g ((d, G)) for every divisor of M, where G = p"Ml/F = M/F. Thus f and g split M as required, and (0.99) holds when w(M) = k. This completes the induction and the proof of the lemma. Now let Xd be the characteristic function of the set of non-multiples of d. We notice first that x,,, is multiplicatively non-increasing, and second that, provided M (E Z+) is a common multiple of the elements of .sad, we have Xd(n) = xd ((n, M)), (n E Z+).
(0.106)
0.4 The Heilbronn-Rohrbach and Behrend inequalities
21
For each divisor d of M there are cp(M/d) congruence classes (mod M) in which (n, M) = d whence
t(d) = M r, T
(i)
X.d(d) = E(X,,v, M)
(0.107)
dIM
and we deduce (0.82) from (0.99), taking M to be the least common multiple of the elements of both Q and -4. It remains to establish the last part of the theorem concerning the cases
of equality. Let d and . be finite and (a, b) = 1 for all a E d, b E A Then X.j and Xa split M (as above): we take F to be the least common multiple of the a's and G that of the b's, whence (0.99) and (0.82) hold with equality. The extension to the infinite case with (a, b) = 1 is clear. Next let d and -4 be finite primitive sequences and for some a and b let (a, b) > 1. Let M = FG where (F, G) = 1. Then either (a, F) < a or (b, G) < b, whence Xd ((a, F)) * X,1 (a) or Xa ((b, G)) * X,a(b), because C and -4 are primitive. Hence M is not split and (0.99) and (0.82) are strict. This completes proof (ii) of Theorem 0.12.
We mention one further consequence of Behrend's inequality at this point, concerning taut sequences. We recall that d is taut if and only if
t(d \ {a}) > t(d) for every a E sd. We count d _ {1} among the taut sequences.
Corollary 0.19 The non-trivial sequence d is taut if and only if it is primitive and does not contain a sequence 64 in which . is Behrend.
Proof Let cR c a with
.
Behrend.
Then 8(4l (a) n cZ) = 1/c,
and this is unchanged if we remove an element of cam. This cannot affect n (Z \ cZ) whence is unchanged and d is not taut. Conversely if 4 is not taut, there exists a E 4 such that t(d1) = t(sd) where d1 = d \ {a}. By Theorem 0.8 the sequence v1 (a)
aM a)
:
a0) E
/1 }
(0.108)
is Behrend. By corollary 0.14 (extended to a finite union of sequences), at least one of the sequences c
:
a() E A1, (a(), a) = c} , c I a
(0.109)
is Behrend, and we denote it by -4. Hence d 2 cJ as required. This proves the corollary.
0 First ideas
22
0.5 Total decomposition sets
Some branches of number theory inevitably involve calculations with highest common factors and least common multiples. When just two integers occur we can write a = (a, b)a', b = (a, b)b' in the obvious way, but even with three integers the business can be somewhat tiresome. Total decomposition sets provide a mechanism which sometimes copes with these difficulties. They occur in Chapters 2 and 3 of the present book. Part of Theorem 0.20 below was proved independently by Ruzsa (1988). We begin with the example of three (positive) integers a, b and c. By analogy with the factorization of a and b above, we can write
a = a'vwd, b = b'uwd, c = c'uvd,
(0.110)
where d = (a, b, c) and, for example, (a, b) = wd. That is, the highest common factors are the products of the visibly common elements. It is not quite obvious that we can always do this but at any rate if a, b and c are square-free the reader will see that (0.110) works by arranging their prime factors in a Venn diagram. In this scenario the h.c.f.'s become intersections and the l.c.m.'s unions. The general case of (0.110) is covered by the theorems which follow. The rule for lowest common multiples is to be that we multiply together all visibly different elements. For example [a, b] = a'b'uvwd, [a, b, c] = a'b'c'uvwd.
The decomposition into factors is not the end of the matter since in applications we often require to know whether the various elements a', b', ... , d are relatively prime. We emphasize that in the general case some, but not all, pairs of these elements are necessarily coprime. The decomposition (0.110) involves in all seven factors and for n integers a, b, c.... there will be 2" - 1 factors. Our notation needs some care and we make this quite formal. In applications some shorthand is usually appropriate. Theorem 0.20 Let V = {al, a2, ... , a } be a set of positive integers which may contain repetitions and 1's. Then sl possesses a unique total decomposition set {d(S)}, comprising 2" - 1 positive integers d(S) labelled by the non-empty subsets S of {1, 2, 3, ... , n}, with the following properties:
(i) For every non-empty subset T c {1, 2,..., n} we have h.c.f.(a;
:
i e T) = [I{d(S) : S 2 T},
(0.111)
0.5 Total decomposition sets
23
(ii) For every non-empty subset T c { 1, 2,..., n} we have l.c.m.[ai
:
i E T] = [J{d(S) : S fl T
0}.
(0.112)
The proof in Hall (1989) is a bit clumsy and we give a new one. We use the notation vP(a) = max{a : p" I a}
(0.113)
for primes p and non-zero integers a. Proof of Theorem 0.20 We write down a formula for d(S) and verify that these numbers satisfy (0.111) and (0.112). Let p be a fixed prime and let the distinct values of vP(ai), for 1 < i < n, arranged in decreasing order be v1i v2, ... , vk. Thus k < n and vl>V2>...>vk>0.
(0.114)
Put
Z3(p) = {i : pviIai},
1 < j
(0.115)
so that the sets Zj(p) satisfy (with strict inclusion)
Z1 cZ2cZ3c...cZk.
(0.116)
Let u j = v j - v j+l if j < k, uk = vk. We put d(S , p) =
pui 1
if S = Zj(p) for some j, else.
(0.117)
This defines d(S, p) for every non-empty S s { 1, 2,..., n} and every prime p, and we put d(S) =11 d(S, p) P
where the product is over all primes (or if we prefer, all prime factors of ala2 ... an).
We have to show that the set of numbers {d(S)} so defined has the properties (i) and (ii) claimed in the statement of the theorem. In each case we consider, for each prime p and subset T the exact power of p which divides the left- and right-hand side of (0.111) or (0.112). We begin with (i) and we have vP (h.c.f.(ai
:
i c T)) = min{vP(ai) : i E T j = Vj(T) _
ul, j(T)
where
j(T) = min{j : Zj(p) 2 T}.
(0.118)
0 First ideas
24
The power of p dividing the right-hand side of (0.111) is 11{d(S,
T} = fJ{p"' : Zi(p)
p) : S
T} = 11{p"' : j(T) < 1 < k}
and this agrees with (0.118). This establishes (i). We proceed to (ii) and we have vp(l.c.m.[ai
:
i E T]) = max{vp(ai) : i E T} = Vh(T) _
ui,
(0.119)
h(T)
where
h(T) = min{j : Zi(p) fl T
0},
(0.120)
(notice the minimum is required on the right). The power of p dividing the right-hand side of (0.112) is fJ{d(S, p) : S fl T
* O}
= fl{p"' : Zi(p) fl T * O}
= fJ{p"' : h(T) < l < k}
(0.121)
which agrees with (0.119). This completes the proof of Theorem 0.20. The next result classifies the total decomposition sets.
Theorem 0.21 Let {e(S) : S c {1, 2, ... , n}} be a set of 2"- 1 labelled positive integers. This is a total decomposition set (associated with a suitable d) if and only if, for every pair of sets R and S such that R S and S R we have (e(R), e(S)) = 1.
Proof We begin by showing that the condition stated is necessary. Let {e(S)} be a total decomposition set and pI (e(R), e(S)) for some pair of sets R, S. By (0.116) this requires R = Zi(p), S = Zm(p) for some l and m. By (0.116), one set lies inside the other. This establishes necessity.
For 1 < i< n let ai = 11{e(S) : i E S};
(0.122)
we claim that if {e(S) : S c 11, 2,..., n} } satisfies the condition stated in the theorem then this set of numbers is the total decomposition set of SY = {al, a2, ... , a"}, defined by (0.122). By Theorem 0.20 Q possesses a total decomposition set {d(S)} and we show that d(S) = e(S) for all S.
(0.123)
Let p be a prime. If ple(R), ple(S) then either R c S or S c R whence the sets S for which pie(S) are ordered by inclusion; suppose they are
S1= S2= S3c...=Sh.
(0.124)
0.5 Total decomposition sets
25
We distinguish two cases. If Sh = {1, 2, ... , n} we put It = k. If not we put h = k - 1 and set Sk = {I, 2, ... , n}. For 1 < j < k let U j = vp (e(Sj)) ,
(0.125)
so that in the second case above, Uk = 0. For 1 < i < n let m(i) = min{j : i E Sj}.
(0.126)
Then we have, from (0.122), (0.125-0.126),
vp(ai) = E uj.
(0.127)
m(i)<j
Thus vp(ai) takes exactly k distinct values
vj = E ul, 1 < j< k,
(0.128)
j
as in (0.114). We find that the sets Zj(p) defined in (0.115) are the present Sj whence by (0.117), d(S, p) =
p"j
if S = Sj for some j,
I
else..
(0.129)
By (0.125) we now have vp (d(S)) = vp (e(S)) for all S, p
(0.130)
and (0.123) holds. Therefore {e(S)} is a total decomposition set. This completes the proof of Theorem 0.21. This theorem appeared in Hall (1990a) with a different proof.
1
Besicovitch and Behrend sequences
1.1 Introduction We recall two definitions from §0.3: an integer sequence c is a Besicovitch
sequence if its set of multiples #(.d) possesses asymptotic density. d is a Behrend sequence if 1 d and .'(d) has logarithmic density 1. It is then a consequence of Theorem 0.2 that d,#(d) = 1, so that any Behrend sequence is Besicovitch. In this chapter we are concerned with general criteria to decide whether or not a sequence 4 has one or other of these properties. We may restrict
our attention to non-trivial infinite d, indeed we may assume that 1 a
aEW
oo,
(1.1)
since, as in (0.81), this is necessary for d to be Behrend, moreover if (1.1) is false, d is automatically Besicovitch by Theorem 0.1.
1.2 Erdos' criterion
Erdos (1948a) (this paper and Behrend (1948) are adjacent in Bull. AMS) gave a necessary and sufficient condition for a sequence to be Besicovitch. We suppose that d _ {al, a2,. ..} is strictly increasing, and note that al > 1. Theorem 1.1 Let W (d) denote the set of positive integers n such that
a;jn, aj%n if j < i. Let
M,(x) = card{n : n E & (sl), n < x}. 26
(1.2)
1.2 Erdos' criterion
27
Then d is a Besicovitch sequence if and only if
lim lim sup x-1 E Mi(x) = 0. x-00
E -O
(1.3)
x'-e
The proof which follows appeared in Erdos, Hall and Tenenbaum (1994).
Proof of Theorem 1.1 We recall from the proof of Theorem 0.1 that di(d) is Besicovitch for every sl and i. From (0.24) and (0.67) we have CO
d-di(m) = S
(sl).
(1.4)
i=1
For each fixed k, we have k
/
k
\
I > dJli(d) I x
Mi(x)
(1.5)
\ i=1
i=1
and by the selection principle and (1.4) we can allow k = k(x) to tend to infinity as x -+ oo so slowly that
Mi(x) - S#i('W)x.
(1.6)
i5k(x)
So far we have not used any hypothesis. Suppose now that d is Besicovitch. We have dl#(d) = S./GZ(d), whence from (1.6),
E Mi(x) = o(x)
(1.7)
i>k(x)
which implies (1.3). Hence (1.3) is necessary.
Suppose that (1.3) holds. To show that s.,2 is Besicovitch, it will be sufficient to prove that for every e > 0 there exists T = T (E) such that, uniformly Mi(x) << ex. T
<x1-E
We write n = lm where 1 is the product of the prime factors of n which do not exceed y. We recall from Divisors §0.3 the function
O(x,y,z)=card (n<x: JJ p°>z} l p,11 n,p<_Y and the uniform bound, for 2 < y < z < x,
O(x,y,z) «xexp -c log z
logy)
(1.9)
JJJ)J)
( 1.10 )
28
1 Besicovitch and Behrend sequences
(with a suitable c > 0). We put z = xE'2, y = xE2 so that
O (x, y, z) «xex p
-c
2c
«Ex,
(1.11)
and it follows that to establish (1.8) we may restrict our attention to integers n < x for which l « x`'/2. We employ the following result.
Lemma 1.2 Let Mi(x, s) denote the number of integers n = lm < x such that ailn but aj%n if j < i, and in addition such that y=xE:2.
1= 11 pV <xE/2,
(1.12)
P° II
Then uniformly for ai <
xi-E
we have
Mi(x, E) << d./#i(d)x. Proof of lemma
(1.13)
Let us write
n = lm = ain = ail'm'
(1.14)
so that (whatever the prime factors of ai may be) we have I'll, m'Im. Thus 1' < x812 and x/ail' > X,12. Since n E so does ail'. Hence Mi(x, a) <
card
{m': m' <
t
x ail
(1.15)
where the only constraint on m' is that its prime factors exceed y. Since y < xE/2 < x/ail' this yields (see for example Halberstam and Richert (1974), Theorem 3.5)
M,(x,8) «E X H Cl ail
t'
P<-v
1
(1.16)
P
where we require on the right that ail' E .Gli(d) and that the prime factors of l' do not exceed y. For y as in (1.12), Mertens' formula implies
Mi(x,v)<xs 2E !'
For n e
1
ail' P<x
H(1-1).
(1.17)
P
PV(1.18)
we may write n = airs where
s=H
P,IIn,P>x
1.2 Erdos' criterion
29
Clearly (s,aj) = 1 for every j < i, whence n E sli(d) if and only if air E Gli(sl). Therefore
d,#i(d) = E 1 11 r
air p5x
(i - 1)
(1.19)
p
where we require on the right that air E sli(d) and that the prime factors of r do not exceed x. Every P in (1.17) is an r, whence
Mi(x, s) < 8-2 d-#i(d)x.
(1.20)
This proves the lemma. We deduce from the lemma that
Mi(x,e) < (E d/Ili(d))x << vx, T
a;>T
/
(1.21)
if T = T (s) is sufficiently large. As shown above, this is all we need for (1.8), and d is Besicovitch as required. This proves Theorem 1.1.
Corollary 1.3 (Erdos (1948a)) Let d be such that
card{d fl [1,x]} << lox. gx
(1.22)
Then 4 is a Besicovitch sequence. Moreover this is best possible in the sense that the right-hand side of (1.22) cannot be multiplied by fi(x) for any function fi(x) -> co.
The first part of the corollary is a straightforward exercise in partial summation employing no more than the upper bound Mi(x) < x/ai. We shall not prove the second part here and refer the reader to Erdos'
original paper or to Halberstam and Roth (1966). We give another corollary of Theorem 1.1 as part of Theorem 1.6 below. A sufficient condition for d to be a Besicovitch sequence is that its elements should be pairwise coprime, because in this case (1.1) implies that Q/ is Behrend. The next theorem provides a similar but considerably weaker condition for 4 to be Besicovitch. Theorem 1.4 Let k and s be fixed positive integers and .4 be such that for
w((aio,ai,,...,ais))
(1.23)
Then d is Besicovitch. This is best possible in the following sense: if
1 Besicovitch and Behrend sequences
30
l; (x) -+ oo as i --> oo then we can find a non-Besicovitch sequence such that
for all i, S2(ai) < g(i).
(1.24)
This result is in all essentials due to Erdos, Hall and Tenenbaum (1994)
- in the paper s = 1 - I am grateful to Tenenbaum for correcting my proof of the generalization. We require the following lemma.
Lemma 1.5 Let k E Z+ and h E (0,
be fixed, and Vk(z,j) denote the 2) number of integers n < z free of prime factors < zn and such that w(n) < k. Then for z > zo(r1) we have 1k-1
Vk(z, 1) «k
Proof of lemma addition, exceed
(1.25)
(log - I
log Z
Let Vk(z, j) count the integers as above which, in and are square-free. We have z1-n
P-2 <<
Vk(z,11) -
(1.26)
P>zn
and suppose that z > zo(r1) implies zn > log z so that this is « z/ log z. Let P+(n) denote the largest prime factor of n. If n is counted by Vk(z, n) then P+(n)k > lz- whence log Z Vk(z,
2k
n) <
*1 2(n) log P+(n)
(1.27)
n
where the asterisk denotes that n satisfies the conditions of the lemma. The primes contribute « z on the right. If n is composite put n = mp where w(m) = j < k and p = P+(n). The contribution of these integers is
loge « m
p
pj)-1
m << z
(PI P2 ... j
m
o(m)
<< ZE
-
j
zn
«k
z
(log -
(1.28) //
Thus
k-1
Vk(Z,h) «k
logz (log !)
and together with (1.26) this gives the result stated.
(1.29)
1.2 Erdos' criterion
31
Let (1.23) hold. We employ Theorem 1.1, and we
Proof of Theorem 1.4
require an upper bound for the sum
E Mi(x).
(1.30)
xl-E
Let n be counted by Mi(x) so that n = aib, say, moreover aj%n for j < i. xE2 Put y = and a1 = uivi where the prime factors of ui do not exceed y, and the prime factors of vi do exceed y. We first count the integers n for which ui > xE: by (1.10) we have
card n< x : Hpv > xE} = O(x,y,xc) «xexp
c (--)
,
(1.31)
a
JJJJJJ
p"IIn pSy
which contributes zero on the left of (1.3). Next we have ai > x1-8 and ui < xE whence vi > x1-2a. Let us split these ai into two classes according as w(vi) < k or not, and let S1 and S2 be the contributions of these classes to (1.30). Then S1 <
card{n < x : vIn},
(1.32)
x1-2E
P (v)>y, w(v)
where P-(v) denotes the smallest prime factor of v, whence by the definition of Vk(z, q) in Lemma 1.5 we have x
S1 < x
in which z'i < y for
x1-2e <
x1-2EE
z-1dVk(z, q)
(1.33)
z < x: we put h = E2. We apply the lemma
which yields 1
S1
«k x { log x + <
fXIX-2E
(1_-__+) x
dz z log z } (log 91)'
/
(log
l
k-1
(1.34)
which also contributes zero on the left of (1.3). We turn to S2, in which w(vi) > k.
For integers v such that w(v) > k let v = g(v) be the product of the k + 1 largest distinct prime factors of v. If ai = uivi where w(vi) > k then
g(vi) = g(ai), and if io < ii < ... < is then g(aio) = g(ail) = ... = g(ai) is impossible, by (1.23). Thus the function v -* g(v), restricted to these vi, is at most s onto 1. Let V denote the set of vi such that w(vi) > k
1 Besicovitch and Behrend sequences
32
arising from ai E (x1-E, x] and V (z) be the counting function of V ; from the above argument we have V (z) < sVk+1(z, E2) whence
<
S2
x E -= x J " 1
uEV
<
z-1dV(z)
x1-2c
ll
V(x)+x
x V(z) z72
1)k
log x
log
E
xpx
+ sx
\ x(log+El
1-2s
(log E )kdz
z log z
E)k
(1.35)
(log gx which again contributes zero on the left-hand side of (1.3). Hence by Theorem 1.1 d is a Besicovitch sequence as required.
In order to prove the second part of the theorem we modify the construction of a non-Besicovitch sequence given in the proof of Theorem 0.1, and we need a generalization of Lemma 0.3 due to Tenenbaum (1984) and contained in Theorem 21(iii) of Divisors. Let H(x, y, z) denote the number of integers n not exceeding x possessing at least one divisor in the interval (y, z]. Let 2y < z < y22 and put z = yl+". Then we have, uniformly for y > 2 and x >_ z2,
H(x,y,z) <<usx,
(1.36)
where 6 = .086071 ... is defined in (0.37). (We have given a slightly weakened form of this result for convenience. Some new results on the function H(x, y, z) are contained in Chapter 7, where we are interested in `short' intervals (y, z] for which z < 2y.) Let T1 > 2 and Ti j+> T 3 for j = 1, 2, ... ; let g j = j-2/6 for all j and
dj={a: Tj
(a)>T'(1.37)
finally (with J at our disposal),
JJJJJJ
d=adj.
(1.38)
j>J
By (1.36), ignoring the condition on P-(a), we have El K J-1.
(1.39)
j<J
Now let n E (Ti,
and put n = uv where P+(u) <
P-(v). We
1.2 Erdos' criterion
33
allow v = 1, (we may formalize this if we wish by defining P-(1) = oo as in Divisors), but our object is to show that usually v * 1, indeed v > Tj, so that v E dj and n E #(d). If not, either n < or u > T8''2, and the number of such integers n E (Ti,Tj'') is, employing (1.10), <
Tj,
Ilk) +
O"(T1+ei
J
T"i J
J
(Ti2+exp(__))
(1.40)
as j --> oo. Thus dJf(d) = 1, and by choosing J large enough in (1.39)
we may ensure that d is not Besicovitch. Now let a E d so that for some j, a c sij. By (1.37) we have Q( a )
< (1 +Ej)
( 1 . 41)
On the other hand if a = a, then (for large j)
i > E I,41iI > Ti-1,
(1.42)
J
since every prime in (Th, Th+"] lies in Qh, and we can arrange that Tj increases so fast that (1.41) and (1.42) imply (1.24). This completes the proof. The next theorem also appeared in Erdos, Hall and Tenenbaum (1994); it answers a question which might well have been asked fifty years ago but apparently was not. The result is a little surprising too. Theorem 1.6 Let s11 and sd2 be Besicovitch sequences. Then d1 U 'Q/2 is Besicovitch. There exist primitive Behrend sequences 41 and S12 such that d1 n d2 is not Besicovitch.
The first part is a straightforward deduction from Erdos' criterion, Theorem 1.1, and we deal with this before discussing the second part.
Proof of part (i)
Let us write 41 U d2 = fail where a1 < a2 <
and Mi(x) denote the number of integers n < x such that ailn, but for all j < i, ajjn. By Theorem 1.1 it will be sufficient to show that
lim lim sup x 1 >Mi(x) = 0. 8-0 x-,oo
(1.43)
x'-'
We consider the ai belonging to d1. For such i, let M,1)(x) denote the number if integers n < x such that a, n but n has no smaller divisor in
1 Besicovitch and Behrend sequences
34
d1. It may have such a divisor in d2 \ Q1 and we see that M,(x) < MP)(x).
(1.44)
If ai E 42 \ d1 we define M; 2)(x) similarly, and Mi(x) < Since d1 and d2 are Besicovitch sequences (1.43) applies to M(1)(x) or M(2)(x) and the conclusion follows.
It is not difficult to construct an example to show that 41 and 42 may be Besicovitch sequences while c1 n Q/2 is not. Let c be any nonBesicovitch sequence and put 41 = {2} U 2d. Then &(d1) = 2Z+ and d1 is Besicovitch. Also Q2 = {a : a > 3} is Besicovitch, indeed it is a
Behrend sequence. We have d1 nd2 = 2d, and since t(2d) = 2 #(d) this is not Besicovitch. Consideration of this easy example might suggest that the extra con-
dition that d1 and d2 should be primitive would be enough to make d1 n d2 Besicovitch, but this is not the case. Proof of part (ii) The idea of the proof is as follows. We begin with a primitive, non-Besicovitch sequence -4 which is actually going to be d1 n d2. To construct d1 and Q2 we extend -4 in two ways, disjointly, into a Behrend sequence. Of course the awkward part is to arrange that d1 and .12 are primitive, since it is an easy matter to extend any sequence into a Behrend sequence for example by adding primes. The construction of . involves a slight modification of the sequence sl used in the proof of Theorem 1.4 and defined by (1.38), after which we set 9 = Y(d). The modification occurs in (1.37) which becomes
'j={a:
P-(a)
(1.45) mak))))))es
The extra condition on the greatest prime factor of a d thinner so that (1.39) still holds: we may choose J so large that d.%'(d) < gWe . claim that 9, whence q is non-Besicovitch. To see this, we consider the integers n < Tj'1+E', writing n = uv where, as in the argument
following (1.39), we have P+(u) < Tj8' < P-(v). Suppose that n has no divisor in s l'. Then there are three possibilities, namely n < T,'+(Ejl2) (ii) u> (iii) P+(n) > Ti /2. (i)
The number of integers in cases (i) or (ii) may be estimated as in (1.40), and for j sufficiently large does not exceed g T +E'. Otherwise v = u > Tj
1.2 Erdos' criterion
35
whence v E d1, unless P+(v) > Tj'. But then n is in case (iii). The number of such integers n does not exceed
1 < (log 2+Ej n
(1.46)
pin P>
T;
(inverting summations on the left) and this is less than 4 T +E (assuming >1 j is large). This leaves at least I ] - 8g T J' integers n 9 [T,'
Tj'
in .1#(dj), whence dill(d) >- 9. Our claim is valid and 4 is nonBesicovitch. As stated above, we put .4 and sd2. For l = 1, 2 (and p, q primes), put
Next we construct d1
W1 = {pq : q > exp ((log p)2) , q - (-1)1 (mod 4)}
(1.47)
'Q/1 = 14 U W1.
(1.48)
and set
Clearly `V1 and '2 are disjoint whence d1 r1 S12 = 2. We have to show
that .1 and d2 are primitive, and since -4 is primitive and each W1 is primitive (its members all have the same number of prime factors) this reduces to showing that b%cl, c1%b (with obvious notation). The former assertion is clear because from (1.45), KI(a) 3 for every a E c/'. whence fl(b) >- 3 > SZ(c1). Also (1.45) implies, for a E sl'j,
log P+(a) < 282 log P-(a) < (log P-(a)) 2
(1.49)
J
if, as we may assume, log Tj > j8Is = z 'J. 4. This justifies the latter assertion that c1Xb. It remains toz show that 41 and Q2 are Behrend sequences: in fact '1 and `'2 are already Behrend. Consider W1 for example. Let Q+(n) denote the greatest prime factor - 1 (mod 4) of n; if n has no such prime factor then Q+(n) = 0. We have to show that Q+ (n) > exp ((log P-(n))
2) P.P.
(1.50)
i.e. that the number of integers n < x for which the above inequality is false is o(x). Let fi(x) be at our disposal. For such an integer n, either P-(n) > exp fi(x) or Q+(n) < exp (x)2, whence the number of these integers is
«xfl (l-+x11 p q=1(mod 4)
q
(1.51)
36
1 Besicovitch and Behrend sequences
Let x be the non-principal Dirichlet character (mod 4). The product on the right of (1.51) is
I
<<
1
II (i_ q 1
K
j
x(q)
1
q
expg2
«
1--
--
q
2
(1.52)
og x
because partial products of the convergent product 1/2
C1 -
x(q)
q
= L(1,
)
x)-1/2
=
(1.53)
are bounded. By (1.51) (and Mertens' theorem) the number of exceptional integers n not exceeding x is
<<x1 +
x
<< x(logx)-1/4
(1.54)
0-0-9x
if we choose fi(x) optimally. Therefore W1, and similarly 2, is Behrend. This proves part (ii) of Theorem 1.6.
1.3 Behrend sequences
Erdos (1979) wrote "It seems very difficult to obtain a necessary and sufficient condition that if a1 < is a sequence of integers then almost all integers n should be a multiple of one of the a's." This problem had concerned Erdos for many years and is merely formalized by our definition at the end of §0.3 of a Behrend sequence. Erdos himself, and other authors, have given many examples of (what we now call) Behrend sequences, some of them far from easy, but of course special cases.
In this section we make a start on this problem, and it is taken up again at other points of the book. We begin with a general, necessary condition for a sequence to be Behrend, and we show by examples that it is in some respects best possible.
Theorem 1.7 (Hall (1990b)) Let d be a Behrend sequence. Then for each
y, (0y-l-logy, we have 00
Ear 1(loga,)flyn(a,) = oo. i=1
(1.55)
1.3 Behrend sequences
For every y and
37
< y - 1 - logy there exists a Behrend sequence d(y, fl)
for which the series in (1.55) is convergent.
It is likely that the condition on /3 can be weakened to /3 > y-l-log y, but we are unable to settle this point. The theorem would then include, as the special case y = 1, the only other general condition of this sort which we have, Corollary 0.10. This is 00
ail=co.
(1.56)
i=1
We make two remarks at this point. The first is that y - 1 - logy > 0 for 0 < y < 1, whence we must have /3 > 0 and (1.55) is weaker than (1.56) if we have no information about S2(ai). We should expect this to be the case because (1.56) is both necessary and sufficient for .4 to be Behrend when (for example) all the ai are prime. Our second remark concerns the status of the variable y. It will emerge that in an important special case we may choose y = 1 optimally. However this could be misleading: we show by example that if any open interval of the range (0, 1) be omitted then there exists a non-Behrend sequence which the theorem consequently fails to detect. We prove the first part here, and then we describe the special sequences ,(y, /3) which establish that the condition on /3 is, Proof of Theorem 1.7
apart from the moot point concerning equality, best possible. Then we justify our second remark above about y. We require a lemma which is a useful variant of Erdos' law of the iterated logarithm discussed in detail in Divisors, Chapter 1. It is a similar result to that indicated in Exercise 10 of Divisors.
Lemma 1.8 Let S2(n, t) denote the number of prime factors not exceeding t of n, counted according to multiplicity. For each K E (0, 1) and to >- 3 let .(K, to) denote the sequence of integers n for which sup to
c(n,t) log log t
> K.
(1.57)
Then we have
d-4(K, to)
(1.58)
K = (1 + K) 10g(1 + K) - K.
(1.59)
where
1 Besicovitch and Behrend sequences
38
Proof of lemma We use a result from the proof of Lemma 0.3, that is (0.38). For fixed yo < 2 we have uniformly for 0 < y < yo and 2 < t < x
that E yO(n't) KYo x(log t)Y-1.
(1.60)
n<x
We fix yo = 1 + K. Let t E [to, x] be fixed and 0 < 2 < K. By (1.60), the number of integers n < x such that S2(n, t) >- (1 + A) log log t
(1.61)
does not exceed
E(1 +
2)f)(n,t)-(l+2) log log t
<
(1.62)
n<-x
Similarly, the number of integers n < x such that
fl(n,t) < (1 -2)loglogt
(1.63)
does not exceed
E(1 -
«K x(log
t)-2-(1-2)log(1-2).
(1.64)
n<x
Put 2 - (1 + 2) log(1 + A) = -L. The exponent of log tin (1.64) is < -L, and we have shown that for 0 < A < K and each fixed t we have
card {n<x:
fl(n, t) log log t
>- Al
«K x(log t)-L.
(1.65)
This inequality applies for fixed t. To obtain a uniform result such as (1.57) we introduce a sequence of discrete checkpoints tk E [to, x] and employ the monotonicity of n(n, t) and the slow variation of the function log log t to interpolate between these checkpoints.
Let 6 > 1 be such that l og o
<
1
+K
lo g lo g 3
(1 . 66)
and let tk satisfy log tk = 9k log to, k = 1, 2, 3, ... ;
(1.67)
then if t E (tk,tk+1) and IS2(n, t) - log log tI >_ K log log t
(1.68)
1.3 Behrend sequences
39
we have either Q(n, tk+1) > (1+ic) log log tk or fl(n, tk) < (1-K) log log tk+1, whence
fl(n, t') - 1
>K-(1+K)log0
log log tj
(1.69)
log log tj
for at least one of j = k or j = k + 1. Since to > 3, the right-hand side of (1.69) is positive, by (1.66), and we denote it by Xj. We put Lj = (1 + )L) log(1 + 2j) - )j, and we note that 0 < A j < K so that (1.65) is applicable. Let K be as in (1.59). By the mean value theorem, we have K - Lj < (K - 2j) log(1 + K) whence (K - Lj) log log tj < 1 (uniformly). We apply inequality (1.65) with t = tk, X = ;.k, L = Lk for every k, and we deduce that the number of integers n< x for which (1.57) holds is
<
x(log tk)-Lk
k=0 00
<<
0-kK
x(log to)-K
(1.70)
k=0
after choosing 0 suitably in (1.66). This proves the lemma. (In Divisors, Exercise 10 the specified checkpoints are tk = tok, that is 0 = 2; this works because we have K < ?K2 < K and so we may assume that K log log to is large, else (1.58) is nugatory. We replace 3 by to in (1.66).)
We proceed to the proof of the theorem. We may assume y < 1, and we have /3 > y -l-logy so we may fix K > 0 so that /3 = y -1- (1 + K) log y
(making /3 smaller if this would involve K > 1). We next fix to in Lemma 1.8 so that d,4(K, to) < 1, and we let l be such that a1 >- to. We consider the tail sd i = {a,, a1+1 i ...} of c. We argue by contradiction, showing that if the series (1.55) is convergent and 1 is large enough then d1 is not Behrend. Let X(n) = min(1,i(n,sd,))
(1.71)
so that if Q1 were Behrend we should have X(n) = 1 p.p. whence
lim (a - 1) E n °X(n) = 1.
Q-. 1+
(1.72)
n=1
By the construction of M(K, to) we have
limsup(a - 1) E n 'X(n) < U-1+
nE.1(K,to)
1 3
(1.73)
1 Besicovitch and Behrend sequences
40
since x(n) < 1. For n
V(K, to) we have
X(n) < i(n,dl) < 7'
yi2(n,a;)-(1+K)logloga,
(1.74)
adn i>1
because y < 1 and the exponent is negative by (1.57). Therefore n-oyfl(n,a;)
ri-6x(n) < Y(logaj)-(l+k)logy nO.PJ
(1.75)
n=O(mod a;)
i>_1
and we put n = ma, in the inner sum, to obtain 00
<
m oyo(m,a;)
ai '(log ai)-(1+rc)logyyo(a;)
(1.76)
m=1
i>_1
The inner sum in (1.76) is
C1- P/ Q)-111
p<_a; \
p>a;
\
C1- Q P
I
= (a)[J
1
pSa;
/
\ /
< C(a)expl (y- 1)>p 6 1. \\\
pSa1
We have
>
E1(v1)ElogP P psa; P
p
loglogai-(a-1)logai+O(1)
(1.77)
and (1.75)-(1.77) imply that
n 6x(n) « (a - 1)-' E a7 '(log ai)flyn(aj) iZl
nom
(recalling that y - 1 - (1 + K) logy = /1). By hypothesis the series (1.55) is convergent whence if 1 is sufficiently large we have ri aX(n) < 1.
limsup(a- 1) 6-.1+
no-4
3
(1.78)
This, together with (1.73), contradicts (1.72) whence ./ is not Behrend. This completes the proof of the first part of the theorem. It remains to describe the sequences A(y, f'). For this purpose we introduce a family of special Behrend sequences d`(t) depending on the parameter t E (0, 1). Let 11 ull = min{lu - ml : m E Z}, (u E R),
(1.79)
1.3 Behrend sequences
41
and
µ(t) = -t log t - (1 - t) log(1 - t),
t E (0,1);
(1.80)
furthermore let l; (a) -> oo as a -> co. We define
-' 12 I = {a > 1
:
11 log all < (log a)-1og2 exp( (a)
log log a)} (1.81)
and for t < 1, t > 1 respectively ,d*(t)
= {a > 1
:
logloga,
S2(a) <
11 log all < T-
log-log a}
log log
{a > 1 : Q(a) > t log log a - I; (a)
(1.82)
log log a,
11 log all <
T = exp µ(t). We prove in Theorem 4.23 that these are Behrend
sequences. Some remarks about their structure may be helpful here. First, µ(t) is symmetric about 1 where it takes its maximum value log 2; therefore as t increases from 0 to 1, T first increases from 1 to 2 and then decreases to 1, so that the condition on 11 log all in (1.81)-(1.83) is most stringent when t = 12 . This condition is relaxed progressively as 1 t - 12 increases, but there is a complementary condition on S2(a) in (1.82) and (1.83) which we observe may be written in the form 1
S2(a) - 2 log log a
t-- log log a - I; (a) 1
log log a.
(1.84)
Normally, a random divisor of a large random integer will have about log log a prime factors. We explain this statement heuristically here: it will be made precise in Theorem 5.4. If aln then S2(a) is (essentially) binomially distributed with mean S2(n)/2, (because we can assume that n has few repeated prime factors). Also S2(n) will have about log log n prime factors by the Hardy-Ramanujan theorem, moreover we may expect that log log a and log log n will be almost indistinguishable. In view of this, we see that (1.84) becomes more taxing as It - 11 increases. We wish to apply the first part of Theorem 1.7 to the sequences d'(t) to show that both the theorem and the specification of these sequences are in some respects best possible. We require the following result.
Lemma 1.9 Let 0 < y < 1. Then we have
Eyw(n) «w(logw)Y-l x-w
uniformly in y, and for x > w > 2.
(1.85)
1 Besicovitch and Behrend sequences
42
Proof We may assume x > 2w else (1.85) is an easy application of the Halberstam-Richert inequality (Divisors, Theorem 00). We follow Shiu (1980) very closely. Put
z = w 113 v = log z. log log z, (1.86) assuming as we may that z > 16: we need that log log z > 1 later. For each n E (x - w, x] we write n = bndn where bn is maximal, subject to the constraints bn < z, P+(bn) < P (dn). (1.87)
Notice that n > x - w > w = z3, whence do > z2 > 1. We split the integers n E (x - w, x] into four classes, and we write E(`) for summation over the ith class. These classes are: (i) (ii) (iii) (iv)
P-(dn) > V `, P-(dn) < /z, bn>< /'" P-(dn) < v, bn v < P-(dn) < Jz-, bn > /Z--
We note that y < 1. We have (1)Y"J(n) <
EYcu(b) E 1, b<-z
(1.88)
(x-w)/b
and we may apply Selberg's upper bound sieve to the inner sum because (w/b) > z2. The above is yw(b)
<<
<<
w
blogz
w 11 « loges
1+
P
p-Y 1
w(log w)Y-1
(1.89)
as required. Next, let n E class (ii). Let P-(dn) = q and gsIldn. By
the maximal property of bra we have brags > z, whence qs > For each prime q < Iz- let s(q) denote the smallest integer such that qs(q) > ,,,fz-. Then s >_ s(q) >- 2, moreover qs(q) < z. Hence IIZ-.
(2)Yw(n)
<
q
q
W
(E z-1/2 + E q-2 ) q<-z114
q>zt/4
K wz-1/4,
(1.90)
43
1.3 Behrend sequences
well within (1.85). Next (3)yw(n)
<
(3)1
b < iT(z,v).
(1.91)
f b
P+(b)
Shin's Lemma 1, which is an application of Rankin's method, gives og log z}, again more than we require.
'I'(z, v) < exp{3 log z/
In order to cope with class (iv) we require the following inequality
(valid by the way for all y > 0), which is a special case of Shin's lemma 4. As Shiu points out, this derives from Wolke (1971). We have, uniformly for z > 6 and 1 < r < log z/log log z, that yw(b)
b>f
< (log z)y expf-
b
1 rlogr}.
(1.92)
P+(b)
This again depends on Rankin's method. Let
_ logz log v
r0
logz < log log z'
(1.93)
(because z > 16). Then vro+l > z, i.e. zl/(ro+t) < v. We split the range for in class (iv) into the sub-ranges zl/(r+i) <
2 <_ r <_ ro,
z11r,
(1.94)
and we note from (1.87) that P-(bn) < z11r in this sub-range, whence ro
(4)yw(n)
<
EE r=2
yw(b)
ft
P+(b)Szi/r
1
(x-w)/bzU(r+1)
E [ yw(b) (r +
L E r=2
lz-
1)W
blogz
« low- E(r + 1)(logz)y exp {_jrlogr} g
(1. 95)
r=2
by the sieve, in line 2, and (1.92) in line 3. The sum over r is 0(1), and (1.95) together with (1.89), (1.90) and (1.91) yields the desired bound. This proves Lemma 1.9. We consider the infinite series (1.55) when d is one of the sequences
d*(t). We note that we may replace w by 0 in (1.85) because y< 1.
1 Besicovitch and Behrend sequences
44
Let us set
Xk =
{a i(loga)fy11(°)
Ia - ek <ekk-1*)exp(q(k) log k),
f1(a)tlogk+q(k) logk}, in which q(k) have
(1.96)
oo as k - oo, and t < 2. We consider whether or not we 00
E Xk < 01D.
(1.97)
k=2
For 0 < z < 1 at our disposal, we have Xk «Xk(z) where
Xk(z) = ekkfl-tlogzZ-q(k) logk j:{(yz)n(Q) la - ek I < ekk-µ(`) exp(q(k)
log k)}
(1.98)
and we estimate the inner sum by means of Lemma 1.9, which yields Xk(z) << k#`10gz-1(`)+Yz-1 exp {(1 - log z)q(k)
logk 1.
(1.99)
We choose z = t/y to minimize the exponent of k : we require t < y at this point. The exponent is
/3+tlogy+(1-t)log(1-t)+t-1
(1.100)
which, as a function of t, has a minimum /3+log y-y, at t = 1-y. If y > 2 then t < 1, z < 1. If /3 < y-1-logy the exponent of k is < -1 and (1.97) holds. The series (1.55) converges although A(y,/3) is Behrend. Next, let
Xk be as in (1.96) except that now t > 1 and the condition on )(a) is )(a) >- t log k - q(k) log . We have Xk « Xk(z) provided z > 1, where XX(z) is as in (1.98), with the exponent of z (outside the sum) changed to q(k) og We apply Lemma 1.9 - we need yz < 1 - and we obtain Xk(z) <<
kf`1Ogz-µ(`)+Yz-1 exp{(1
+ logz)q(k) logk}.
(1.101)
As before we choose z = t/y to minimize the exponent of k, which is in order provided 1 > t >- y. If y < 1 we choose t = 1 - y > 1 and the exponent, given by (1.100), is < -1. The conclusion is the same as before. We leave the case y = 1 as an exercise. This completes the proof of Theorem 1.7. We draw two further conclusions from the argument given above. The
first is that T = expp(t) with µ(t) defined by (1.80) is the best possible T in (1.81)-(1.83). For if µ(t) were larger we could make the exponent of k in (1.99) and (1.101) < -1, with /3 > y -1- logy, y = 1- t. By the
1.3 Behrend sequences
45
theorem, the corresponding sequences W"(t) would not be Behrend. In this sense the specification of d (t) is best possible. Finally we may now justify the second of our remarks following the statement of Theorem 1.7.
Let Y E (0, 1) be an open interval. Choose t such that 1 - t E Y and put µ1(t) = µ(t) + b, so that the sequence c1 obtained by replacing y(t) by pi(t) in the definition of d'(t) is non-Behrend. Let us consider the exponent of k in (1.99) or (1.101), with z = t/y optimized, y 4 Y. This is
/3-tlogt+tlogy-µ1(t)+t-1
>y-l-logy -tlogt+tlogy-µ1(t)+t-1 >-1+e-b, /i>y-l-logy, where > > 0, and depends on Y only. We put b = a and we see that we cannot detect that 51 is non-Behrend. (The loss in the inequalities Xk < Xk(z), Xk < Xk(z) with z optimized is not significant here.)
We give two corollaries of Theorem 1.7. The first of these was the original motivation behind this theorem. Corollary 1.10 Let sad have the form 00
d = U(Z+ n (Tk, Tk + Vk])
(1.102)
k=1
where T1 > 1, Tk+1 >- Tk + Vk for every k, and for some c = c(d) > 0 we have
Vk > T.
(1.103)
Then a necessary condition for sad to be Behrend is that for every y > log 2 - 1 we have T. a-' (log a)Y = c. (1.104) aEd
Proof We may assume that Vk < Tk always; if this condition is not met we may insert extra T's as necessary. Then by Lemma 1.9, we have (for
any /3, and0
1: a' (log a)'y°(a) Tk
« Tk ' (log Tk)f
y0(a) T, Tk
Tk '(log Tk)'3 Vk(log Vk )y-1
«y,c Tk 1 Vk(log
TW+Y-1
a ' (log a)#+y-'.
«y,C Tk
(1.105)
1 Besicovitch and Behrend sequences
46
We apply Theorem 1.7 and deduce that for y E (0,1] and /3 > y-1-logy, we have a ' (log
oo.
(1.106)
aE.c
Put y =#+y-1. We require y > 2y-2-logy and we choose y = optimally. This completes the proof. There is a heuristic explanation for the condition y > log 2 - 1. By the Hardy-Ramanujan theorem we have (log n) 1,g 2-E < T(n) < (log n)bog 2+Ep.p.
(1.107)
and we may expect that if sY is Behrend, and in some sense random, then
an integer d < n should have a probability at least 1/i(n) of belonging
to d, say card {,q/ n (1, n]} >
n (log n)nOg2 E
(1.108)
This leads to (1.104). The `randomness' of sy may be interpreted as meaning 'multiplicatively structureless', and the corollary shows that sufficiently long blocks of consecutive integers fulfil this albeit somewhat vague criterion. These blocks of integers occur frequently in the theory for the reason given. Their significance was first understood by Erdos, and they made their appearance in the conjecture due to Erdos which begins §5.3. We can cope with much shorter blocks than envisaged in (1.103), with a weaker outcome.
Corollary 1.11 Let sI be as in (1.102) and suppose that for some a = a(d) > 0 we have log Vk > (log Tk)".
(1.109)
Then a necessary condition for sl to be Behrend is that (1.104) should hold for every y > log(1 + a) - a. The proof goes through as before, except that we use (1.109) in (1.105).
We arrive at the condition y > (1 + a)(y - 1) - logy, and we choose y = 1/(1 + (X).
Notice that in general (1.104), with y < 0, is not a necessary condition for d to be Behrend. If the a, are pairwise coprime then (1.56) is both necessary and sufficient. On the other hand Corollary 1.10 is best possible in the sense that the constant log 2 - 1 cannot be reduced, even if c(d) > 1 - E, by reference
1.3 Behrend sequences
47
to the sequence d" (2) in (1.81). Another example is provided by the sequence Go
SIA = U (Z+ n (eke, 2e"]) .
(1.110)
k=1
Erdos conjectured that this is Behrend for some A > 1 and this was proved by Hall and Tenenbaum (1986), (1992) for A < 1.31457..., 2 < 1/(1 - log 2) respectively. The second of these results, due to Tenenbaum,
depends on the technique of Maier and Tenenbaum (1984). It is best possible, by reference to Corollary 1.10, moreover it lends some support to the hypothesis that (1.104) holds for y = log 2 - 1, and that we may take fi = y - 1 - logy in Theorem 1.7. Definition 1.12 A block sequence is a sequence of the form (1.102) for which (1.103) holds, for some c = c(sd) > 0. A weak block sequence satisfies the condition (1.109) only, for some a = a(sd) > 0. We refer to Z+ n (Tk, Tk + Vk] as a short block if Vk < Tk, else it is a long block, and we write Tk + Vk = Hk Tk in this case, so that Hk > 2. We require that Tk+1 >- Tk + Vk for all k.
Sometimes we shall assume (after splitting some blocks if necessary), that Hk < Tk.
Let d be Behrend and comprise long blocks. We can apply Corollary 1.10 and we have log Hk k=1
(log
To 1 -log 2-c
= 00.
This is useful only if Hk is much smaller than Tk and we want a better result. This depends on the following lemma, which has some independent interest.
Lemma 1.13 Let K > 0, z0 > 16 and B > B(ac) where
B(x) = 2Q(l
K
+ >c)
Q(2) = A log A - 2 + 1.
(1.112)
Then we have, except for n E -4(K, B, zo), that
S2(n;w,z)-log
(log
lo < xlog (-) +Blogloglogz,
(1.113)
g
for all w, z such that 2 < w < z < n, z > z0, where the sequence R(K, B, zo)
1 Besicovitch and Behrend sequences
48
of exceptional n satisfies d1(x, B, zo) -> 0,
zo -> oo, ic, B fixed.
(1.114)
This is a further development of results of Erdos (1969), (Theorem 1) and Hall and Tenenbaum (1992), (Lemma 2.1). Albeit (1.112) may not be best possible the second term on the right of (1.113) (or one very like it) really is necessary: this is best understood from Erdos' paper, in which he states the following theorem. There exist functions f+ : (0,00) -- > (1, co) (decreasing continuously from oo to 1) and f- : (0, co) (0, 1) (= 0 on (0,1] and increasing continuously to 1 on (1, oo)) such that for almost all n, max T2(n;w,z)
- f+(x)log I
min S2(n; w, z)
- f -(x) log
log z log w
log z wl log C
(1.115)
each extremal being subject to the constraint log
(logz
gw
> xlogloglogn.
(1.116)
Erdos does not specify f±, and we leave it as an exercise for the reader to show from Lemma 1.13 that f±(x) = 1 +O(x 113), (x -* co), f+(x) «x 1 (x
0).
(1.117)
Proof of lemma Let q > 1/ (Q(1 +,c)), 2(1 + K)77 < B. We set up checkpoints at the points tk = exp exp(gk log k). Let (1.113) be false, say
S2(n;w,z) > (1 +K)log (lo gw) +Blogloglogz
io-g
(1.118)
where w e (tj, tj+i], z E (tk, tk+i]. (We may assume w > 3:) Thus k > max (j, ko(zo)) moreover (1.118) implies that S2(n; tj, tk+i)
>_
(1 + x) log
>
(1 + x) log
log tk 1 + B log log log tk log ( g +i J log tk+i tog t j
provided B log log log tk >_ 2(1 + ic) log
1.119 (
)
log(1.120) (t). log tk
We have B > 2(1 + Ic)q and (1.120) then follows if k > 17, as we may
1.3 Behrend sequences
49
assume. Let y = 1 + K. The number of integers n < x such that (1.119) holds is
E
(1+k)logy yi2(n;t,,tk+i)
n<x
<
(
y-1-(1+K)logy
log tk+1
logtj
log tk+1 log t j
(1.121)
)
by the Halberstam-Richert inequality (Divisors, Theorem 01). The exponent is -Q(1 +K), moreover we have log log tk+1 - log log t j > q(K + 1 - j)(1 + log j). Hence the number of exceptional integers for which (1.118) holds for some w,z is
<< x x E (e
(1.122)
j)-Q(1+ic)n(k+1-j).
j>1 k>max(j,ko)
Suppose next that, instead of (1.118), we have
S2(n; w, z) < (1 - x) log (_)
- B log log log z.
(1.123)
This is impossible if the right-hand side is negative, and we deduce that
j+1
) ( log tk+1-B 109 log log tk+1, log t j
(1.124)
(we may assume that zo is so large that the right-hand side of (1.123) increases for z > zo), whence by (1.120), fl(n; tj+l, tk) <- (1 - x) log (lo109 tk
g ti+1)
.
(1.125)
We proceed as before, but now set y = 1 - 1c, and arrive at a sum like (1.122) with k >- max(j +2, ko) and -Q(1- ]c)q(k -1- j) in the exponent. We note that Q(1 - 1c) > Q(1 + K), moreover Q(1 + 1c)rl > 1, whence each of these sums is o(x) as zo and therefore ko tends to infinity. Hence (1.114) holds as required.
Theorem 1.14 Let W be a weak block sequence, and Behrend. Let e > 0, 0 < s' < 1 - log 2. For the short blocks, Vk < Tk, set Wk = Vk (log Tk
Tk)1og(1+ak)-ak+I
(1.126)
where log Vk = (log Tk)"k.
(1.127)
1 Besicovitch and Behrend sequences
50
For the long blocks, Vk > Tk, Tk + Vk = Hk Tk, set
Wk = min (log log Tk)A
log Tk GlogHk)
log2-1+E'togT '
a
( log Hk Y
(1.128)
where A > A(e'), with A(e) as in (1.156) below, and
o= 1
086071.... -log 2 + log 2 log (log 2) =
(1.129)
Then 00
E Wk = co.
(1.130)
k=1
We assume in the above that Hk < Tk, splitting blocks if required.
This is a development of Theorem 1 of Hall and Tenenbaum (1992), which was restricted to block sequences. It has a rather technical appearance and we begin with some explanatory remarks. First, notice that this result is much better than what we should obtain from a direct application of Corollary 0.13, that is 00
6,,# (z+ n (Tk, Tk + Vk]) = c0.
(1.131)
k=1
For example, Theorem 21(iii) of Divisors implies that for the long blocks JVk = Z+ n (Tk,HkTk], we have
(+C « S t(dk) « log Tk)
(log
Tk)
'
(1.132)
and Wk is smaller than this whenever log Hk < log Tk/(log log Tk)c(E
For the short blocks 4k = Z+ n (Tk, Tk + Vk] we have, for Vk < Tk(log
Tk)1-iog4-E
S#(-Q1k) ti Tk
(1.133)
by Theorem 21(i) of Divisors. Of course the point is that the 'probabilities' B-1#(dk) are not independent. Technically, the improvement comes because we apply Lemmas 1.8, 1.13 once only to each integer n c .#(V).). We want the exponent of log Tk in (1.126) to be negative, i.e. s < a(d) - log (1 + a(d)). A small e' may not be optimal in (1.128) because A(e') is a decreasing function of Z. The minimum value of C(E') occurs
when e'x6,andisz504.
1.3 Behrend sequences
51
Let d = db U,0 where d comprises the
Proof of Theorem 1.14
short, and c# the long blocks. At least one of these sequences is Behrend by Corollary 0.14, and so it will be sufficient to show that the corresponding sub-series in (1.130) diverges. First we suppose that s1' is Behrend, following the proof of Corollary 1.11 except that we are going to make y depend on the individual blocks. We go back to the proof of Theorem 1.7 to see how a variable y may be introduced. Let K > 0 be at our disposal, and fix to = to(K) in Lemma 1.8 so that d_4 (K, to(K)) < 1. Let ko be such that Tko > to, and consider the tail do of db of blocks for which k > ko. do is of course Behrend. Put Wk = Wk if dk is a short block, Wk = 0 else. We suppose that 00
Wk < co
(1.134)
k=1
and seek a contradiction. Instead of (1.74) we write 7y(k)n(n,a)-(l+K)1ogloga
x(n) < i(n,do) -<
(1.135)
amn
aE1/0'
where y(k) is to depend on the block Qk. Here x is the characteristic function of .1i'(d0). After (1.75)-(1.77) we arrive at
n ax(n) «(6 -
1)-1
nER(K,to)
E a-' (log aEd
«(6 -
a)y(k)-I-(l+K) log y(k)Y(k)D(a)
0b
T (log Tk)Y(k)-1-(1+K)logY(k)(log Vk)Y(k)-1
1)-1
k>-ko
k
(1.136)
employing Lemma 1.9. Eb is restricted to short blocks. We substitute log Vk = (log Tk)"k from (1.127) and set y(k) = 1/(1 + ak). Provided K log 2 < a we may deduce that
nax(n) « (Q - 1)-1 > Wk
.
(1.137)
k>ko
If (1.134) holds then we may take ko so large that
lim sup(a - 1) T, n ax(n) < a-+1+
3
(1.138)
nOR(K,to)
as in (1.78). We deduce that d is not Behrend by the argument used
1 Besicovitch and Behrend sequences
52
in the proof of Theorem 1.7; this is the required contradiction and we conclude that the series (1.134) is divergent.
Next, we suppose that Q is Behrend. We denote by c the union of the blocks for which the first term on the right of (1.128) is the minimum,
and by AR the union of the remaining blocks. If AR is Behrend then (1.130) follows from (1.132) and there is nothing more to prove; so' we suppose that A is Behrend. Let .'' denote the sequence of integers k for which the block dk is contained in d , and write .N'k = (d k) Since any tail of AL is Behrend we have for every k1,
d( U .N'k
= 1.
(1.139)
kE.' k>kl
We apply Lemma 1.13. Let K E (0, 1), B > B(K). Let zo be so large that da(ic, B, zo) < a ; and let us take k1 in (1.139) so large that Tk, >_ zo. We write . / = .N'k \.4(K, B, zo) and we deduce from (1.139) that
dNk > 4
(1.140)
kE.' k>kl
Next, let .A/ denote the subsequence of A -k'
fJ pl < Hkk,
n E .A/
which (1.141)
,
P"Iln
P
with vk at our disposal. By Theorem 0.7 of Divisors, we have
d(Z+ \ X") << exp(-cook)
(1.142)
where co is an absolute positive constant. We put Vk
=
co
log
(1.143)
)
loges g k/
whence we have d(
k\
log HTk
k) «
kE.7C
kE.Y'
k_ki
k_ki
to g
(1.144)
k
Either (1.130) holds or we may take kl so large that the sum on the left of (1.144) is < 4, so that we have, from (1.140), dN'k >- 2. kE.)r
k_kj
(1.145)
1.3 Behrend sequences
53
Let n E and aIn, a E 4k. We write a = st where P+(s) < Hk < P-(t), whence s < Hkk in view of (1.141). Since Tk < st < HkTk we have
TkHk°k
(1.146)
Since n 0 .(,c, B, zo) we have from (1.113) log Tk
+ B log log log Tk
92(n; Hk, Tk) < (1 + x) log ( \ log Hk
(1.147)
whence for y < 1, and large x,
E1< n<x
log Tk
(l+K )logy
(lH)
yO(n;Hk,Tk)
n<x
(log log Tk)-B logy
*
1
tin
(1.148)
where the star denotes that t satisfies (1.146), moreover P-(t) > Hk. We put n = mt and invert summations, employing the inequality yO(m;Hk,Tk)
m<x/t
-t
<x
log Tk )Y-I
( 1.149)
log Hk
(see Divisors, Theorem 01). Hence log Tk y-1 -(1+k)logv
d Vk « (logHk)
logTk)-BlogyZk
(log
(1.150)
where Zk =
-1 ya(t;Hk,tk)
We estimate Zk by partial summation. Provided u >_ Tk, we have yn(t;Hk,Tk) << u(log Hk)-Y(109 1: t
Tk)y-l
(1.152)
P-(t)>Hk
(see Divisors, Theorem 01) whence by (1.146)
Zk«(1 +
vk)(log
Y-1
Tk
log H k)
(1.153)
We insert this into (1.150), and choose y = (1+K)/2 optimally, which yields
ayk << (1 +vk)
( log Tk log Hk
log 2-1-Q(l+x)+x log 2
(log log Tk)Blog(,+K) (1.154)
1 Besicovitch and Behrend sequences
54
The function K log 2 - Q(1 + K) increases from 0 to 1- log 2 as K increases
from 0 to 1; we choose K to be the unique root of the equation K log 2 - Q(1 + K) =d
(1.155)
and then we define A(E)
= B(K) log
(---).
(1.156)
By (1.143), (1.154)-(1.156), provided A > A(E') in (1.128) we can choose B > B(K) so that we have d.N'" < Wk, (k (=- A-).
(1.157)
Since k1, in (1.145), is unbounded, (1.130) follows. This deals with the long blocks and thereby completes the proof. The next result is a refinement of Theorem 3 of Hall and Tenenbaum (1992).
Theorem 1.15 Let .4 be a union of long blocks
dk = Z+ n (Tk, HkTkj, (2 < Hk < Tk).
(1.158)
Then Q is Behrend if either 00
log Hk
_
(1.159)
klogTk or, for some e > 0 we have
(loH)s+E tog Tk
=1
k
= oo
(1.160)
where 6 =.086071 ... is defined in (1.129) and in addition, for all k > 2, log(Hk_1 Tk_1) <
log Hk
C(E) + (26 + 3E) log("logHk
(1.161) )
Here C(E) is a sufficiently large constant.
The idea of the proof which follows is that we `thin down' the dk
to subsequences 4k in such a way that the events n E A (sl') are independent, when it will be sufficient to have 0 S./Il("ik) = 0C).
(1.162)
k=1
To do this we associate with each dk a set of primes 91k in such a way that the sets 'k are disjoint. Then for every k,
,4k={aEs4 : pla=> pEYk}.
(1.163)
1.3 Behrend sequences
55
This is simple but effective: if we compare (1.160) with (1.128) we see that b, as an exponent, is best possible in both instances. It will emerge that (1.159) is best possible too, as it stands. Proof of Theorem 1.15
We begin with (1.159). We put
slk = 9k = {p E dk, p prime}
(1.164)
and we have
log Hk
(I-p)»
11
tk
1
gTk
(1.165)
whence (1.162) holds and .4 is Behrend. We turn to the alternative condition comprising (1.160) and (1.161). In this case we let
9k={p: Hkk
(1.166)
have to be disjoint
where ?1k is a small number at our disposal. The so we require that Hk_iTk_i < Hkk.
(1.167)
Let dk be as in (1.163) and (1.168)
.c/k = (Hk-ETk, HkTk] n Z
where we have assumed that k is so large that this is non-empty. Let n have a divisor a" E sdk and put a" = A where P-(a') > Hkk > P+(b) Then either a' E sd'k and n E(d'k), or b > Hk-E. Therefore Sm("Ik) > B_#('wk) - 6
fl p° > Hk-E}.
n
(1.169)
p"Iln
p
By Theorem 21(iii) of Divisors, there exists cl(c) > 0 such that
&#(d) >- ci(s)
Iog Hk Tk
log
Y+&
Hk_ETk
(1.170)
(provided, as we may assume from (1.161), Hk > 2), whence
oJf(d) ? c2()
(FJ)o+C Tk
g
(1.171)
k
because Hk < Tk. To estimate the second density on the right of (1.169)
1 Besicovitch and Behrend sequences
56
we employ Theorem 07 of Divisors with an explicit value of the constant, co = z, provided by Tenenbaum (1990), (p.437 Ex.5). This density is
11-s\
«exp -2.
(1.172)
11k
and we put rik = {C(8) + (26 + 3E) log
log Tk
1
Clog Hk) }
(1.173)
and deduce from (1.169), (1.171)-(1.173) that if C(E) is sufficiently large then &#/(d ) >_ I C2(E)
S+e
(log
(1.174)
log Tk
Together with (1.160) this gives (1.162); moreover (1.167) follows from
(1.161) and the .1k are disjoint. Hence c is Behrend as required. An example of a Behrend sequence of this sort is furnished by
2
Tk=exp{(k!)A}, Hk=exp{(k
(1.175)
The proof that (1.159) is sufficient for the long block sequence d to be Behrend is easy and this might suggest that with more work, a weaker sufficient condition could be found. We show next that, given oo as any function F(c) oo, we can construct a sequence d of long blocks, as in (1.158), such that °O
YF k=1
log Tk) log Hk _ log Tk - o0 C log Hk
(1.176)
but d is not Behrend. Our construction follows Hall and Tenenbaum (1992), (eq. (1.17) et seq.).
We assume as we may that F is non-decreasing, F > 1. We put f (t) = F(log t) and we consider blocks (Sh, GhShl,
Gh = e8 (h) ,
g(h)
- [loSh] f(Sh)
(1.177)
in which we let Sh -> oo so fast that both 00 E.f(Sh)-b
< c, Sh+1 > Sh,
(1.178)
h=1
where 6 is defined in (1.129). By (1.177)-(1.178) we have
h=1
log log Sh J
(1.179)
57
1.3 Behrend sequences
so that
00
d = U {Z+ (1(Sh, GhShJ }
(1.180)
h=1
is not Behrend, by Theorem 1.14. Let k(h) be the sequence such that k(h + 1) = k(h) + g(h), It > 1, k(1) = 1, and put
= =
Tk(h)
Tk(h)+1
Sh,
h = 1, 2, 3, .. .
Tk(h)e3, j < g(h),
Hk = e Vk
(1.181)
so that we also have 00
d = U {Z+ n (Tk, HkTkl }.
(1.182)
k=1
For k(h) < k < k(h + 1) we have Tk < S2h, and
F
log Tk log Hk
= f(Tk ) -> f(S
h ),
(1
.
83)
whence
F (log Tk" logHk > g(h)f(Sh) k(h)Sk
log Hk
log Tk
2 log Sh
(1 . 184)
Then (1.177) and (1.184) imply (1.176), and the required construction is complete. It is not clear whether such a sequence can be constructed if both the function F and a sequence {Hk : Hk > 2} are given in advance. This problem is left open. In a sense the above construction is a trick, in that most of the blocks are contiguous or nearly so (obviously we could space them out a little, e.g.
by striking out alternate blocks). We may draw the conclusion
that, to make progress with block sequences, we should for the most part concentrate on the case of 'well-spaced' blocks. Condition (1.161) achieves this but is rather strong. Notice that we may assume log Hk = o(log Tk) else (1.159) and (1.160) are indistinguishable; but then (1.161) requires log Tk_1 = o(log Tk), indeed if (1.159) does not apply, rather more. An innocuous looking problem, again left open*, is to determine what extra spacing condition, together with log Hk
llog
m
(ioTk)
o0
(1.185)
implies that W is Behrend. Here m e Z+ is fixed. We have in mind a * See Tenenbaum (199x) (note added in proof).
1 Besicovitch and Behrend sequences
58
possible analogy with (1.159), i.e. the case m = 0, in which we restrict to a subsequece sl' of sad comprising numbers with m + 1 prime factors. In this circle of ideas there is the following (straightforward) generalization of a result of Erdos (1959).
Theorem 1.16 Let k be a positive integer and for 1 < i < k let ei(p) be a positive valued function of the prime p. Let
d = {PPlP2 ... Pk : P < Pi < p"i(P), (i< k)}.
(1.186)
Then 4 is Behrend if and only if we have 1
k
1 H min (1, ei(p)) = oo.
(1.187)
i=1
P
Erdos proved this in the case k = 1 and stated that 4 is in any event Besicovitch. The second part of his assertion also holds in the general case and is a consequence of Theorem 1.4. Proof We begin with the proof of necessity, which is by contradiction. Let us suppose that the series (1.187) is convergent. We begin by splitting the primes into 2k disjoint sequences labelled by the subsets of {1, 2,3,..., k}. The prime p belongs to a particular sequence if and only if ei(p) < 1 for precisely those i belonging to the label subset. We denote these subsets by I, and associate with I the sequence
(I) _ P R Pi : P < pi < p1+e;(p)
i E I }.
(1.188)
iEI
Since the series (1.187) converges by hypothesis, we have 1
a
: aE d(I)
(1.189)
(it clearly does not matter whether or not we count repetitions at this point), because
p = log(1 + e) + O w
( log w )
.
(1.190)
Hence none of the 4(I) is Behrend, and neither is d. We turn to sufficiency. We may assume si(p) < 1 always, simply by disallowing any pi > p2. We have to show that x + o(x) integers n < x
belong to ol(d), and we restrict 4 by the condition p < x" where a = 1/(4k + 2), so that we have 17p1P2 ... Pk <
always. Let f (n) denote
1.3 Behrend sequences
59
the number of divisors of n of this form, counting repetitions multiply. We employ Turan's method, that is we consider the sum (f (n) - A(x))' = S2(x) - 2A(x)Sl (x) + [x]A(x)2
(1.191)
n<x
say, where we define A(x)
pl+Ei(P)
}.
(1.192)
= xA(x) + o(x)
(1.193)
p < x", p < pi <
pplp2 ... pk
=
Then we have
S1(x) = E f (n) = E
[
n<x
x
pp1P2 pk
the range of summation on the right being as in (1.192). By similar reasoning we have S2(x) = xB(x) + o(x)
(1.194)
where
B(x) _ EE
1
[pplp2 ... pk, gglq2 ... qk]
(1.195)
and the primes q, qi satisfy the same conditions as the p, pi; we have q < x", q < qi < q1},j(q) (The error term in (1.194) is 0 (fzk+2(x)) where
fl,(x) is the number of integers m < x such that S2(m) < 1: for fixed 1 this is « x(log x)-' (log log x)1-'.) We split B(x) into k + 2 subsums Bm(x), 0 < m < k + 1; the mth subsum contains the terms from (1.195) for which f([pp1 p2 ... pk, qq1 q2 ... qk]) = k + 1 + m
(1.196)
and we introduce the auxiliary sum, over these same primes 1 Bm(x) = E E pp1p2 ... pkgglq2... qk
(1.197)
so that Bm(x) < Bm(x), with equality if m = k + 1. Also k+1
Bm(x) = A(x)2,
(1.198)
M=0
whence we have k
A(x)2 < B(x) < A(x)2 +
Bm(x). M=0
(1.199)
1 Besicovitch and Behrend sequences
60
We claim that for 0::5:, r n< k, we have (1.200)
Bm(x) <
This is clear in the case m = 0 when the q's are a permutation of the p's. For 1 < r n< k let us write
Bm(x) «EE
1
(1.201)
ppip2...pkrrlr2...rm
where the dash means that the rj are primes such that [ppip2 ... pk, ggiq2 ... qk] = pp1p2 ... pkrrlr2 ... rm.
If we demand that rl < r2 <
(1.202)
< rm then these primes are uniquely
determined, moreover any particular product r1 r2 ... rm only occurs <
(1 . 203)
q:!
p2
because every si < 1. Hence
< rj < p4,
1<j<m
(1.204)
and
(1.205)
We insert this into (1.201) to yield (1.200). Then we assemble (1.191), (1.193)-(1.194), (1.199)-(1.200) to obtain
(f (n) - A(x))2
(1.206)
n<x
Let n 0
Then f(n) = 0, whence from (1.206),
card{n : n < x, n 0 mo(d)} « Ax)
(1.207)
oo by (1.187) and (1.190). The right-hand side of (1.207) is o(x) and so c is Behrend. This completes the proof. since A(x)
1.4 Witnesses
We introduce the topics to be discussed in this section with a numerical problem. Suppose that we are given non-trivial sequences d and . such that
S-Z(2d u 34) = 6,# ({2, 3}) =
3.
(1.208)
1.4 Witnesses
61
Can we infer that c and 9 are Behrend? We answer this question affirmatively and then consider its generalizations.
A ({2, 3}) comprises four arithmetic progressions .N' = 61 + j, j = 0, 2, 3, or 4. Each of the four sequences -&(2V U 3.) n . 1 has upper logarithmic density < 1 and the union of any three has upper density < 2. By (1.208) the lower logarithmic density of each sequence is therefore > 6, whence for each j, we have u 3M) n _/1} = 6.
(1.209)
Let j = 2. The multiples of 3. contribute nothing to the intersection in (1.209) and so in the case .N = 61 + 2 we have
8{.t(2d) n -4-} = 6.
(1.210)
We divide throughout by 2, and we deduce that
n Y} = 3,
(1.211)
in which I denotes the arithmetic progression 31 + 1. We require the following result.
Lemma 1.17 Let 4 be any integer sequence, and q, r be coprime integers. Let qZ + r denote the arithmetic progression {n : n = r(mod q)}. Then the sequence mo(d) n (qZ + r) possesses logarithmic density, moreover we have the formula
S{JI(B) n (qZ + r)} = q-'S # (d*(q))
(1.212)
where Q/* (q) comprises the elements of d prime to q. In particular, if d is non-trivial and the density on the left is equal to 1/q, then sd is Behrend. If we assume this for the moment then we may complete the solution of our initial problem. By (1.211) and the last part of the lemma, 'Cl is Behrend. A similar argument shows that . is Behrend. The last part of the lemma follows from (1.212). If the left-hand side equals 1/q then ,'(q) is Behrend, and so is d. It remains to prove that S {. #(d) n (qZ + r)} exists and that (1.212) holds. We may replace d by d*(q) because any remaining elements of d contribute nothing to the intersection. Hence we may assume that (a, q) = 1 for every a e 5. Let us begin with the case when 4 is finite, say d = {a1, a2, ... , aN}. < ih, < N. The integers n such that Let 1 < ii < i2 < n = 0(mod aig), 1 < g < h;
n - r(mod q)
(1.213)
1 Besicovitch and Behrend sequences
62
have asymptotic density equal to 1/[ai,,... aih]q by the Chinese remainder theorem, and we employ the inclusion-exclusion principle in the usual fashion to deduce that
d{.#(d) n (qZ + r)} = q-ld.#(d).
(1.214)
We may replace d by 6. Next, let . be infinite and put d(N> _ {a1,a2,...,aN}. Then 6
f'&(',j(N))
n (qZ + r)} = q-ldf(d(N))
(1.215)
n (qZ + r)}
(1.216)
whence by (0.67),
On the other hand we have
8{M(d) n (qZ + r)} < S{,#(sd(N)) n (qZ + r)} + 8{ Gl(d) \ jl(,,V (N)
< q-1#('4) + v
(1.217)
by (0.67), if N > No(v). This is all we need.
Theorem 1.18 Let 9 = {si, s2, ... , sk } be a (finite) primitive sequence and
8.l(s1d1 U s2d2 U
U skdk) = 84'({s1, s2, ..., sk}).
(1.218)
Then each of the sequences di, 1 < i < k, is either trivial or Behrend.
Proof
It will be sufficient to consider d1, which we may assume to be non-trivial. We put s = [s1, S2.... sk], v = s5.l ({sl, s2, ... , sk 1), so that the set of multiples on the left-hand side of (1.218) is contained in the union of v arithmetic progression . V = sZ + jµ, 1 < p _< v. By the argument employed in our initial example we have for every p,
S{./01d1
1 s
(1.219)
We put .K = sZ + s1: since {s1, s2, ... , Sk } is primitive .K contains no multiples from sidi, i > 1, and we have 6 {'/#(s1 d1) n .N} = 1
(1.220)
6{#(d1) n Y} = s
(1.221)
s
and s
where 2 = (s/sl)Z + 1. We apply Lemma 1.17 to deduce that d1 is Behrend. This complete the proof.
1.4 Witnesses
63
We note that the hypothesis (q, r) = 1 is essential for the final conclusion of the lemma. For example, 8{.J/({3}) n 3Z} = 1, but {3} is not a Behrend sequence. If every element of d is prime to q, that is d = d*(q), then (1.212) holds for every r. (This can be seen by an examination of the proof of Lemma 1.17, where the extra hypothesis (q, r) = 1 was just used at the outset, to enable us to strike out d \ d`(q).) Theorem 1.19 Let 4 be a Besicovitch sequence, all of whose elements are prime to the integer q. Then for every r, n (qZ + r)} exists and is equal to q-id. (d).
Proof When c is finite the proof of Lemma 1.17 already gives this. Let c be infinite, and d(N) = {al, a2, ... , aN}. Then we have d{,l(,,V(N))
n (qZ + r)} = q-1d#(d(N))
(1.222)
and we apply (0.67) to obtain
d{ ./#(d) n (qZ + r)} > q-'d h'(d).
(1.223)
This holds for every r (mod q) and so the opposite inequality applies to the upper densities by subtraction. This completes the proof. The last part of Lemma 1.17 demonstrates a special property of the
arithmetic progressions qZ + r with (q, r) = 1, and suggested to the author the following definition. The name comes from the world of law, in which a witness is a person who must testify, but only to first-hand knowledge.
Definition 1.20 Let 0' be an integer sequence possessing positive logarithmic density. If the relation
nrY} = S'Y
(1.224)
implies that 4 is either trivial or Behrend, then 'IY is called a witness.
Thus we may draw a conclusion about the behaviour of ,, although 0 -sees' only a part of 11(d). By Lemma 1.17, any arithmetic progression (qZ + r) n Z+ is a witness. The union of two such progressions may or
may not be, and a suggested exercise for the reader is to write down the necessary and sufficient condition for a finite union of arithmetic progressions to be a witness. From (1.224), the set of multiples #(-4) is not a witness unless -4 is Behrend or trivial - in some sense the class of witnesses is distant from the class of sets of multiples.
1 Besicovitch and Behrend sequences
64
In the Oberwohlfach (1994) list of problems Tenenbaum made a conjecture which, if true, would be a very interesting result, analogous to Theorem 1.18 but with 9° infinite. We conclude this chapter with a proof. We employ the notation Yd = {sa : s E 5°, a E d}.
Theorem 1.21 Let d, 9, 9- be sequences such that
> SW(J) ),
U .T)
(1.225)
U.) = &/#(Y U g-).
(1.226)
Then d is Behrend or trivial. We employ the following result.
Lemma 1.22 (Tenenbaum) Let s e Z+ and ,, J be arbitrary. Then 6_/#({s} U .f) - S./Ol(sd U J) > (1 - &W(d)) (&#({s} U .T) (1.227)
This inequality gives an alternative proof of Theorem 1.18, not involving arithmetic progressions and therefore avoiding Lemma 1.17.
Proof of lemma Let n E 1Gl({s} U .%) \ 1&(ssl U J). Then n = sm where m Also, sm h'(g) so that in the notation of §0.4, m #l (9-'(s)). The density of such m is
1 - 8Al (d U 9-(s)) > (1 - sJI(d)) (1- 6M (5'(s)))
(1.228)
by Behrend's inequality (Theorem 0.12). The second factor on the right is t ( 1, and go = 0. Also, for every v, _ {sv+1, sv+2, }. For arbitrary gal, and v we may write (actually an identity between sets),
8
({sl} U
(y# U T)) - SAl(sid U
+ 8#({s2} U (slW U A U J")) + SJ({S3} U (Y2`1/ U
U
-
U .%
U (sid U
U .T
Y-)) - SA'(S3d U (9'24 U 5°3 U J
+ &W({sv} U (w_1,4 U 4# U
)) -
U J)
(1.229)
1.4 Witnesses
65
The expression in the last line of this identity is equal to U Y# U .%) -
U -)
(1.230)
and this is non-negative. Every other line has the same form as the left-hand side of (1.227), and we apply the lemma, which yields
s.I(Y U .T) - sdI(5d U J) > (1 -
{&#(9 _1d U Yµ_1 U g-)
U Yµ U )}.
(1.231)
< 1. Suppose now that Q, Y, .T satisfy (1.225)-(1.226), and that The sum on the right of (1.231) is zero, and since every term in this sum is non-negative, these terms are all zero, in particular the last. That is, for every v
U 9'#_1 U 9-) - S I(9v_1d U 9P# U 9-) = 0.
(1.232)
We apply Theorem 0.8, and deduce that for every v the sequence
( _1d U Y# U
(1.233)
is either Behrend or trivial, in which we have employed the notation introduced in (0.74). We deduce that for every v, the sequence
(w-1 U Y# U
(1.234)
is either Behrend or trivial. We may assume that 9' is primitive, and since Yv_1 is finite this implies that for every v, (Y# U .%)'(s,,)
(1.235)
is either Behrend or trivial; applying Theorem 0.8 again we deduce that for every v, we have 8./Gl(y,_1 U g-) = 8./Gl(9# U .J-).
(1.236)
6-&(9 U.) = s Y(r ),
(1.237)
Therefore
contradicting (1.225). We conclude that &#(d) = 1 as required.
2
Derived sequences and densities
2.1 Introduction
Let s$ = {a1ia2.... } be an integer sequence and .#(d) be the set of multiples of 4. Then n belongs to
if and only if
i(n,d) > 0
(2.1)
where i(n, d) denotes the number of divisors of n which belong to d. For each positive integer k, there is a (possibly empty) subsequence of ,#(d) on which tc(n, d) > k
(2.2)
and we notice that this subsequence is itself a set of multiples and as such possesses logarithmic density, by Theorem 0.2, the Davenport-Erdos
theorem. For (2.2) holds if and only if there exist ij, 0 < j < k such that a;, In,
0 < j
(2.3)
equivalently [a;0, a;, , ail, .... aik) In.
(2.4)
Let us write ',1(k) = { [aio, ai..... aik ]
1 < ip < l1 < ... < ik }.
(2.5)
and, to fix our ideas, order d(k) as a non-decreasing sequence. It may contain repetitions, but we do not exclude this case. Then we see that (2.2) is equivalent to n E 66
2.1 Introduction
67
Definition 2.1 The kth derived sequence of d, denoted by 4(k), is given by (2.5) above. We write
tk(d) = 1 -
1 - d q(f(k)),
(2.6)
equivalently tk(d) is the logarithmic density of the sequence of integers n for which
i(n, d) < k.
(2.7)
Sometimes we refer to these sequences as derivatives (but the reader should note that derivatives of derivatives are not generally derivatives: fortunately we are not concerned with these). We have the following immediate result, which is simply an equivalent alternative form of Corollary 0.15.
Proposition 2.2 Every derivative of a Behrend sequence is Behrend.
We remark, en passant, that the sequence of integers n such that i(n,d) = k, which is of course .#(S1(k-1)) \ G#(Sl(k)) also possesses logarithmic density, and we denote this by tk(d). We have to(d) _ to(d) = t(d): we use all three of these expressions according to the context.
This chapter contains four main parts. In the first of these, §2.2, we are concerned with inequalities satisfied by the tk(d), particularly upper
bounds for tk(d) in terms of t(d), which are quantitative forms of Proposition 2.2. Some of this work is quite intricate and we mention at the outset that there are harmless looking problems in this area which we are unable to solve. We draw these to the reader's attention as they arise. In the whole of this section we are concerned with a single sequence S/. In §2.3 we take two sequences 4 and R and we generalize Behrend's inequality in various ways. The results are lower bounds for the densities
tk(d U A. One of the basic difficulties in the study of sets of multiples or sieves is that, often driven back to the inclusion-exclusion principle, we are forced to write down sums involving a large number of terms of opposite sign to represent small positive quantities such as t(d). Thus for a finite sequence d we have the familiar formula
t(om) = 1i
1+E ai i<j
[ai, ai]
(2.8)
which could hardly be more ill-suited to most applications, indeed it is not obvious that the right-hand side is positive. There is a parallel here
68
2 Derived sequences and densities
with the exponential series 2
e-x=1-x+Zi -... (x>0); again the right-hand side is not obviously positive and for some purposes
the `expansion' (we are not concerned here with the definition of the exponential function) is quite useless. In §2.4 and §2.5 we describe a different method of dealing with quantities like tk(d): this is still combinatorial in nature, but leads to formulae involving sums of positive and therefore necessarily small terms. This technique is very successful with the problem encountered in Chapter 3, for which the last sections
of this chapter are an essential preparation. We do not know to what extent it applies to the open problems mentioned above, but this would appear to be a possible avenue of further study. 2.2 Upper bounds for tk(d) In this section we seek an upper bound for tk(,4), defined by (2.6), in terms of t(d). Ideally we should like this to be simply a function of t, that is we suppose there is no a priori information about d. Our second theorem is a result of this sort but is (presumably) far from best possible. A complete solution of the problem is presented on the strong hypothesis
that the elements of d should be pairwise coprime. This is the content of Theorem 2.9. We begin with a preliminary result. Theorem 2.3 We define
lk = inf{t(d) : IdI < k}.
(2.9)
Then we have
nk=fl(1i=1
\
11
(2.10)
Pi /)
where p, denotes the ith prime.
Proof We have to show that t(d) is not less than the right-hand side of (2.10). Let P+(a) denote, as usual, the greatest prime factor of a, and put
P = {p : p = P+(a) for some a E d},
(2.11)
so that P is a set of at most k primes. For each p belonging to P let Slp = {a : a E 9, P+(a) = p}.
(2.12)
2.2 Upper bounds for tk(s'ad)
69
Then M(sdp) c pZ+ and t(s1p) > 1 - (1/p). We apply Behrend's inequality, (Theorem 0.12) which yields
(_!)
t(d) ? fl t(dp) ? 1
(2.13)
p
pEP
pEP
and since IP I < k, the result follows.
It is important to notice that the functions tk(d) are not monotonic in the strong sense that t(d) < t(R) would imply tk(sdO < tk(P). A counterexample is d = {3,5,7}, M = 12,41 when we have
t(`d) = 35' t(a) = 2, tl(d) =
ti(a) =
105'
4.
(2.14)
Definition 2.4 For each positive integer k and a E (0, 1] let Pk(a) = Sup{tk(d) : t(SI) < a}.
(2.15)
This is certainly an increasing function of a, in the weak sense. Notice that if a > nk then d may have only k elements, when i(n, sad) > k is impossible and tk(d) = 1. Hence WPk(Irk) = 1, or
irk < a < 1
implies
cpk(a) = 1.
(2.16)
A moment's thought will convince the reader that the converse implication is not obvious; indeed we are unable to say whether or not it is true.
Definition 2.5 For each positive integer k let Pk = inf{a : cpk(a) = 1}.
(2.17)
Pk = irk for all k.
(2.18)
Conjecture 2.6
Evidently pk < Irk, and we prove later that if we restrict sad in (2.15) by
requiring that the elements of d should be pairwise coprime then the quantity corresponding to Pk equals irk. For the moment we proceed as far as we can without any side condition on W. Theorem 2.7 For each positive integer k we have
a < (MO < (k +
i1k+i
2)a.
(2.19)
70
2 Derived sequences and densities
A corollary is that pk > 1/(k + 2)k+1 This is (presumably) very far from the truth since Mertens' theorem (and the fact that log pk - log k) imply
Irk -
e -T (2.20)
log k
The lower bound is simple: we have tk(d) > t(d) and we can find a sequence , (for example containing primes only) such that t(d) = a. For a lower bound of the correct order of magnitude see (2.78). We turn to the upper bound. Let t(d) < a. We are going to split Proof of Theorem 2.7
d into k + 1 disjoint subsequences dj, 0 < j < k. If n E(Vj) for all j then r(n, d) > k, whence
tk(d) L t(dj).
(2.21)
j=o
We may assume that a11k+1 < 1 else (2.19) is trivial, and we set Uj = 6(J+1)I(k+1) 0 < j < k,
(2.22)
2>uo>u1>...>uk-1>a>t(d).
(2.23)
so that
For each j, (0 < j < k) let mj denote the largest integer such that t({a,,a2,...,am1}) > uj.
(2.24)
Since t({at}) = 1 - 1/a1 > 2 > uj > t(d) by (2.23) we have 1 < mj < cc and put
-4j = {al,a2,...,am,}, 0 < j < k.
(2.25)
By the maximal property of mj we have (2.26)
t(R j U {amp+1 }) < U j
and the Heilbronn-Rohrbach inequality (Theorem 0.9) implies
uj>t(qj)(1-
1
amp+l
)>
fi
1
J1
- 1). ai
(2.27)
We may assume that d has no repeated elements for repetitions would not affect the value of t(d) and would not increase tk(d). Hence the
2.2 Upper bounds for tk(d)
71
product on the right of (2.27) is at least 1/(m1 + 2) > 1/(amp+i), whence (2.27) implies firstly that amJ+l z 1/uj and then that 1
t(a,) < u1 11-
\
1
am;+1
< u- uj
,
(0 < j < k).
(2.28)
1
Now we may specify the d1 appearing in (2.21). We put do = P4o, .W1 = -41 \ °Ro, d2 = -42 \ -41, ... ,. 4k = d \ P4k-1 Behrend's inequality (Theorem 0.12) gives t(P41)
>-
t(P4o)t(-41), t(P42)
> t(P41)t(d2), ... , t(d) > t0k-1)t(dk)
(2.29)
and since we have both lower bounds (2.24) and upper bounds (2.28) for the t(-4,) we may deduce from (2.29) that uo/(1 - uo),
t(do)
t(dj) < uju,-.it/(1- uj), 1 < j < k, t(dk) < auk I1
(2.30)
whence by (2.22)
t(d,) <
a1/(k+1)
0<j
1 - a1/(k+l)' with strict inequality when j > 0. Thus (2.21) implies
(2.31)
a l/(k+1)
tk(d) < (k + 1) 1 - al/(k+1) This is a uniform bound, and when a1/(k+1) < 11(k + 2) it implies (2.19). For larger a, (2.19) is trivial. This completes the proof.
The reader will have observed that our choice of the uj in (2.22) is not optimal. However, it is clear that once we have adopted the strategy
of this proof, simply adding the densities in (2.21), the upper bound envisaged cannot be better than (k + 1)a1/(k+1) Theorem 2.8 For every k, the function (Pk(a) is continuous.
Proof By Theorem 2.7, (pk(a) is continuous at a = 0. Since (pk
is
non-decreasing it will therefore be sufficient to show that for a > 0 we have
(Pk(a) - (Pk(xa) < 1 - x,
0 < x < 1.
(2.32)
We may assume (pk(a) > 0 else there is nothing to prove. Let 0 < s < (pk(a). By definition, we can find a sequence ,(e) such that t (d(8)) < a
72
2 Derived sequences and densities
and tk (d(e)) > (pk(a) - e > 0. Hence t (d(E)) > 0 by Theorem 2.7, that is W(E) is not a Behrend sequence. Therefore
Y'
p
:p
< oo,
and we deduce that there exists a set P(e) of primes, not intersecting d(E) and such that
ri (i_!'\=. P
PEP(C)
Put M(e) = P(e) U d(e).
By Theorem 0.12, we have t (R(E)) =
t (P(E)) t (d(E)) = xt (d(E)) < xa. If i (n, d(e)) < k < i (n, P4(E)) then n E .#(P(s)), whence the density of such integers n does not exceed 1 - x, that is tk (d(c)) - tk (a(s)) < 1 - x. This implies (2.32) because E is arbitrarily small. We obtain a much better result than Theorem 2.7 if we assume that the elements of Q are pairwise coprime. In the context of sets of multiples
this is a rather unwelcome hypothesis, but it leads to an interesting problem in probability theory which we are able to solve, and the final result which we state next is essentially best possible. Theorem 2.9 Let 9k(a) be as in (2.15) with the extra hypothesis that the elements of d be pairwise coprime. Then for each k, cpk(a) is a continuous, non-decreasing function of a, such that for each n > 1 and nn ? a >- ltn+1 we have
k(a) < P
(a
- 7En+1)tk,n + (nn - a)tk,n+l
(2.33)
7n - TTn+1
with equality at the end-points, where (2.34)
,Pn})
tk,n = tk({P1,P2,
is the density of the integers divisible by at most k of the first n primes. In particular, for irk ? U ? irk+1 we have (pk(a)
-1- 1-
k
6 7rk
1
l l pj
(2.35)
Corollary 2.10 For each k E Z+ we have (ple(a) < 1 if and only if a < Irk.
This is immediate from (2.35), which is simply (2.33) with the values tk,k = 1, tk,k+l = 1 - 1 /(P1P2 . . . Pk+1) inserted.
2.2 Upper bounds for tk(sl)
73
We see that the function (pk(o) is determined precisely at the points 7rn, and within quite narrow confines between these points. However, the exact behaviour of this function within the intervals Ocn+1, 7Cn) remains
somewhat mysterious. We shall see in Theorem 2.13 below that it is constant in an interval [in, 7Ln + lk,n], where rlk,n > 0 for every pair k, n > 1. We do not know whether (2.33) or (2.35) holds with equality at any intermediate point. Of course we can write down an explicit formula for tk,n (see the proof of Theorem 2.13) and thereby derive an asymptotic formula, and bounds
for this quantity. We do not undertake this here and so Theorem 2.9 suffers from the defect, in comparison with Theorem 2.7, that it does not These can provide us with explicit, applicable analytic bounds for be very useful and we pursue the matter by an alternative, direct method in Theorem 2.12.
Lemma 2.11 Let Yj (1 < j < m + 1) be independent Bernoulli variables, (random variables taking the values 0 or 1) such that 1 > Prob(Y1 = 1) > Prob(Y2 = 1) > . . . >_ Prob(Ym+1 = 1)
(2.36)
and let
Y =Y1+Y2+"'+Ym+1 Let Xj, 1 < j < oo be a further sequence of independent Bernoulli variables such that
Prob(X2 = 1) > Prob(X3 = 1) > ... Prob(Xj = 1) < Prob(Yj = 1), 1 < j < m X r,j Xj < co almost surely Prob(X = 0) <_ Prob(Y = 0).
Prob(X1 = 1)
(2.37)
(2.38) (2.39)
(2.40)
Then for every positive integer k we have
Prob(X < k) <_ Prob(Y < k).
(2.41)
Proof Put aj = Prob(Xj = 1),
f3 = Prob(Y3 = 1).
(2.42)
Condition (2.39) is equivalent to 00
E aj < co, 1=1
(2.43)
2 Derived sequences and densities
74
and we begin by supposing that aj = 0 for j > n (say). Consider the case n< m first. We have aj < /3, < 1 by (2.35) and (2.38), whereas from (2.40) we have n M+1 fl(1- c) <
fl(1 - /3i).
(2.44)
J=1
.1=1
Therefore aj = flj, 1 < j < n, f3j = 0, n < j < m + 1, that is aj = f3i for all j, and (2.41) is trivial. We may therefore suppose that n > m, moreover an,+l > 0. Let k E Z+ be given: we may suppose k < m because the event Y < k is otherwise certain and (2.41) holds. Let a denote the vector (al, a2 .... an) and Pk (a) = Prob(X < k) : we have to show that
the supremum of this function subject to the constraints (2.37), (2.38) and (2.40) is attained when a = fi := (/31, /iz, ... , /3n,+i, 0, 0, ... , 0)- Pk (06) is continuous and attains its maximum: let us suppose for some a fi. Put XW = X - Xi. Then Pk(a) =
Prob(X < k)
Prob(X(') < k - 1) + Prob(X(') = k, Xi = 0)
= Prob(X(') < k - 1) + (1- ai)Prob(X(') = k) whence
8Pk
aai
(a) = -Prob(X(') = k).
(2.45) (2.46)
Now ak+1 >- an,+l > 0, by our earlier assumption, that is at least k + 1 of the aj are non-zero and Prob(XO = k) 0. Hence from (2.46), a kk (a) < 0, (1 < i < n).
(2.47)
Let h be the least integer such that ah
Ph. We must have h < m + 1 else X > Y almost surely and (2.41) follows. Further, let 1 be the greatest integer for which al > 0. We see from (2.40) that l > h. We have ai < ah by (2.37) and it may be that ai < ah. In this case we consider the vector a in which ah = ah + E, ai = al - E, aj = aj else,
(2.48)
where E is small and positive. Since ah < Qh < Ph-i = ah_1 and ai > 0 = a,+1, moreover Prob(X = 0)
= PO(a)
PO(a) (
1-
E
1
-5h
)
< Po(a) < Prob(Y = 0)
E
(I + 1
al (2.49)
2.2 Upper bounds for tk(d)
75
so that (2.40) still holds, this is permissible ifs is small enough. We have Pk(a)
Pk(a) + E
(aOPk
ah
(a)
Bat
(a)) + O(E2)
(2.50)
and, by (2.46), the coefficient of E on the right is Prob(X<<W = k)
- Prob(X(h) = k).
(2.51)
Now
Prob(X(i) = k) = Prob(X = 0) ) II i,
1
a"j -ai
(2.52)
i. Since ah > ai this implies that (2.51) where 7,t`t denotes that ij is positive, that is Pk(a) > Pk(a) if E is sufficiently small. This is a
contradiction. We deduce that al = ah. In this case we again vary a slightly, by increasing ah and decreasing al. As before, the constraints (2.37), (2.38) hold, but the relative movement of Po(a) and Pk(a) is more subtle. Rather than embark on a calculation involving higher variations we make ah and ai functions of a parameter u in such a way that Po(a) is independent of u, i.e. constant. We put 1 - ah = (1 - ah)e u,
(2.53)
X - Xh - Xi. We have
where u > 0. Put Pk(a) =
1 - al = (1 - ai)eu
Prob(X < k) Prob(X(h^i) < k
+
- 2)
= k - 1).Prob(Xh + X1 = 1) + Prob(X(h"i) = k).Prob(Xh = Xi = 0). Prob(X(h,i)
(2.54)
Only the middle term on the right depends on u; we have Prob(X(h") = k-1) > 0, because at least k+1 of the aj are non-zero, and the right-hand factor is
(1 - ai)ah + (I - 401
=(1-ai)eu+(1-ah)e--u-2(1-ah)(1-ai) = 2(1 - ah) cosh u - 2(1 - ah)2
(2.55)
because ah = ai. Since ah < 1 by (2.35) and (2.38) this is a (strictly) increasing function of u, and this again is a contradiction. Hence a = as required. It remains to consider the case when the series (2.43) does not terminate, that is aj > 0 for every j. As before we may assume that for some h < m+
76
2 Derived sequences and densities
1 we have ah < Ph else X > Y almost surely and (2.41) holds. Let s be the greatest integer for which fl j = Ph for h < j < s, so that either s = m+ 1 or s < m and /3S > is+i We decrease Ph, Ph+1, ... , f3 by a small quantity 6 in the definition of the random variables Yj, leaving the remaining variables unchanged. Thus Y becomes a new random variable Y(b) for which Prob(Y(6) = 0) >- Prob(Y = 0)(1 + 8)S-h+i
(2.56)
Of course (2.35) is still satisfied. Let 0 < e < 6/2, and n > m be so large that T, IXj
(2.57)
j>n
We write X = X'+ X" where X' = X1 + X2 +
+ Xn and we have
Prob(X" * 0) = Prob(X" > 1) < a by Markoff's inequality, whence Prob(X = 0) < Prob(X = 0) Prob(X' = 0) = Prob(X" = 0) 1 -S
(2.58)
(2.59)
Since e < 6/2, and clearly S < 1, we deduce from (2.56) and (2.59) that
Prob(X' = 0) < Prob(Y(S) = 0).
(2.60)
Since ah < Ph we can make S so small that (2.38) holds with the new Yj, and we deduce from the first part of our proof that for all k,
Prob(X' < k) < Prob(Y(6) < k)
(2.61)
and clearly Prob(X <- k) <- Prob(X' < k) since X > X. The right-hand side of (2.61) is a continuous function of 6 and (2.41) follows. This completes the proof of Lemma 2.11. Proof of Theorem 2.9 Let d _ {al, a2, ...} be any sequence of pairwise coprime elements arranged in increasing order, and such that t(d) < a. It will be sufficient to show that for each k, tk(d) does not exceed the righthand side of (2.33). We may assume that t(d) > 0, for otherwise the series 00 1
E a;
(2.62)
would be divergent, and for a sufficiently large N we should have N
(2.63) ai
But then r(n, sd)> r(n, d') and tk(d)
2.2 Upper bounds for tk(d)
77
Accordingly let t(sc/) = p, 0 < p < Q, and let nm > p >
7rm+1 , so that
m >_ n. We consider a sequence of Bernoulli trials in which
XJ _
1 if ajIn 0 else
(2.64)
and we have Prob(Xj = 1) = 1/aj; moreover the Xj are independent because the a j are coprime. The series (2.62) is convergent, and so conditions (2.37) and (2.39) of Lemma 2.11 are satisfied. Now we consider another sequence of Bernoulli trials in which
Prob(Yj = 1) =
l
1<j<m
,
(2.65)
Pi
where p j is the jth prime, and
Prob(Ym+i = 1) = 1 -
P
1
<
7Em
(2.66)
Pm+1
Then (2.38) is satisfied, because it is easy to see that aj >- pj, moreover (2.40) holds with equality. As in Lemma 2.11 we put m+1
00
X = > X j, Y = > Yj,
(2.67)
j=1
j=1
the left-hand series terminates if d is finite, and we apply the lemma to obtain
Prob(X < k) < Prob(Y < k).
(2.68)
The right-hand side is a linear function of p, equal to tk,m when p = ?Cm, and tk,m+1 when p = nm+1 Hence it equals the right-hand side of (2.33) with m and p substituted for n and a. We leave it to the reader to check the intuituve proposition that Prob(X < k) = tk(sl). Since the expression on the right of (2.33) is non-decreasing the conclusion follows. This completes the proof of Theorem 2.9.
j=0
Theorem 2.12 For positive integers k and 0 < a < 1 we have
')
J
Clog-')' og
Q
Q
J
<_ (pk(a) <_ eC a E _i j=0
where
C=
/
{p
1
1 + log 1 1- p
\l I
(log 1 J 6
} _ .45743...
(2.69)
(2.70)
2 Derived sequences and densities
78
We note that ec = 1.58000... (> o
This factor may, possibly, be replaced by a function of k tending to 1 as k -> oo: this question could no doubt be settled by a careful analysis of the numbers tk,n defined by (2.34). The convergence cannot be too fast because 9Z(71k) = 1. As it stands, (2.69) implies that gpk(a) < 1 only if a is extravagantly small compared to 7Ck, indeed we require a < exp (-k - co(k),Ik- ) where !).
0 0 it may be shown by straightforward combinatorial arguments that
tk(SI) = t(
SS (d)
(2.71)
j=o
where Sj(sd) denotes the jth elementary symmetric function of the num-
bers 1/(ai - 1), So(d) = 1. For j > 1, we have Sj(d) < S1(d)j/j!, moreover 00
00
S&4) =
< - 1: log (1
1
ai
- 1) + C ai
i=1
(2.72)
because ai > pi, the ith prime. If t(d) = p this gives k
((log!)
p
C\
+
j (2.73)
I
j=0 -I
and since p < u < aec and the right-hand side of (2.73) increases with p, this gives the upper bound in (2.69). Next, by Lemma 13 (p.147) of Halberstam and Roth (1966) we have
j (d) >
S
i
S1
(d) { 1 -
()Si (d)_2V
Let w, z
(ai
i=1
ll
1
1)2
}
.
(2 . 74)
JJJ
oo together in such a way that
fi
-
(i_i)
o-0,
(2.75)
w
and put 4 = {p : w < p< z}. We have
Si(d) = E p-1 - log 16 w
(2.76)
2.2 Upper bounds for tk(d)
79
by (2.75), applying Cauchy's convergence criterion to the series (2.70), whence by (2.74) SS(d) > (1 + 0(1))
j
U)
(log 1 !
(2.77)
and the lower bound in (2.69) follows from (2.71). This completes the proof of Theorem 2.12. Evidently we also have k
(pk(6) > 6 > j =o
j
(log 1 \ 6
,
a< 1
(2.78)
but there is a substantial gap between this and the upper bound contained in Theorem 2.7. The central problem in this section is therefore to obtain an upper bound for c0k(6) comparable with Theorems 2.9 and 2.12: of course the strongest conjecture would be that for all k, we have (pk(6) = 9k(6),
0 < o < 1,
(2.79)
which in view of the corollary to Theorem 2.9, would certainly imply (2.18), pk = irk. There is a heuristic argument in favour of (2.79) as follows: if the elements of 4 are not coprime then there is a tendency for r(n, sd) to be large, given that it is positive. That is, for a given value of t(sd) we might expect tk(,W) to be smaller, in the unrestricted situation.
All our upper and lower bounds for cpk(a) and (ple(a), provided by Theorems 2.7, 2.9 and 2.12 are concave functions of a, and this might suggest that these functions are themselves concave. Our next theorem shows that this is not true of (pk(a), and additionally that this function is not differentiable at the points 7[,,, n >_ k. We cannot prove any similar assertions about c0k(6), but even if (2.79) were false we should not expect the two functions to behave completely differently.
Theorem 2.13 For each k and l > 1 there exists rjk,j > 0 such that (i (6) = tk,l, (iti < 6 < it, + qk,1),
(2.80)
where tk,l is defined by (2.34).
Thus 'pk has infinitely many intervals of constancy, and cannot be concave. In particular it is not concave in any neighbourhood of 6 = 0. If q were differentiable at a = ni then we should deduce from (2.80) that dcpk/d6 = 0, in contradiction to (2.33) whenever 1 >_ k.
80
2 Derived sequences and densities
Proof of the theorem We may assume that l > k because 9k(6) = 1 for 6 > 7Ck. We want to find a > 7Li such that t(d) < a implies tk(A) < tk,i, and we begin with the restriction that a < 7Ci-1 (1 - (1 /pi+1)), where pj
denotes the jth prime. Since the upper bound for tk(d) follows from Theorem 2.9 if t(d) < mi, we may restrict our attention to sequences _4 (of coprime elements) such that 7Ei < t(SI) < 6 < 7E1-1
1
C1 \\ Pi+1
).
(2.81)
Since t(d) < 7C1_1 this implies IdI > 1 by Theorem 2.3. As usual, we write
d = {a,,a2.... } and we have al > pi, since t(sar) > iti. Hence al >_ p1+1 because the elements of sad are coprime. The right-hand inequality in (2.81) is impossible with just l elements, since the Heilbronn-Rohrbach inequality (0.80) and Theorem 2.3 would then imply in turn t(sd)
> >
C1 -
1-
-I) t({ai,a2,..., a,-1})
1 Pi+1
i1
Therefore I sd I > l + 1. We are going to apply Lemma 2.11. Let i+1
Y=1: Yi l=1
where the Yj are independent random variables equal to 0 or 1 (Bernoulli trials), such that
Prob(Yj = 1)
= pi 1, j < 1, = Pi+1>
Prob(Yi+l = 1) = 0p1+1,
j = 1, 0< 0 < 1.
(2.82)
We determine 0 = 0(6) in such a way that
Prob(Y = 0) = a, thus 0 < 0 < 0(iri) where C
1- 1 1-0(700 Pi+1 ( Pi+1)
1- i.
(2.83)
Pi
We note that 0(iti) < 1 by Bertrand's postulate. By Lemma 2.11 we deduce that for the sequences d satisfying (2.81) we have
tk(d) < Prob(Y < k).
(2.84)
2.2 Upper bounds for tk(d)
81
We evaluate the right-hand side. Let us write Prob(Yj = 1)/Prob(Yj = 0) = yj, and denote by sh and sh respectively, the hth elementary sym-
metric functions of the y j, for j < l + 1 and for j < l - 1. Then we have
Prob(Y
and since Sh = Sh + (yl + y1+1)Sh-1 + YlYl+13h_2 for h > 1 (with the convention that so = 1 and 3_1 = 0), this becomes Prob(Y < k) = a{Sk + (yi + yl+l)9k_1 + y1y1+1Sk_2}
(2.85)
where Sh = So + 31 + + Sh. We denote the right-hand side of (2.85) by F(a) - notice that within the curly brackets only yl+l depends on a and in view of (2.84) we have now shown that q *(a)< max (tk), F(a)) ,
a < ir1-1(l - p +1).
(2.86)
Since F(a) is continuous, our result will follow if we show that F(761) < tk,l.
Recall that tk,l is the density of the integers divisible by at most k of the first l primes. We may write
tk,l = Prob(Y' < k) where
Yj, Prob(Yj' = 1) =
Y'
1 p/
j=1
and the Y.i are independent Bernoulli trials. By (2.82) we have Prob(Yj' = 1) = Prob(Yj = 1), j < 1, whence by a similar calculation to that which led to (2.85) we have tk,l = 7Cl{Sk + Zsk_1},
Z=
1
PI-
(2.87)
We put a = 7Cl in (2.85). By definition, yl = 1/(p1+1 - 1) and y1+1 = 6(iri)/ (p1+1 - 0(ir1)) where 0(7rl) is determined by (2.83). We notice that we may write (2.83) in the form YI+1)-1
(1 + YO-1(1 +
= (1 +
z)-1
so that in fact YI + YI+1 + YIYI+1 = Z.
(2.88)
Since 3k-1 > 0 we have Sk-1 > Sk_2, whence (2.85), (2.87) and (2.88) imply that F(7cl) < tk,l as required. This completes the proof.
2 Derived sequences and densities
82
We may pursue this matter a little further. Let flk,l be taken to be as large as possible in (2.80), that is cpk(a) > tk,l for a > itl + qk,l. Then we obtain a lower bound for 11k,, by solving the equation F(a) = tk,l. We have, for l > k > 1, '1k,l >
Sk-1
(p1+1 - 1)Sk + Sk-1
1 7El-1
(1
I.
-1
- 7[l } .
P1+1
(2.89)
)J
Theorem 2.13 shows that the function goZ(a) is not concave. However, we do have the following result concerning the tk,l. Theorem 2.14 For every k and 1 we have tk,l >-
Itl - ltl+l
tk,l-1 +
M-1 - 7r1+1
II-1-77l
tk,l+1,
(2.90)
it1-1 - 7r1+1
moreover the inequality is strict if and only if l >- k. Thus the function gpk(a), if it be restricted to the points a = its, is concave.
Proof Put yj = 1/(pj - 1) for every j, and let sh denote the hth elementary symmetric function of Yi, Y2, -j1-1, and Sh = 30+31+- +sh. As in the proof of the previous theorem, it is understood that so = 1, 3-1 = 0. We have tk,l-1 tk,l
tk,l+1
itl-ISk, 1I(Sk +YISk-1), it1+1{Sk + (YI + Yl+1)Sk-1 + YIYl+1Sk-2}.
Let uk,l denote the difference between the left- and right-hand sides of (2.90). We find after some algebra, that uk l =
Sk-1
> 0,
PI(pl + pl+l - 1) -
moreover we have 3k-1 > 0 when l >- k, as required. This completes the proof.
These theorems provide us with a fairly detailed description of the function cpk. Our information about (Pk is, by comparison, vague, and fresh progress in this direction would be of great interest. 2.3 Generalized Behrend inequalities
We consider two sequences d and 4, and the densities U-4), and we are concerned with inequalities involving these densities and the densities tg(d), th(P4), for various combinations of g, h and k. As in the classical
2.3 Generalized Behrend inequalities
83
Behrend inequality (Theorem 0.12) t(d U -4) >_ t(d)t(9), our results will be for the most part lower bounds for tk(d U -4). We shall be able to concentrate on the finite case, that is IdI, I-4I < oo; because firstly the tk were defined as logarithmic densities, so that we
can apply the results about sequential density from Chapter 0 to the derivatives W(k), g1(k) and (d U _4)(k) in the infinite case, and secondly
because essentially all the difficulties with which we have to contend already occur when 4 and . are finite. An exception to this rule already arose in Theorem 0.12 when we considered the cases of equality - for example see (0.103) - but it is enough at this stage to be aware that such things can happen. The tail might wag a little at infinity. We begin with a simple and intuitive result concerning the densities
tk(d) = S{n : i(n,d) = k}.
Recall that d and . are said to be coprime if (a, b) = 1 for all a E Proposition 2.15 Let 4 and . be non-trivial and coprime. Then for every k,
tg(d)th(1M).
tk(OW U -4) _
(2.91)
g+h=k
We give two proofs. Each uses ideas which we need later. In view of our previous remarks we may assume that d and -4 are finite.
Proof (i)
The hypotheses imply that q and . are disjoint and so we
have
r(n, d U M) = T(n, c) + i(n, -4).
We may therefore suppose that i(n, d) = g and i(n,,4) = h, where g + h = k, and sum over the possible values of g and h. Thus (2.91) expresses the fact that, in probabilistic terminology, the events T(n, d) = g and T(n, -4) = h are independent. Let A = l.c.m. [a : a E d] and B be defined similarly. Then T(n, d) _ i((n, A), d), that is the value of z(n, sad) depends only on the congruence
class (mod A) to which n belongs. There are Atg(d) classes in which i(n,d) = g and similarly Bth(PJ) classes (mod B) in which z(n,.1) = h. By the Chinese Remainder Theorem, there are ABig(d)ih(c) congruence
classes (mod AB) in which both events occur, because A and B are relatively prime. This is all we need.
2 Derived sequences and densities
84
Proof (ii) We employ the inclusion-exclusion principle to write down a formula for tk(sl). This is
E(-1)l-k(k)fl(d)
tk(d) =
(2.92)
lak
where
fl(y) = E [ail,ai2,...,ai,
1.
it
We may drop the summation condition l >- k if we define the binomial
coefficient to be 0 when l < k: also fl(d) = 0 when I > IdI. Since s and 9 are disjoint and coprime we have
fl(sl U,4) = E fr(d)fso)
(2.93)
r+s=l
By (2.92), the right-hand side of (2.91) is
EE
E(-1)r+s-g-h
r
g+h=k
(r) ()fr(d)fs()
s
and since
(r gh=k \g/
\h/ -
k
s/
we may perform the` outer summation to obtain
L J 1:(-1)r+s-k ( r r
s
S
k )
We employ (2.92) and (2.93) to see that this is ik(d U -4). This completes the proof.
Notice that whenever d and 4 are disjoint the left-hand side of (2.93) is at least as great as the right-hand side, but this inequality seems difficult to use. Similarly we do not obtain an inequality from the Chinese Remainder Theorem in the first proof: recall that if (A, B) = d > 1 then the congruences n = u (mod A), n - v (mod B) are consistent, and define a unique congruence class (mod [A, BI), if and only if dl(u - v). In the situation which arises in Proof (i) we should not in general know whether this condition held. The next result is our first inequality of Behrend type. It generalizes the classical inequality but falls short of our expectations in one respect.
2.3 Generalized Behrend inequalities
85
Proposition 2.16 For all ad,-4,g and h we have tg+h(s"" U -4) > tg(s1)th(R)
Proof We may assume, by the limiting argument described previously, that , and -4 are finite. We shall employ Lemma 0.18, and we define, for any d, finite or not, and g >_ 0,
if i(n, d)
80(g)(n) (n) =
0
g,
(2.94)
else.
This function is multiplicatively non-increasing, that is nln' B(g)(n) z 0(19)(n'). Let A = l.c.m.[a : a E d] and B be defined similarly. Then as in (0.107) we have
tg(d) = E(0(9); S/ M) =
ME
W
dIM
M/
(0(g)(d)
(2.95)
for any multiple M of A. This is because i(n, sad) = z((n, M), d) and there are co(M/d) congruence classes (mod M) in which (n, M) = d. We put M = [A, B], and apply Lemma 0.18 to obtain
M)
E (B(g)8(h); M) Z E (0); M) E
(2.96)
.
Since i(n, sad U -+) < i(n, sad) + i(n, -4) we have
e.u-,(n) >
(2.97)
and our result is an assembly of (2.95) applied to sad,. and 4U. and (2.97). This completes the proof.
,
(2.96)
Proposition 2.16 includes Behrend's inequality (Theorem 0.12) as a special case g = h = 0. However the reader will have noticed that we have not mentioned the cases of equality. Of course there are such cases, a trivial example being given by g >_ IdI, h ? I-4I, but they are necessarily a little artificial, because (2.97) is not generally an equation except when
g=h=0. Let us return to the case when d and . are non-trivial and coprime. By Proposition 2.15 we have
tg+h(d U 9) =
tiooi(°a) i+jSg+h g+h
tg(d)th(I) +
ti(d)tg+h-i0) i=g+1
g+h
+
tg+h-j(d)tj(4) j=h+1
86
2 Derived sequences and densities
The last two sums are an excess, moreover the t-factors are non-zero, except in the trivial case when sad or -4 is Behrend. However, it is
possible to arrange that t;(d) = 0 in a non-trivial, albeit somewhat artificial, fashion. We explore this phenomenon briefly here, to warn the reader that such odd cases can arise. Let pl, p2, ... , p,,, be distinct primes and, for fixed l < m, let
sat={pi,p,Z... pt, :1
i(n,sl) _ (l), with the usual convention if k < 1. Thus r(n, d) has omitted values. A simple example is given by sad = {6,10,15} when r(n, d) * 2. If p, q, r, p', q', r' are distinct primes and .4 = {pq, qr, rp}, .q = {p 'q', q'r', r'p'} then indeed
t2(d U -4) = ti (d)ti (A with none of the three densities equal to 0 or 1. We should like to have a generalization of Behrend's inequality which maintains the property that there is equality when d and 4 are coprime. A possible strategy is to consider identities like (2.91), and see whether they generalize to inequalities. If we intend to employ Lemma 0.18, which involves multiplicatively non-increasing functions, it would seem that we must begin with an identity clear of the densities tk, (k > 0). We obtain an identity of this type from Proposition 2.15 after two summations which yield: for sad and .1d non-trivial and coprime, k
k
th(sI U R) _ E th(d)tk-h(am). h=0
(2.98)
h=0
Examination of Theorem 0.12 suggests that even in the finite case coprimality is unnecessary for this to hold: for k = 0 the relevant extra condition is that sad and 9 should be primitive. An appropriate generalization is as follows. Definition 2.17 The sequence ,z/ is said to be k-primitive if sad has at least < ik the least common k + 1 elements, and for any indices io < ii < i2 < multiple [a;0, a;,, ... , ask] is not divisible by any other element of W (if such exists).
2.3 Generalized Behrend inequalities
87
Thus a 0-primitive sequence is primitive. We note that any sequence of exactly k + 1 terms is k-primitive. If IQ/1 >- k + 2, the condition a;%[a;O,a1,,...,a;k], i ih for any h
aik]
>
(2.99)
which leads to a useful description of k-primality in terms of total decomposition sets in the case when sad is finite. If {d(S)} denotes the total decomposition set of sd then .4 is k-primitive if and only if for any k + 2 distinct indices i, io, it ... ik we can find S c { 1 , 2, ... , IdI } such that i E S, ih S, 0 < h < k, and d(S) > 1. We leave the proof to the reader. Theorem 2.18 For all W and A we have k
k
E th(d U -4) > E th(d)tk-h(°4) h=0
(2.100)
h=0
with equality if d and R are non-trivial and coprime. If .4 and -4 are finite sequences, and if there exists an h, 0 < h <_ k, such that d is h-primitive and 4 is (k - h)-primitive, then it is necessary for equality in (2.100) that d and . should be coprime.
Proof We begin with the inequality: as usual we may assume d and -4 finite, moreover we let A = l.c.m.[a a E v], B be defined similarly, :
and M = [A, B]. We let 8(g), O be as in (2.94) so that in the notation of Lemma 0.18, the right-hand side of (2.100) equals k
E(B
M)E(0(1-h)
;
; M).
(2.101)
h=0
Put
0 ur(n) = (k + 1 - i(n, d U 9))+. For all n, we have i(n,.d U -4):5 i(n, d) + r(n, -4) whence k
B (n)O(k-h) (n). h=0
We also have k
th(d U -4) = E 7O h=o
u M)
(2.102)
88
2 Derived sequences and densities
and in view of (2.102) this is k
z
E
(e.ae9-");M)
h=0
We apply Lemma 0.18 to deduce that each term in this sum is at least as great as the corresponding term in the sum (2.101), and (2.100) follows. This proves the first part of the theorem. Let the hypotheses of the second part hold and in addition suppose
that .4 and JJ are not coprime. We have to show that there is strict inequality in (2.100). It will be sufficient to show that for some h, we have
E (o B(k-h); M) > E (o ; M) E (B(k-h0; M)
(2.103)
and we consider the h specified in the statement of the theorem. We claim, in the terminology of Lemma 0.18, that B(h) and 8('-h) do not split M, whence (2.103). By hypothesis A and B have a common prime factor p. Let p" I I A, p1 B. Let a(0),a('),...,a(h) be elements of 4 such that p" lat0>, and put d = [a(0), a('),..., a(h)]. Then i(d, d) >- h + 1. On the other hand r ((d, Ap "), d) < h, for a(0)%(d, Ap ") and no other element of divides d. Thus O 9(d) < 0(h) ((d,Ap ')). By a similar argument we can find e such that 0(k-h) (e) < O' ((e, Bp #)), moreover d and e are divisors of M. If M = FG where (F, G) = 1, either p%F or p%G, and either 0(h) (d) < 9(h) ((d, F)) or 0(1-h) (e) < 8(k -h) ((e, G)). Therefore the functions h B1) and 8< do not split M, whereby our claim is substantiated. This completes the proof.
2.4 Multilinear functions
When , is finite we can write down explicit formulae for the densities tk(d) by means of the inclusion-exclusion principle; for example (2.8) is the familiar expression for t(,c/). These formulae are almost never applied
because of two drawbacks: the alternating signs and the occurrence of the l.c.m.'s in the denominators. Partial sums of (2.8) give, alternately, upper and lower bounds for t(d), by similar considerations to those employed in the early stages of Brun's sieve, and of the two snags the l.c.m.'s are usually the more crippling. We know from Chapter 0 that any finite sequence sd = {al, a2i ... , an) of positive integers possesses a total decomposition set {d(S)} of 2n-1 pos-
itive integers d(S), indexed by the non-empty subsets S s {1,2,3,...,n}.
2.4 Multilinear functions
89
The h.c.f. and l.c.m. of any sub-sequence of W have a representation as a product of certain d(S): we do not need to write these representations down here since all we require at this point, as a glance at (0.105), (0.106) will confirm, is that each d(S) which occurs in such a product does so linearly, that is the S are distinct. The d(S) may not be distinct but this is irrelevant to the present discussion in which we regard the numbers d(S) as variables. Let f (XI, x2, ..., xm) be a polynomial and suppose
a2f ax,
0
1
We call such a polynomial multilinear, and we see from the foregoing observations that t(d) and t(k)(d) are multilinear functions of the 2" - 1 variables x(S) = d(S)-1. For example, if W = {al, a2} and the total decomposition set is d({1}), d({2}), d({1,2}) so that a,
= d({1})d({1,2}),
a2 = d({2})d({1,2}),
(a1, a2) = [al,a2]
d(11,21),
= d({1})d({2})d({1,2})
then
t(d)= 1-x({1})x({1,2})-x({2})x({1,2})+x({1})x({2})x({1,2}). (2.104)
We now have an expression which we can think about analytically, for example it may be differentiated; and maybe we can establish inequalities in this way. Now (2.104) suffers from appalling notation, it still contains terms of opposite sign, and we now have 2" - 1 variables instead of n. In fact this last remark is the tip of an iceberg because we shall have to consider not just the subsets S 9 11, 2,..., n} but all the possible families of such subsets. We overcome the first difficulty by writing d({1,2}) = d12, x({1,2,}) = x12 etc.,
and it is the object of this section to overcome the second difficulty of alternate signs. If we consider (2.104) again we can make two observations. We know
that the variables x(S) on the right satisfy 0 < x(S) < 1, (it will be convenient to allow 0 < x(S) < 1 and to regard x(S) as a continuous variable), and we also know that t(d) >- 0. (For number theoretic
2 Derived sequences and densities
90
reasons, of course t(d) > 0 if c is finite and the aj exceed 1: we allow our functions to be zero in this section, moreover a sequence sd containing l's and repetitions still possesses a total decomposition set.) When a function is non-negative it is not unreasonable to ask it to be written in such a way that this property is obvious. Thus if 0 < x, y < 1 we can write
1-x-y+xy=(1-x)(1-y).
(2.105)
A similar example, with 0 < x, y, z < 1, is
2-x-y-z+xyz = 2(1-x)(1-y)(1-z)+(1-x)(1-y)z + (1 - x)(1 - z)y + (1 - y)(1 - z)x. (2.106)
In this case it is not so obvious that the left-hand side is non-negative. The multilinear functions (we restrict our attention to polynomials) of m variables may be viewed as a vector space of dimension 2m, with basis xi' x2 ...
Ei = 0 or 1,
1 < i < m.
We are concerned with real functions and so the field of scalars is R. Guided by examples (2.105) and (2.106), we seek another basis with the property that any non-negative function is a linear combination of the base functions, with non-negative coefficients. This is m
fl(1 - xi)1-'ixi', Ei = 0 or 1,
1 < i < m.
(2.107)
i=1
We have to show that this is a basis and that it has the positivity property required. Clearly there are 2m polynomials (2.107); moreover they are independent, since if we set xi = Ei, 1 < i < m, all the polynomials are zero with one exception, equal to 1. Hence (2.107) is a basis. Furthermore, the coefficient A(E1, E2, ... , E,,,) of each base polynomial in (2.107) may be
determined by setting xi = Ei, 1 < i < m, that is A(E1,82,...,Cm) = f(E1,E2,...,Em)
(2.108)
where f is the multilinear function to be expanded. We have used the fact that such a function is non-negative throughout the m-dimensional cube [0, 1]m if and only if it is non-negative at the corners of the cube. The reader will have noticed that the right-hand side of (2.106) may be simplified, for example to (1 - 01 - y) + 0 - 01 - z) + (1 - x)(1 - z)y
2.5 Formulae for the densities tk(d)
91
which has the same `obviously positive' property. Such an expansion is not unique, indeed there are clearly 3m products in which each variable contributes one of the factors 1, x, or 1- x, and we make no use of such expansions here, tempting as such reductions may occasionally be. We refer to the polynomials (2.107) as the elementary multilinear functions (of m variables) and we have proved the following result. Theorem 2.19 Every multilinear function of the m variables x1, X2.... , x,,, has a unique expansion as a linear combination of the 2m elementary multilinear funtions (2.107), moreover (2.108) is a formula for the coefficients.
The coefficients are all non-negative if and only if the function is nonnegative throughout the cube [0,1]"'.
2.5 Formulae for the densities tk(d) In this section we develop the ideas introduced in §2.4. We require that d be finite, but it need not be monotonic and we allow repetitions and and {d(S)} denote 1's amongst the elements. Let sd =
the total decomposition set of d; thus S is any non-empty subset of So := { 1, 2,..., n} and the cardinality of {d(S)} is m = 2" -1. We introduce m real variables x(S), 0< x(S) < 1 and write down multilinear functions of these variables taking the values t(k)(d) when
x(S) = d(S)-1 for all S c So.
(2.109)
Furthermore, we expand these functions as linear combinations of the 2"' elementary functions defined by (2.107).
It may be helpful to work out a simple example in ad hoc fashion to begin with, so that we can visualize the shape of the formulae to which
we are heading. Let d = {al, a2} and the total decomposition set be {dl, d2, d12} - we streamline our notation when convenient by indexing with suffices instead of sets - where d12 = (al,a2), and a1 = d1d12, a2 = d2d12. Now consider the polynomial to = 1 - x1x12 - x2x12 + xlx2xl2.
When we make the substitution (2.99) we obtain to = t(sd), and of course to is multilinear. In this example m = 3, the suffix 12 corresponding to S = {1,2} replacing 3. There are eight elementary functions (2.107): if we denote these by f (x, E) where x = (xl, x2, x12), E = (81, 82, 812) then by
2 Derived sequences and densities
92
Theorem 2.19 we can write to =
2(E)f (x, E)
where, by (2.108), 2O = to(81, E2, E12)
Evaluating these eight coefficients we find that to
=
(1 - xi)(1 - x2)(1 - xiz) + (1 - x1)(1 - x2)X12
+ (1 - xi)(1 - x12)x2 + (1 - x2)(1 - x12)xi
+ (1 - x12)xlx2,
whence
t(d) =
(i_k) (i_k) (i_L)+(i_) (i_)_ d12
+ 1- I
1
1
1
+ I 1 - dlz)
d12
1
d2
+ 1- d2 1
I
1
1
dlz
dl (2.110)
dldz
Notice that the coefficients equal either 0 or 1 and that there are five out of eight equal to 1. It would be foolhardy to extrapolate too far from a simple example, nevertheless the reader might also observe a characteristic of these five terms not shared by the missing three: in each term both integers 1 and 2 appear somewhere within suffices of variables inside brackets. We require some notation and terminology. We put So = { 1, 2, ... , n} throughout, R, S, T etc. are subsets of So. Curly letters, F, W etc. denote families of such subsets and we define
Span. = U{S : S E F}. We say that F is complete if Span. deficiency of F as
(2.111)
= So; more generally we define the
6(F) = n- ISpanF1.
(2.112)
Cardinalities of sets or families are written ISO or I. 1, thus J. l is the number of sets in F. We write 1
if F is complete,
0
else.
(2.113)
2.5 Formulae for the densities tk(d)
93
It is understood, unless stated otherwise, that subsets of So are nonempty. Families may be empty, so there are m = 2" - 1 subsets and 2m families. Summation over sets or families with no further instructions are over these ranges. A typical elementary multilinear function as in (2.107) therefore has the form
11 (1 - x(S)) rl x(R). SE.y
(2.114)
RAF
We can now state our first theorem in this section. Theorem 2.20 Let , be finite, with total decomposition set {d(S)j. Then t(om)
= 1:
(1
S
d(S)) RI-f d(R).
(2.115)
Proof We follow the steps which led to (2.110). For T c So = {1,2,3,...,IdI} we have, by (0.106) l.c.m.[ai :
i E T] = fl{d(S) : S n T* 0}.
(2.116)
In view of the expression (2.8) for t(d), we consider the polynomial
fJ{x(S) : S n T
to =
0}
(2.117)
T
where the sum includes T = 0; the product is 1. Let ),(F) denote the coefficient of the elementary function (2.114) in the expression afforded by Theorem 2.12: we have to show that .1(F) = °l'(F ). By (2.108) we obtain 2(S) by evaluating to at the point x(S)
0
if S E
1
else.
(2.118)
whence by (2.117),
A(te)=E{(-1)ITi : S EF =SnT=O},
(2.119)
T
since the product on the right of (2.117) is zero unless x(S) = 1, S
i.e.
F, whenever S intersects T. We can rewrite (2.119) in the form {(_l)ITI : T c So \ SpanF},
A(te) =
(2.120)
T
where \ denotes set subtraction, so that So \ Q is the complement of Q.
2 Derived sequences and densities
94
If F is complete, the complement of SpanF is empty and 2(F) = 1. If . has deficiency k, the sum on the left of (2.120) is k
E
(h)(-1)h = 0
h=O
whence 2(F) = X(F) as required. This completes the proof of Theorem 2.20. Next we extend this result to a formula for the density tk(S). Theorem 2.21 Let d be finite, with total decomposition set {d(S)}. Then we have
k(cfl 11 11 R.
tkW)
SE.F
(2.121)
d(R)
where etk is the characteristic function of the families F with deficiency
6(F) < k. The deficiency of a family F of subsets S { 1, 2, ... , lc/ I } was defined by (2.111) and (2.112). By (2.112), 6(F) < 1d with equality if and only
if F is empty (recall that subsets S are not allowed to be empty but families of subsets are so allowed). This corresponds to the fact that tk(d) = 1 whenever k > IdI. If k = s1l - 1, only the empty family is excluded from the sum in (2.121), and we obtain 1
dR) = 1 - l.c.m.[a :
1-
aE
R
by (0.106).
Notice that to obtain ik(d) instead of tk(d) in (2.121) we need only replace IfTk(F) by Gt"k(F), the characteristic function of the families with deficiency k. Notice that such families exist for all k < 1.-Q/I, even though
we may have ik(d) = 0 (see the discussion following (2.97) in which non-trivial examples are constructed). An exercise for the reader is to see how the new formula for ik(d) can be correct in such cases. Proof of the theorem It is convenient to work with ik(d) rather than tk(d), and we begin with the formula for this function provided by the inclusion-exclusion principle, which is
[ail ai2,... lair*
)1-k (k) !>k
ti
l <12
1
(2.122)
2.5 Formulae for the densities tk(sI)
95
We recall formula (0.106) which states that
[a;: iET]=fl{d(S): S n T
O}
and we put x(S) = d(S)-1 so that (2.122) becomes
tk(d) = ECk (IT 1) ff{x(S) : S n T * 0}.
(2.123)
T
where k
Ck(l)
(1)
with the usual convention that ck(l) = 0 if 1 < k. Let us think of the x(S) as independent variables, 0 < x(S) < 1, and consider the polynomial on the right-hand side of (2.123), denoting this by ik. We may expand ik as a series of elementary multilinear functions, viz
tk = E 2(F) rj (1 - x(S)) [J x(R) SE.°r
(2.124)
RO.F
and we claim that 2,(F) = e'k(F ). We evaluate 2(F) by choosing {x(S)} as in (2.118) and we obtain, similarly to (2.120),
2(F) = >{ck (I T 1) : T c So \ SpanF}.
(2.125)
Let the deficiency 6(F) < k. Then T s So \ SpanF implies I T I < k and Ck (ITI) = 0. Now let 6(F) = r > k. The sum on the right of (2.125) is
I
Ck(l) (r)
!=k
(-1)1-k (i) (r) (;)(-1)1-k (1-k ) 1
0
ifr=k ifr>k.
Therefore 2(F) _ 'k(F) and by (2.124),
tk = E 'k (F) [f (1 - x(S)) 11 x(R). JV
SE.°
When x(S) = d(S)-' we have ik = tk(d), and this is equivalent to (2.121).
3
Oscillation
3.1 Introduction
Let mil = h(d) be a set of multiples and
M(x)=card{m: m<x,mE./!il}
(3.1)
be the counting function of A. We are interested in the oscillations of the function M(x), in other words in irregularities in the distribution of .t, specifically in showing that usually these are not too small. We distinguish two cases according as 4, or rather Y(sd) if Q is not assumed to be primitive, is finite or not. In the finite case Wl (sl) is a finite union of arithmetic progressions and is of course Besicovitch, moreover the discrepancy
E(x, d) := M(x) - dM(d)x
(3.2)
is a periodic function of x. We are interested in the extreme, or suitable mean, values of E(x, d) over its period, and we seek lower bounds or what are often called S2-theorems. In the infinite case 4 may not be Besicovitch and JE(x, sd)I >> x for arbitrarily large x. We might instead consider the logarithmic discrepancy, that is I
: m< x, M E
x
(3.3)
J))111
but this idea is not taken up in the present work. Instead we consider, in §3.5 below, the problem of what we call perfect sequences, for which M(x) = 8./#(d)x + 0(1). This highly restrictive looking condition leads to all sorts of ramifications. 96
3.1 Introduction
97
We assume until then that Q is finite and we write d = {al, a2, -,an }. We do not demand initially that 4 should be primitive; indeed we allow repetitions and 1's. We recall that 4 is non-trivial if there are no 1's. We denote the (minimal) period of E(x, sad) by X0 so that always (3.4)
XoI
Note that X0 = [al, a2, ... , if (but not only if) W is primitive. It is convenient to make a slight formal change. Let M(x) = [x] - M(x) denote the counting function of that is the set of integers which
remain after Z+ is sifted by d, and put
E(x, d) = M(x) - t(d)x.
(3.5)
We work with E(x, d) rather than E(x, d) because some of our various formulae are cleaner: the change is unimportant because
E(x, d) + E(x, d) = [x] - x.
(3.6)
We note, en passant, that E(x, d) is an odd function in the sense that
E(Xo - x, d) = -E(x, S/).
(3.7)
The restriction to finite d is a serious one and so we insist on the most general results which we can achieve within this framework. The obvious measure of the oscillations of M(x) is
maxIE(x,d)l =maxE(x,d)
(3.8)
but this is not very easy to handle. We define xo
(.d, sl) = 12X6-' I E(x, d)2dx
(3.9)
and we begin with the following simple result. Theorem 3.1 Provided d is non-trivial, we have (3.10)
Proof Since d is non-trivial the complement .% (d) of .#(d) is nonempty: we denote the elements of 9-(,4) not exceeding Xo by b1, b2, ... , bN. Then N = t(d)Xo and we have, splitting the range of integration at
3 Oscillation
98
the points bj, that
r
xo
E(x, d)2dx
= (b2 - bi) + 4(b3 - b2) + 9(b4 - b3) + ... + N2(Xo - bN)
-t(d){(b2
- b2) + 2(b2 - b2) +
+ N(XX - b2 )} +
N
= 1Xo +
t(4)-' E (t(sl)bj - J + 2) I
3 t(S1)2X03
2, (3.11)
whence sl, s l
(3.12)
12N-
I
This completes the proof. In general formula (3.12) is not very helpful because we have too little
information about the bj. Our next task is to write down a (somewhat cumbersome) expression for (d, d) in terms of the elements a, of d. By the inclusion-exclusion principle we have
M(x)=[x]-[x]+>[ x
-
(3.13)
«j
<
and we define, for real x, (3.14)
P(x) = [x] - x + 2.
We then have
E(x,d)=P(x)-1: P (x) + E p ( at
i<j
x [ai, aj] )
-...;
(3.15)
of course the 2's all cancel. We require a formula of J.Franel (1924), Kluyver (1903) (see Landau, Vorlesungen uber Zahlentheorie Satz 484).
Lemma 3.2 Let a, b be positive integers. Then i
1 (a,b)
p(ax)p(bx)dx =
J0
12 [a, b]
equivalently if X be any multiple of [a, b], x x
X-iJo
_
((x
p\a)p\b)dx
1 a,b 12[ab]
(3.16)
(3.17)
3.1 Introduction
99
We obtain (3.17), with X = ab, from (3.16) by the substitution x - x/ab, and then we note that the integrand in (3.17) has period [a, b]. One way to prove (3.16) is by means of the Fourier series e(8) = P(x) - E Zrx) Sri
e21tiO
(3 . 18)
r*0
but Landau avoids this. There has been some interest in these integrals recently, see Greaves, Hall, Huxley and Wilson (1993), Wilson (1994). Proposition 3.3 We have
(SI,d) E
1/
b
X([ai',az,...,an],[ai1,az2,...,an"]) 61, 62'. e2' al,a2...,an],[al a2..>anS]] [[E1
En
(3.19)
* denotes that summation
where every Ei, bi takes the value 0 or 1 and is over all 2n choices.
This is an immediate consequence of (3.15), the definition of (d, sad)
and (3.17). The next result breaks our rule about generality but is a useful precursor of developments to follow.
Corollary 3.4 When the elements of d are pairwise coprime, we have
(d, sad) = 2nt(d).
(3.20)
This formula is implicit in Perelli and Zannier (1989) and Erdos (1946).
Proof When the ai are pairwise coprime the interior l.c.m.'s on the right of (3.19) are simply products. The ratio h.c.f./l.c.m. may then be viewed as a symmetric quotient, that is the analogue for products of the symmetric difference of sets. The factor ai appears in both numerator and denominator if Ei = bi = 1, and so cancels; so it survives as a factor in the denominator if and only if Ei # b . We put Ai = IEi -'5j for each i and we have Di = 0 or 1: in either case there are two choices of the pair lei, bi} giving the same value of A,. Since Ai - Ei + Si (mod 2) we deduce from (3.19) that *
2-
0
o' = 2nt(d).
Al Dg . al a2 " an
(3.21)
3 Oscillation
100
The notation (,) arose from the idea of an inner product. If a and -4 are finite the function E(x, d)E(x, ,V)
(3.22)
is periodic with period say X. We define x 12X-'
I E(x, d)E(x, _4)dx.
(3.23)
This has not been developed very far. We leave the proof of the following proposition, which indicates some of the possibilities, to the reader.
Proposition 3.5 Let d = {al, a2,..., a,n}, -4 = {bl, b2,..., (a,, bj) = 1 for all i and j. Then we have
be such that
t(d)t(R) = (d, d)t(°..A) (d U -4, d U m) = (d, d) (., R)
(3.24) (3.25) (3.26)
We want to make progress without such coprimality conditions. The type of sequences a which we have in mind are sail
= {a+l,a+2,a+3,...,2a},
-42
= {PlP2...Ph : Pl < P2 < ... < Ph <- Y}.
(3.27)
Our results yield in these particular cases: 2
/1 = 1 - logo
- a#,
(d2, d2)
>-
27r(y)
g2
-h
(1 p
- -)
i
g
(EpSY p
1
P
6) = .28197... ,
(3.28)
gt
g
The second inequality above is non-trivial whenever ic(y) - h is large.
notice that when h = 1, M(x) = 1(x, y) is Buchstab's function. If we write
(i_ -P p
E(x, d2) = (D(x,Y) - x f 1 - - =: lo, y)x f 1
PsY
P
1
then we have, for each fixed u > 1 that lim ri(y", y) =
Y-00
1
where y is Euler's constant and w is defined by a differential-difference equation. The right-hand side oscillates about, and converges to 0 as
3.1 Introduction
101
u -+ c; (cf. Maier (1985), Friedlander and Granville (1989), Hildebrand and Maier (1989). It would be of interest to know whether (3.28) is applicable to this area of research). We employ the machinery involving total decomposition sets introduced in §2.4-5 to cope with the l.c.m.'s appearing in the formula (3.19) for (d, .sad). Let .F be a family of non-empty subsets S c So = { 1, 2,..., n}, where n = Id . For each subfamily f of S (including the empty family) we define
a(.9, F) _
{(-1)15L '(W) : &
Wc
(3.29)
}
where X, as in (2.113), is the characteristic function of the complete families. We also define
b(F) _
{la(.ff,F)I :'sue
(3.30)
{a(8,F)2: SsF}.
(3.31)
The main result of this section is the formula for (d,
which follows.
c(am) _ If
Theorem 3.6 Let d = {al, a2,..., an} and {d(S)} be the total decomposition
set of sl. Then
F c(am)
s1
(1 -
(3.32)
d(S) ) x11 d(R)
with c(,) as in (3.31). This formula has a similar shape to (2.115), for t(,Qt). As an example we write out (3.32) in full in the case , = {al, a2}, as before using the shorthand {d1, d2, d12} for {d(S)}. We have
(Qt, si) = 4 1 - dl
\1
+211-d1 l(1
d2
\1
4121
+2 (1 - d12) d1d2.
-
W12
+ (1 - dl d2+2(1-d2 )11
/
\1
d2
d12
d12) dl (3.33)
Comparing this with (2.110), we observe that the same five elementary multilinear functions appear on the right-hand side: only the coefficients
3 Oscillation
102
have altered. An immediate deduction from (2.110) and (3.33) is that for n = 2,
2t(d) < (d, d) < 4t(d).
(3.34)
Moreover it is not difficult to isolate the cases of equality. Equality on the left is equivalent to d1 = 1 or d2 = 1, that is a1la2 or a2Ia1. This means
that d is not primitive. Equality on the right is equivalent to d12 = 1, that is (a1, a2) = 1, when (3.33) is a special case of Proposition 3.3. It will emerge that the left-hand inequality in (3.34) generalizes to all n. However, it is not true that for all n (sail, a) <
2"t(,,/).
(3.35)
This holds if and only if n < 3; for n > 3 the pairwise coprime case is not extremal. We must take care not to extrapolate too far. Proof of Theorem 3.6 We begin with formula (3.19). Let Q and T (which we allow to be empty) be the subsets of (1,2,..., n} on which, respectively Ei = 1, S1 = 1. We recall from (0.112) that l.c.m.[ai : i E Q] =11{d(R) : R n Q * 0}
(3.36)
whence we have i E Q], l.c.m.[aj
\(-1)IQI+ITj(l.c.m.[ai
(mil, d)
Q
E Q
T
[l.c.m.[ai
:
j E T])
i E Q],l.c.m.[aj : j E T]]
T
(fJ{d(R) : R n Q " [fl{d(R) : R n Q
0}, 11 {d(S) : S n T* 0})
0}, fj{d(S) : S n T# 0}]
(3.37)
Let P be a subset of So = 11, 2, ... , n} which intersects both Q and T. Then d(P) is a factor of both the numerator and denominator on the right of (3.37) and so cancels. On the other hand let R and S be subsets such that
RnQ*O, RnT=O, SnT*O, SnQ=O.
(3.38)
Then we cannot have R S or S s R. For example if R s S then R n Q * 0 implies S n Q * 0, contradicting (3.38). We deduce from Theorem 0.21 that for R, S satisfying (3.38) we have d(R) and d(S) relatively prime.
Now consider the quotient on the right of (3.37). Once the factors
3.1 Introduction
103
d(P) described above have been cancelled, the h.c.f. remaining in the numerator is equal to 1. We define
0(Q, T) = {W : W intersects one and only one of Q and T}
(3.39)
and we deduce that
(.4 d) =
1:(_1)IQI+ITI Q
JJ{d(W)-1
: W E 0(Q, T)}.
(3.40)
T
Thus (d, d) is a multilinear function of the variables d(S)-1. We consider the corresponding polynomial
J(_1)IQI+ITI fl{x(W) : W E V(Q, T)}
f= Q
(3.41)
T
and by Theorem 2.12, f has an expansion as a linear combination of elementary functions. Let 2(F) be the coefficient of
(1 - x(S)) 11 x(R) SE3
(3.42)
R¢.F
so that we have to show that 2(.F) = c(F). To obtain 2(F) we evaluate f at the point x(S) = 0, (S E F ), = 1, (S 0 F ).
(3.43)
Hence
2(F) = E T{(_1)IQI+ITI : F n V(Q, T) Q
=E Q
= O}
T E{(_1)IQI+ITI
: 0(Q, T) 2 F}
(3.44)
T
where
0(Q, T) = {W : W intersects both or neither of Q and T}
(3.45)
is the complementary family to V(Q, T). We have F c 0(Q, T) if and only if F possesses a subfamily 6 such that Q, T E -*'(&, F), where °(f, F) denotes the family of sets H which intersect no W E and every W E F \.ff. Therefore 2
({(_l)IQI. Q E JY(&, F)})
(3.46)
Q
and, by (3.31), our proof will be complete if we can show that the sum over Q in (3.46) equals +a(&, F ). We employ the inclusion-exclusion principle to evaluate this sum. For any family of subsets of So let
3 Oscillation
104
A'(9) comprise the sets H which do not intersect any member of !21, that is H is a subset of the complement of Spang, defined in (2.111). As in (2.114) and (2.115) we have
E{(-1)lHI : H E i''(W)}
(3.47)
0{(-')IQ' H
The inner sum in (3.46) is
E{(_1)IQi : Q E Q
:QE
(W)}
Q
=
(3.48)
by (3.47) and (3.29); and (3.46), (3.48) give
2(F) = E{a(off, F)2 : f s F } = c(F)
(3.49)
as required. Hence the polynomial in (3.41) is
f = E c(F) 11 (1 - x(S)) fl x(R) SEF
(3.50)
R F
and we substitute x(S) = d(S)-1, when f = (d, sad) by (3.40). This proves Theorem 3.6.
In the next section we investigate the coefficients c(.fl. We conclude
this section by stating an analogue of Theorem 3.6 for the derived sequences 'Cl(k) defined in Chapter 2. We recall from (2.112) the definition
of the deficiency S(F) of a family .F. As in Theorem 2.21 we put
.'k(F)
1
if S(F) < k,
0
else.
(3.51)
Now we generalize (3.29)-(3.31) by defining
ak(fo°, F) =
(3.52)
bk(") =
(3.53)
ck(°F) =
E{ak(t',F)2
9
:
fff c F}.
(3.54)
ii
Theorem 3.7 Let d = {al, a2,-,an I and {d(S)} be the total decomposition set of d. Let Sl(k) be the derived sequence defined by (2.5). Then
1-
(d(k), d(k)) = E ck(F) SE-F
d(S)
ROJF
d(R)
.
(3.55)
3.2 A first lower bound for (sad, sad)
105
The proof is similar to that of Theorem 3.6 and is left to the reader. Notice that we could apply Theorem 3.6 to Sl(k) but would by so doing ignore the extra structure of a derived sequence.
3.2 A first lower bound for (d, sad)
We require information about the coefficients c(F) and ck(F) which appear in the formulae (3.32) and (3.55) for (d, sad) and < d(k), d(k) >. When Isadl = n we require k < n in the definition (2.5) of sa&>.
In this section we treat the two cases together and it is convenient to 'o, co etc. instead of X, c in some of our formulae.
write
Lemma 3.8 Let 0 < k < n. Then
'k(F) and ck(F) are zero or non-zero
together; moreover we have
ck(F) > bk(F) > 2°k(F) ).
(3.56)
This result should be compared with the written out expansions (2.104)
and (3.33) of t(,/) and (d, sad) when n = 2, and the remarks following (3.33).
The inequality on the left of (3.56) follows from the definitions (3.53), (3.54) and the fact that ak( ', F) is an integer. Let .'k(F) = 0. By definition (3.51), the deficiency 6(.F) > k whence
'k(g) = 0 throughout (3.52). Hence ak(ff,.F) = 0 for every f g F and ck(F) = 0. Let Z'k(F) = 1. Then ak(.F,F) = ±1 and bk(F) >_ 1. This proves the first part of our assertion. Next we have for all F,
E(-1)Iflak(S,F) _
Xk(g)E(-lPI+I'fI = 0
(3.57)
'f gw
9-91"
because the inner sum is zero unless W is empty, when .'k(g) = 0. From (3.53), (3.54) and (3.57), bk(F) and ck(.F) are even integers, and (3.56) follows. This completes the proof.
So far we have made no use of bk(.F) and it may be that it has no arithmetical significance. It was introduced because of the following result which is vital.
Lemma 3.9 Let F g F'. Then bk(F) < bk(F').
(3.58)
The corresponding inequality for ck(F) is false. For example let
3 Oscillation
106
n = 3, k = 0 and .
b(F) =
= {(12),(13),(23)}, F' = {(1),(12),(13),(23)}. Then
c(F) = 6, c(,) = 8.
Proof of the lemma We may assume F' has just one extra element, the set T c S0. Let us write d' = 61 U { T} for every F. Then by (3.52)
akK " ')
{(-l) 1+1e'kW) : S s W
F}
(3.59)
<JI
whence
ak(ff, F) -
ak(9, F),
(3.60)
Iak(S',. ')1.
(3.61)
and
Iak(9,. )I <-
The result follows from (3.53) and (3.61). This completes the proof. We look for a class of families for which ck(F) is large. Definition 3.10 We say that
.
is k-minimal if the deficiency S(S 1) < k,
but () > kfor every c F 1. Every family such that 8(g) < k has at least one k-minimal subfamily F1 (which may have deficiency < k). We sometimes refer to 0-minimal families as minimally complete.
Definition 3.11 Let 8(F) < k. We define x(F,k) to be the maximum of the cardinalities of all the k-minimal subfamilies 1 of F. Lemma 3.12 Let .tk(F) = 1. Then bk(F) > 2x(°k)
(3.62)
Proof By hypothesis 8(F) < k and so F has a k-minimal subfamily F1 of cardinality K(F,k). For every
F1, (3.52) implies that
akKF1)
(3.63)
whence by (3.53), (3.54), bk (F 1) = ck (F 1) = 21Fi I =
The result follows from this and Lemma 3.9.
2x(F k)
(3.64)
3.2 A first lower bound for (d, d)
107
We may now identify the families for which ck(F) = 2.
Lemma 3.13 We have ck(F) = 2 if and only if
has the form
J =-9 U{S1,S2,...,Sh}
(3.65)
where 8(9) > k and IS;) >_ n - k, 1 < i < h. In particular when k = 0,
c(.) = 2 if and only if
= -9 U {So} where -9 is incomplete.
Proof Let F have the form (3.65) and write W =
c -F. If
S c F we can
' U _* where
C-9
fin_q C vo n {S1,S2,...,Sh} c
)r 9; {S1,S2,...,Sh}.
(3.66)
If A' is non-empty then X k(-*-) = 1 and so 'k(W) = 1 because S 8(g) < 8( ). If '' is empty then .°'k(W) = 'k(') = 0 because and X k(-9) = 0. Thus 'k(W) = 'k(. *') and
'k(.)
ak(fo,, ) =
,
-9,
(3.67)
with ' and '' as in (3.66). The first sum on the right is zero unless .ff n- _ -9, that is & 2 -9. Now let 6' = _9U 2', where Y s {Sl, 52, ... , Sh }. Then _ -9, 2' c A' c {S1, S2, ... , Sh }. By (3.67) we have
ak(e,-)
(3.68) Jr
If 2' is non-empty then 2'k() = 1 always and the sum on the right of (3.68) is zero unless 2 = {S1, S2, ... , Sh}. In this case g = .F. If 2' is empty, the sum is -1 because 2tk(2) = 0 but Xk(i) = 1 for every
1'
Y. In this case & =.9. So there are two families & c akW, ±1, else it is zero. Thus ck(F) = 2.
for which
Conversely let ck(S) = 2. Then by Lemmas 3.8, 3.12 we have . 'k(.) _ 1, K(F, k) = 1. Every k-minimal subfamily of F is therefore a singleton, say F 1 = {Sj}, and 8(S ) < k requires IS;I >- n - k. We may suppose there are h such subfamilies, and let -9 comprise all the remaining sets in F. We must have 8(') > k else -9 would have a k-minimal subfamily -9 1. Since T E -91 implies I TI < n - k, 6(_9 1) < k requires 1-911 >- 2 and this contradicts the assertion that K(F, k) = 1. This proves the lemma.
Theorem 3.14 Let 0 < k < n. Then for all d,
> 2tk(d)
(3.69)
3 Oscillation
108
with equality if and only if 9(d(k)) is a singleton, that is there is an element of ,Al(k) which divides all the others.
This theorem classifies certain extremal sequences. However the right-
hand side of (3.69) is small and we shall have to look further for an applicable lower bound. Proof The inequality follows directly from Theorems 2.21 and 3.7, and Lemma 3.8, so that we just have to consider the cases of equality. Since
tk(SI) = t(d(k)) we may assume k = 0, substituting 5(k) for d in the other cases. Let ai l aj for some i and every j. Then .1d(d) = aiZ+ and t(d) = 1 - 1/a1, (d, d) = 2(1 - 1/ai). The condition stated is sufficient for equality. Notice it involves ai = (a,,a2,...,an) = d(So). Suppose on the contrary that for every i we have ai > d(So), that is there exists S(i) for which i e S(1) c So, d(S(i))
1.
(3.70)
Put F = {S(1), S(2), ... , S(n)}. Then SpanF = So, that is F is complete and .(F) = 1. By (3.70), K(F, 0) > 2, and c(F) >- 4 by Lemma 3.12. By Theorem 3.6 and Lemma 3.8,
d) - 2t(d) > (c(.) - 2) [J
C1 -
d(S(i)))
(3.71)
d(' Rfl and the right-hand side is positive by (3.70). This completes the proof.
Theorem 3.15 Let d = {a1, a2, ... , an} and (at, a2, ... , an) = 1. Let q = q(d) be the least integer with the property that whenever 1 < it < i2 < < iq < n we have (ai aiz, .... aiq) = 1. For q > 2 and r E Z+ let l(q, r) denote the least integer such that l(q -1) > r. Then for 0 < k < n we have (d(k) d(k)) > 21(q,n-k)tk(1/4
(3.72)
Proof Let BSI >- q. If il,i2,...,iq e S then d(S) = 1. We may therefore restrict the sums over F in Theorems 2.21
and 3.7 to families F containing only sets S for which BSI < q. Let F be such a family and e'k(.) = 1. Any k-minimal subfamily F1 of F has cardinality at least l(q, n - k) whence K(F, k) >- l(q, n - k) and by Lemma 3.12,
ck(F) >
21(q,n-k)
'k(, ).
(3.73)
The result follows.
We remark that (3.72) is stronger, when k >- 1, than the lower bound
3.2 A first lower bound for (d, 4)
109
obtained by substituting d(k) for Q in the result for k = 0. We claim that (n
q(d(k))>
K+1
)
- (n-(,Qt+1) k+1
(3.74)
To see this notice that by the definition of q(d) there are q(d) - 1 elements of d, say a1, a2, ... , aq(d)_1 with common factor d > 1. Now let io < i1 < < ik and [a,o, a,...... aik ] be an element of d(k) not divisible by d. Then io > q(d) whence card{a : a E
,(k), d%a}
(n
k(+d1 + 1).
(3.75)
Since q (d(k)) must exceed the number of elements of d(k) divisible by d we obtain (3.74). It follows that if l' is the least integer such that it
{(k+1)
-
(k+1)
(3.76)
then l' > l(q(d(k)), JA(k) 1). On the other hand it is not difficult to check
that l' < l(q(d), n - k). Theorem 3.15 does not apply in the case (a1, a2, -,an) > 1. We rectify this as follows.
Lemma 3.16 Let F be a family of sets, not containing the set So and let F' _ F U {So}. Then
ck(F') = ck(F) + 2 - 22Ck(F).
(3.77)
Proof Let 60, and &' _ f U {So}. As in Lemma 3.9, (3.60) we have ak(fo°, F') - akW, F') = ak(Iff, F).
(3.78)
Now
ak(0°',) =
_ E{(-1)1 : g' s W s F'} since
(3.79)
'k(W) = 1 throughout: So E W. Hence
F 0
else
=
(3.80)
From (3.78) and (3.80), we infer that
ak(9, F )2 = E {ak(g, F')2 + ak(fo', F')2},
(3.81)
3 Oscillation
110
whence
Ck(F') = ck(F) + ak(F, F')2 + ak(.F',.')2 - ak(F,
.)2.
(3.82)
We find that
IakF)I Xk(.F) = 1,
Iak(F,t) =
1 -.Ik(F).
(3.83)
This implies (3.77) as required.
Theorem 3.17 Let d = {al, a2, ... , 141 =
A > 1. Let
and (al, a2, ... ,
Then
+ 2 (A - 1)(1 - tk(Sl))
(3.84)
In these circumstances we apply Theorem 3.15 to d1.
Proof Let the total decomposition sets of c and c/1 be {d(S)} and {d1(S)} respectively. We have d1(S) = d(S) if S * So, and d1(So) = 1, d(So) = A. Let T,* denote summation over families S not containing So. We have ('91(k),SI(k))
_ E*Ck(J)SJJ
C1 - d1(S)) Rf
(3.85)
dl(R)
because for the families which contain So the product over S is zero. We remove the suffix 1 throughout the right-hand side: since So is an R always this gives
A E*ck(. ) .F
(1 - d(S)) SII
d(1
.
Next we consider (d(' ), d(k)). With the convention that F' = this may be written in the form (j(k)' d(k))
(3.86)
R
'117
U {So},
1
E*Ck(J ) 11 ( 1 10,
SEF
+ E*Ck(F') rl SE.y'
- d(S) ) fi d(R) I
f
(3.87)
In the second term on the right, we replace F' by F in the specification
III
3.2 A first lower bound for (,4, sl)
of each product. This removes a factor 1 - 1/0 from the product over S, and introduces a factor I /A into the product over R. Thus ('d(k)"
d(k)) = E*Ck(.F) rj 1 F
SEF
11 d(S)
d(R)
+(A- 1)E*Ck(F')s[J (1-
d(S)) xf d(R)
(3.88)
Now we subtract (3.86) from (3.88), and employ Lemma 3.16 to obtain
(/(k),
i(k))
_
`Sank))
+2(A - 1) j:* (1 - Xk(.f ))
1-
s1
C
d(R). d(S)
(3.89)
R
'k(F) = 1 for the missing families. This
The * is nugatory because yields (3.84).
We conclude this section by working out the second example in (3.27), namely (3.90)
S/2 = {p1p2...Ph : Pi < P2 < ... < Ph <- Y}.
This is best envisaged as a derived sequence with k = h - 1 and
d={p: p
(3.91)
Then q(d) = 2, and l = n - k = ir(y) - h + 1. Theorem 3.15 gives (SI(k)
1(k))
2n(Y)-h+1tk(.W)
(3.92)
We apply the formula (2.71) which implies that
tk(d) > 11 (1 -
1) E P
p
g
The inner sum on the right is at least g
(1: y - (g - 2)! p
P')
(PiP2 ...
Pg)-i.
(3.93)
Pl
-
l g-2
p
1
2
(3.94)
p
with no subtracted term if g < 2. We have E p-2 < 1 whence, summing over g,
1g tk(Si) > -H 1-i Y-) 1
p
g
(3.95)
p
This is all we need. Note that we have a superior result to Hall (1990a) : at that time there were no inequalities for derived sequences.
3 Oscillation
112
3.3 Upper bounds for (d, d) In this section we consider how large (d, d) can be, and we recall the assertion following (3.34) that for n > 3 the inequality (d, d) < 2nt(d) is false.
Definition 3.18 We define
C(n) = maxc(. ) where the maximum is taken over all families .
(3.96)
of subsets S s 11, 2, ... , n}.
Theorem 3.19 For n< 3 we have C(n) = 2n. For n > 4 we have (21n)
Cn < C(n) <
(3.97)
where the sequence Cn is given by (2m)
C2m
C2m+1
+22m-1
4 (2m)2 + 2 m 2m
+22m-1.
(3.98)
Thus Cn > 2n for n > 4. Stirling's formula gives 4n
C" ~ 4nn'
2n n
ti
4n 7rn
(3.99)
so there is a substantial gap in (3.97). No value of C(n), n >_ 4, is known.
We have 20 < C(4) < 70, and the reader will notice that there are 215 candidate families in (3.96) when n = 4. We have not attempted to carry the work in this section over to derived sequences (k > 1), primarily because the C(n) problem itself has not been solved. We require some preparation for the proof of Theorem 3.19. Definition 3.20 Let f and F be families of subsets of { 1, 2,..., n}. We say
that 41 is Ffaithful if
e = {S : S E F, S
Span&}.
(3.100)
Evidently off s F, moreover there is just one F -faithful family of of given span. Thus there are, altogether 2m F -faithful families, where m = I SpanFI = n - b(F ).
3.3 Upper bounds for (sd, sd)
113
Lemma 3.21 Let a((ff, F) be as in (3.29). Then if a(&, F) * 0 we have
S(F) = 0 and S is Ffaithful. Proof of lemma That F must be complete (8(F) = 0) is familiar from Lemma 3.8. We recall from (3.48) that
E{(-1)JHJ : H
F )}
(3.101)
H
where(,
comprises the sets H which intersect every set belonging
to F \ S and no set belonging to 41. Since a(61, F) * 0 we know that l e(&, F) is not the empty family. Now let us suppose (ff is not F -faithful. Then there exists a set S satisfying both
S E F\ 61, S c Spans. Any H E
(3.102)
F) must intersect S, whence it intersects Spans and
therefore intersects at least one member of s". This is a contradiction and so Af (41, F) is empty. Since this is not the case, is F -faithful as required. This proves the lemma. The idea of F-faithful families can be useful in the evaluation of c(F). For example let F = {S : BSI = k}. We claim that
_ C(F)
= (k
1: n
2
- 1)
n
+
(l).
(3.103)
I=k
Let us prove this. Choose l integers xi such that 1 < x1 < X2 < . < xl < n and suppose e F, Spans = {X1, x2, ... , xi }. Let iff be F -faithful, and consider the case l < k. This is impossible unless l = 0 and 01 is the empty family. (O, . ) comprises the sets H which intersect every member of F, that is I H I > n - k, whence from (3.101), we have a(O, F) =
(-1)n-k+l (k
- 1)
(3.104)
This accounts for the first term on the right of (3.103). Next, consider the case 12! k. The F -faithful family S with Spans as above is 61 = {S : ISI = k, S c {x1,x2,...,xi}},
(3.105)
and(, F) has just one member, the complement of {xl, x2, ... , xt }. In this case a(', F) = ±1 and the second term on the right of (3.103) is the number of choices for Spans. Proof of Theorem 3.19 For n < 3 this is by inspection, although the
case n = 3 is tiresome. We assume from now on that n >- 4 and we
3 Oscillation
114
choose k = ["2I] in (3.103) to obtain the lower bound C(n) > C,,. It remains to prove the upper bound. Let H1, H2, H3 be sets such that H1 c H2 H3 { 1, 2, ... , n} and let H1, H3 E (&, F), for some family F and F -faithful S. Then H2 E Let us split all the subsets of {1,2,. .. , n} into Sperner chains '61, W2,- -, (v. We may write E{(-1)iHi
: H E'(Z, F) n 16µ}
(3.106)
µ=1
and we see from the argument above that (&, F) n W,,, is a subchain of IVµ, that is comprises sets H1, H2, ... , Hr say, with Hi c Hi+1, I Hi I + 1 = IHi+1 I, 1 < i < r. Hence the inner sum in (3.106) has absolute value < 1 and therefore la(S, F )l <
Ep0)
(3.107)
µ=1
where sµ(f) = 1 if
F) intersects Wµ, else aµ(6) = 0. Notice that we
have (3.108)
sµ(S) < 1`,1t
because a given set H belongs to g = 1,2,3.... put
F) for at most one f. For
yg = card{µ : Wµl > g}. Let have
(3.109)
be distinct subfamilies of F. By (3.107)-(3.109) we
h
v
h
E la(&i,. )I <_ E E Eµ(&i) µ=1 i=1
i=1
V
<
min(h, IWµl) µ=1
< h card{µ : `', l >_ h} + E g card{µ : II=g} g
<
hyh + E g(Yg - Yg+1) g
Y1+Y2+ +Yh
(3.110)
3.3 Upper bounds for (d, d)
115
Lemma 3.22 Let {xi}N 1, {yi}N 1 be non-negative, non-increasing sequences
such that
xl xl + x2
< <
X1 + x2 + x3
<
x1+x2+...+xN
<_
Y1
Y1 + Y2 Y1 + Y2 + Y3
Y1+Y2+...+YN.
(3.111)
Then 2 xi+x2+...+xN2
(3.112)
We use a simple variational argument. Let
Proof of lemma
X =sup{x1 + x2 +
+ xN}
(3.113)
subject to the constraints (3.111). X is attained, and we claim that this requires xi = yi (i < N). We suppose the contrary, that is xl + x2 +
+ xN = X, xi * yi for some i.
Let g be the least i for which xi
(3.114)
yi. By (3.111) we must have xg < yg and
x1 +X2+'' +xg < Y1 +Y2+'' +yg. If g > 2 notice that yg < yg_1 = xg_1 whence xg < xg_1.
There exists h, g < h < N such that x1 +x2+ +xh = y1 +Y2+ +Yh : if this were not the case we could increase xg a little, leaving the other xi fixed, without upsetting either the monotonicity of the sequence xi or the constraints (3.111). By (3.114) this would give a sum of squares > X which is impossible. Let k denote the first such h: we notice that this implies xk > Yk. Also, by (3.111) x1 +x2+ +xk+l <- y1 +y2+ +yk+l so that xk+1 <- Yk+1 -< yk, and hence xk+1 < xk. (If k = N we have 0 < xk.) We introduce a small variation. There exists k; > 0 such that if we change xg to Xg + E and xk to xk - E all the required inequalities still hold. The sum on the left of (3.114) becomes X + 2E(xg - xk) + 2E2 > X
(3.115)
and this is a contradiction. Hence xi = yi for all i and X = yl + y2 + + yN. This proves the lemma. To complete the proof of Theorem 3.19 we apply this as follows. Let us label the subfamilies .9 of in such a way that la(.91,
)I >- la(-92, F )I >- la(&3, .)I >- ...
(3.116)
3 Oscillation
116
and put xg = Ja(,ffg,. )j, keeping yg as in (3.109). By (3.110), the hypotheses of Lemma 3.22 hold and we conclude that
c(F) < > yg.
(3.117)
However the sequence {yg} is the symmetric decreasing rearrangement (Hardy, Littlewood and Polya (1934)) of the sequence of binomial coefficients (g), and the right-hand side of (3.117) is therefore n
2
nThi
+ (1)2 + ... + (n) 2 = (2n).
completes the proof.
sup{(d, d) : IdI = n} = C(n)
(3.118)
(3.119)
Is/I = n} = C(n);
(3.120)
in (3.120) d is restricted to be non-trivial. The supremum in (3.119) is unattained. That in (3.120) is attained if and only if n < 3.
Proof The inequality
C(n)t(.)
(3.121)
follows from Theorems 2.20, 3.6, Lemma 3.8 and the definition (3.96) of C(n). Since t(d) < 1 it will be sufficient to show that
sup (d, d) >- C(n)
(3.122)
to evaluate both suprema (3.119)-(3.120), moreover it is a consequence of (3.120) and t(d) < 1 that the supremum in (3.119) is unattained. Choose a family F such that c(.F) = C(n). By Theorem 3.6 C(n)
sE.
(1 - d(S)) II
d(R).
(3.123)
ROJ
Let I. F1 = f and PI, p2, ... pf be distinct, large primes, say z < pl < P2 <
< pf. We assign the values d(S) = pi in any order for S E .F and put d(R) = 1 if R F. By Theorem 0.21, this is the total decomposition set of some sequence Q and by (3.123)
(d, d) > C(n) (1 - z
f (3.124)
3.4 Primitive 4
117
Since z is arbitrary, (3.122) follows. It remains to determine under what circumstances we can have equality in (3.121). Suppose we do, and put
po(d) = IS : d(S) * 1}.
(3.125)
The non-zero terms in the expansions of t(d) and (d, d) provided by Theorems 2.20, 3.6 arise from the complete subfamilies of 3 o(d). If d is non-trivial, that is a; > 1 for all i, then for each i there exists SO) such
that i E SO), d(S(O) * 1. Thus 3o(d) is complete if and only if d is non-trivial.
We must have c(F) = C(n) for every complete F c mo(d). Let be minimally complete so that by (3.64) c(.) = 2I But any 1.
minimally complete family has cardinality < n whence C(n) < 2" and, by Theorem 3.19, n< 3. In the case n = 3, (3.20) shows that the supremum is indeed attained. It would plainly be of interest not only to know the value of C(n) but
to have some idea of the form , must have in order that
- C(n)t(d).
(3.126)
3.4 Primitive d In §3.2 we gave a lower bound for (d(k), d(k)) valid for 0 < k < Is11. When we are concerned with derived sequences directly, or as in the example S12 from (3.27) worked out at the end of §3.2, we find it advantageous to treat a given sequence as a derivative, the condition that sP should be primitive is inappropriate because 'Cl(k) and (g(d))(k) may be unequal whenever k >_ 1. Thus Theorem 3.15 carries no condition of this sort: of course if sl is extravagantly unprimitive or bloated with
repeated elements there must be a price to pay, indeed in these and similar cases q(d) will be large. We recall the example
sY1 = {a+ 1,a+2,...,2a}
(3.127)
from (3.27). This is a primitive sequence with a large q - because at least half the elements are even. For such sequences Theorem 3.15 is weak, and we now present a second lower bound for (d, d) motivated by (3.127). We require d to be primitive and we restrict our attention to the case k = 0. We begin by setting
Fo(sl) = {S :
d(S)
1}
(3.128)
3 Oscillation
118
as in (3.125), recalling from §3.3 that mo(d) is a complete family if and
only if . is non-trivial: this must be the case if d is primitive and Id > 2 as we may assume. For subfamilies off of po(d) we define v(4ff) _
f, (d(T) + 1) TES
(3.129)
d(T) -1
and
E (-1P"L o) rI 1 - d(T)
(3.130)
TES
Theorem 3.24 For every d (,cY,sl) _
v(off )a(61,.Fo;sd)2.
(3.131)
This holds whether c is primitive or not. Notice that a(f',. o; d) is to some extent similar to a(4ff, moo), and the right-hand side of (3.13 1) is similar to c(. o): however a(4ff,.f) and c(. F), defined in (3.29), (3.31), were purely combinatorial in structure. Thus we could (to some extent) compute these coefficients without worrying about the total decompo-
sition set of a particular sequence d. This is no longer the case and Theorem 3.24 is, in this sense, of mixed type. Proof For S E o put
y(S) = (d(S) - 1)-',
z(S) = 1 - d(S)-1
(3.132)
so that y E (0, 1], z E [1, 1) and z(S) = (1 + y(S))-1. By Theorem 3.6 we have
(d, d) = E c(. F) [J (1- d(S)-1) fi d(R)-' 9S-=.F0
(3.133)
SEF
because for the remaining families the product over S vanishes. We can rewrite (3.133) in the form
Z JJ (1 - d(S)-i) fJ d(R)-'
(3.134)
SE.Fo
where
Z = E c(-F) ASS 0
fi
y(T).
(3.135)
(Wi)(a2) E 1
(3.136)
TEFoVF
From (3.29) and (3.31) we have c(am) _ , 99 19299
&9SQF I ()W2
3.4 Primitive 4
119
and we insert this into (3.135), to obtain that
E
z =
(-1)IS'II+021X(Wi)'W(W2)
SJ;-p <J2SWp
y(T).
JJ
x21'N'n"'21
(3.137)
0 TE3rp\eF
The innermost sum on the right of (3.137) is
fi
(1+y(T))
(1 -d(S) i)
TT
i
(1 +y(T)) H z(T) rl z(T) TES'2
(3.138)
and we notice from the definition (3.129) of v(e) that
212! JJ (1 + y(T)) = E v(s). TEI
(3.139)
1 nc"2
We assemble (3.137)-(3.139) to deduce that
Z [J (1 - d(S)-') SE.fip
_
!2
(1 )X("2) fj z(T) fj z(T) E v(s)
I, 2s.fp
E
TE'2
(-1)11
fi z(T)}
(3.140)
.
TEg
The inner sum here is and the left-hand side of (3.139) is (d, ) by (3.134) because the product on the extreme right of (3.134) equals 1. This proves Theorem 3.24. We look for families 6 c FO for which a(6",
o ; d) is large.
Definition 3.25 We say that ff is maximally incomplete with respect to F if S c F, X(.9) = 0 and '(,§) = 1 for every W such that .9 c c F.
The definition assumes ..1(F) = 1 which is in order since we are concerned with F o. Let f be maximally incomplete w.r.t. F0. Then (3.130) gives
(-1)1'1a(
TT z(T)
11 z(T) TES
11 (1 - z(T)) - 1
TEyp\d
.
(3.141)
3 Oscillation
120
Since z(T) >- 1 for every T this implies (3.142)
a(,ff ,.Fo;sl)I >- 2 fi z(T) TEA,
for all maximally incomplete !, whence for such 6 we have v(ol)a(&, moo; ,V)2 > 4
II
C1-
TES
d(T)2 ) .
(3.143)
We combine this with Theorem 3.24 to give
Theorem 3.26 For every non-trivial , we have 7-7
>4
11 As.Fo(.4)
TEA
1-
1
d(T)2
,
(3.144)
where T * denotes summation over families 6 which are maximally incomplete w.r.t. F0.
Theorem 3.27 Let 4 be a primitive sequence of length n > 2. Then
si)>4n
11
TEF0(d)
C1-d(T)2).
(3.145)
Proof Let i * j. Then ai%aj if and only if ai/(ai, aj) > 1, equivalently
3S E Fo(d),
i E S, j 0 S.
(3.146)
F(j)={S : SE F, j0S}.
(3.147)
For any family F, let us write
Then by (3.146) and (3.147) we have
SpanFoj)(d) _ {i :
ai%aj}
(3.148)
whence d is primitive if and only if Spank 0(')(s2/) = So \ { j} for every j.
(3.149)
It follows from (3.147) and (3.149) that if 4 is primitive then Fo is maximally incomplete w.r.t. F o for every j, that is there are at least n such families. Thus (3.145) follows from (3.144). This completes the proof.
3.4 Primitive d
121
Definition 3.28 We denote by e(d) the largest exponent e such that for some prime p and some i, pela,. In the usual notation e(sd) = max max vp(a;). P
(3.150)
i
Theorem 3.29 Let d be a primitive sequence of length n >- 2 and e(d) be as defined above. Then 6 1 e(SI)
1
Proof
(3.151)
T2
4
Recall that
[a,,a2,...,an] _ fJ{d(S) : all S}.
(3.152)
The maximum power of p dividing the left-hand side is pe and so for all p'
card{S : pld(S)} < e(d).
(3.153)
For each T E #o(d) put P+(T) = max{p : p prime, pl d(T)}.
(3.154)
Then
11 (1 - d(T)-2) T
fi
>
(1 - P+(T)-2)
T E3 o(sl)
> [J(1 - p 2)e(d)
(3.155)
P
by (3.153). We combine (3.145) and (3.155) to obtain (3.151). This completes the proof.
We work out, as an example, the case given in (3.127). We have Iii I = a and (3.156)
1 + looga
If a = 1 then 91 = {2} and (sail, d) = 1. If a >_ 2 we apply Theorem 3.29 which yields
< di,d »
3
6 -a ( T2
(log a/log 2)
1
> -a#
(3.157)
with fJ as in (3.28), which therefore holds for all a. This method applies to sad _ {a + 1, a + 2, ... , a + b} provided b is not too large compared to a.
3 Oscillation
122
If d is a primitive sequence of squarefree numbers (exceeding 1) then
(d,d) > n.
(3.158)
It seems likely that the condition that the elements should be squarefree is uneccessary. Conjecture 3.30
inf{(d,d) : d primitive, IdI = n} >> n.
(3.159)
This would be best possible if true. We notice that, with p and q distinct primes,
3 = {p n 'p n-2q,P n-3q2 ,...,pqn-2 ,q n-1 } 1
(3.160)
has (d3, d3) Win. This is a very interesting test case since none of our theorems cope with it, both e(d3) and q(,Q13) being large. 3.5 Perfect sequences We consider, albeit somewhat briefly, the case when mo(d) is infinite; that
is Q is not only infinite but contains an infinite primitive subsequence. We assume that d is Besicovitch.
Definition 3.31 We say that d is perfect if 9(d) is infinite and
E(x, sd) = M(x) - d #(d)x = 0(1)
(3.161)
where M(x) = M(x,d) denotes the counting function of .x1(d). Condition (3.161) is strong and it is natural to expect at first sight that we should be able to categorize the perfect sequences relatively easily and proceed to consider the rate of growth of E(x, d). In fact we are only able to achieve the first of these objectives in some restricted cases. The first of these, where a complete solution of the (initial) problem is possible, is that in which 9 is Behrend. This is easy because the error term E(x, d) is negative and, moreover, cumulative: every integer omitted by .mil(d) decreases it. Hence (3.161) holds if and only if N\.ilZ(d) is finite.
Definition 3.32 We say that d is a =sequence or of type 9 if
(i) d omits a finite set of primes P = P(d), (ii) for each p E P, there exists a positive integer /3 = /3(p) such that p#+i
Ed
3.5 Perfect sequences
123
(iii) s may contain some of the divisors of the number m(d) = T7{pa(n)+1 : p E P(d)}. it
For example, we might have
3"= (o \ 15,71) U {125,35,49} where 9 denotes the sequence of primes, in which case N \ mil(d) = {5, 7, 25}.
We leave it as an exercise for the reader to prove the following result.
Theorem 3.33 The Behrend sequence 4 is perfect if and only if it is of type °9'. Let {b1, b2, ... , bk } be primitive, and 4i be Behrend for each i < k. The
sequence
= b1-41 U b2,42 u ... U bkIk
(3.162)
is perfect if every 2i is of type We have t(sd) = t({b1,b2,...,bk}), and so (3.162) provides an example of a perfect sequence 4 for each prescribed value of t(d), provided this value is realized by some finite sequence. We do not know which rational numbers are values of t(R) for finite R: if r is either a rational number outside this set, or is irrational, we do not have any example of a perfect sequence sd such that r. Notice that it is not necessary for d to be perfect that every -4i in (3.162) should be a ,30'-sequence. A counter example is
d=29U3(9\{2}). Here 1l(,4) comprises all the integers divisible by 2 or 3 except the two integers themselves, and 9 \ {2} is not of type 9'. The second category of sequences to be considered here contains none which are perfect. Theorem 3.34 Let s/ contain an infinite subsequence d' whose elements are pairwise coprime, and let t(d) > 0. Then Q is not perfect.
Proof Let d' = {a'1, a2, a3, ...I, and k be any positive integer. By the Chinese remainder theorem, we may find an integer n such that
n--j(mod aj.), 1<j
3 Oscillation
124
whence n + 1,n+2,...,n+k e #(sa'l) and M(n + k) - M(n) = k.
(3.163)
By hypothesis Q is not Behrend, and (3.163) contradicts (3.161) as k is unbounded. Hence d is not perfect. We notice that an apparently weaker condition on sl suffices for the above proof, namely that for every k we can find k elements of d which are pairwise coprime. In fact this implies the existence of d' as stated. For each prime p put
{ai e d : ai - 0 (mod p)}. Then the `weaker' condition implies that Q is not equal to any finite union of the dp. We may therefore construct 4' recursively as follows. Let a1 = a1, and suppose ai, a', ... , d, have been determined. Let P (n) denote the set of prime factors of a'1a'2 ...an , and an+1 = min tai
:
ai c 4 \
p c P(n)}}.
Theorem 3.34 is very simple and we expect that its hypotheses can be weakened. In view of Theorem 3.33 the condition t(d) > 0 is appropriate,
and we may look for a relatively mild condition on d which implies that .#l(d) contains runs of consecutive elements of unbounded length, or less, namely that for every c there exist n and k such that
M(n + k) - M(n) > dJl(d)k + c.
(3.164)
No verifiable condition of this sort, genuinely weaker than the one given above, has been found to date. It is reasonable to suppose that perfect sequences are rare: there are none in the third category of sequences considered in this section. These are sparse, `almost finite' sequences in which the ai satisfy a rapid growth condition, viz, for infinitely many n, an+i > l.c.m. [ai, a2, ... , an].
(3.165)
When (3.165) holds, there is no interference between the multiples of {an+1, an+2.... } and the oscillations within the first period of E(x, dn) where do = {a1i a2, ... , an}. We may therefore apply the results of the earlier parts of this chapter to the finite sequences 4n. Since we have so far been unable to settle the conjecture pertaining to (3.159) we have to impose a side condition on a restricting the exponents of the prime factors of the ai. We write v(a) = ma xvp(a) = max{a : 3p, p"Ia}.
(3.166)
3.5 Perfect sequences
125
Theorem 3.35 Let d be a primitive sequence such that (3.165) holds infinitely often, and in addition 2 ) v(aj) 71
=o(i),
i -oo.
(3.167)
Then d is not perfect.
Proof Let n be restricted to the sequence for which (3.165) holds, and consider fin. Plainly d. t(dn) < d. 4!(d). (3.168) We apply Theorem 3.29: in view of (3.167) we obtain from (3.151) that (dn,dn) >_ fi(n) where fi(n)
(3.169)
oo. It follows from (3.9) and (3.169) that
max IE(x,SO 21
(12))
(3.170)
We recall that E(x, dn) is an odd function in the sense described in (3.7); it is of course periodic with period [al, a2, ... , an]. Hence there exists xn < [al, a2i ... , an] such that
((n))2
(3.171)
E(xn, dn) >-
Let M,,(x) denote the counting function of
(dn). By the definition
(3.5) of E, (3.171) implies
Mn(xn) <- [xn] - t(dn)xn -
((n))
2
) 12)
.
By (3.165), we have M,,(xn) = M(xn), so in view of (3.168) this yields
M(xn)-d/t(d)xn <-
((11)) 2
(3.172)
whence , is not perfect. This completes the proof. Notice that the strategy is to obtain an inequality opposite to (3.164).
4
Probabilistic group theory
4.1 Introduction
In this chapter we describe a method which can sometimes be used to evaluate the logarithmic density
of the set of multiples of a given
sequence c. We may find that d,#(d) exists, that is s./ is Besicovitch,
or that 8.(d) = 1, that is d is Behrend. In particular the method has been used to show that certain sequences d are Behrend when no other way of achieving this is known. An example will be the sequences sd*(t) defined in (1.81)-(1.83) which were used in Chapter 1 to establish
that Theorem 1.7 is sharp. The content of Theorem 4.23 is that these sequences are Behrend.
The method is certainly rather specialized but, when it works, the information obtained is usually very strong and often best possible. It depends on a theorem of Erdos and Renyi (1965) and later developments of this; but the idea of applying such theorems in analytic number theory, and in particular to divisor problems, is due to Erdos (1965). Erdos proved the following result.
Theorem 4.1 Let F(x, D) denote the number of integers n < x such that n has at least one divisor in every congruence class prime to D. Then if a < log 2 is fixed and D < (log x)'
(4.1)
F(x, D) = x + o(x).
(4.2)
we have
This is best possible in the sense that the conclusion becomes false if a > log 2. By the Hardy-Ramanujan theorem, for all but o(x) integers 126
4.1 Introduction
127
n<xwehave (log X)I,g 2-E
< t(n) < (log X)I,g 2+E
(4.3)
and when n is counted by F(x,D) we have r(n) > q(D) >> D/log log D.
Hence if a > log 2 almost all n will have too few divisors to fill the congruence classes. When a = log 2 we need more than (4.3) and we use the Erdos-Kac theorem (Erdos and Kac (1939), (1940), Elliott (1980), Theorem 12.3). As we shall need this later in the chapter we state it for convenience here. Theorem 4.2 (Erdos-Kac) Let f (n) be a real-valued strongly additive function such that If (p)I < 1 always, and let A(x) _ p<x
E f (p)2
B(x) _ +
ff(p)
F
N p<x
p
moreover suppose B(x) -> oo as x -* oo. Then
card In : n < x, f (n) - A(x) S zB(x)} = (G(z) + o(1)) x
(4.5)
where z
G(z)
2. f. e--"Z/2du.
In particular, we have
card {n: n< x,
w(n) - loglogx og log x
< z _ (G(z) + o(1)) x
(4.7)
and a similar result holds for f2(n).
Of course f2 is not strongly additive but the deduction from (4.7) is easy: if fi(x) -> oo then S2(n)-co(n) < fi(x) for all but o(x) integers n < x. Applying (4.7) to both w and K2, with z = 0, we may conclude that card{n : n < x, r(n) < (log x)'°g2} _ 2x. Since we may replace (log x)b0s 2 above by (log x)1og 2 exp{o(
we may conclude that
limsupx 'F(x,D) <
,
og og x)}
D - (log x)b0g2
In fact the limit exists and equals 1. Erdos foresaw a refinement of his theorem, Theorem 4.10, which we state immediately after Theorem 4.8 below.
128
4 Probabilistic group theory
It may be helpful at this point to describe the general method as it would apply to show for example that a given Q is Behrend. We begin by constructing an Abelian group G = (G, +), (for consistency we shall always write the group operation as addition) together with a mapping p : Z+ --> G with the following properties:
(i) for all d, e e Z+ we have p(de) = p(d) + p(e), (ii) there exists a subset A of G such that p(d) E A implies d e d. We then set out to prove that if we have a suitably large set of elements of G, say 91, 92, ... , gk, then we can almost surely find a subset of them whose sum lies in A. There will be exceptional sets of k elements which possess no such subset (else we should have to take k much too large):
for example the gi may belong to a subgroup of G which does not intersect A. The number-theoretic part of our proof will be to establish
that among the prime factors of almost all integers n we can find k distinct primes Pt, P2, , Pk such that the corresponding set of group elements p(pl ), p(p2), ... , p(pk) are unexceptional in the sense described
above. Evidently if p(pi) = gi, 1 < i< k and gi1
for some ii < i2 <
+
gig +
....+ gi E A,
< ih < k then by (i), (ii) above Pii Pie ... Pi, E d
and of course this product is a divisor of n.
The method requires G to be finite. In Theorem 4.1 it is the group of congruence classes prime to D under multiplication, but in many important applications the natural group which arises is (R/Z, +). We have to impose a suitable finite group and make approximations, which unfortunately upsets property (i) above. In this sense Theorems 4.1 and 4.9 are pure and Theorems 4.16-4.23 impure.
We denote the order of G by N. In the following theorems a set of elements gi, 92, ... , gk will be called unexceptional, or simply good, if every
element g E G has at least one representation gi,+giz+...+gih=g
it
rather than just some elements g E A. This requirement imposes the condition 2k > N on k, simply in order that we can write down at least N formally distinct subsums of gl, $2, ... , gk
We now give the first theorem in probabilistic group theory, due to Erdos and Renyi (1965), of the type required for the work in this chapter
4.1 Introduction
129
and which Erdos used to prove Theorem 4.1. Rather than prove this, we state and prove a variant of it (the first of two variants presented in this chapter) which enables us to deal with a conjecture of Erdos which appeared in Asterisque (1979). We require the second variant to cope with the sequences d*(t). Theorem 4.3 (Erdos-Renyi) Let G be an Abelian group of order N, and let k elements 91, 92, ... , gk be chosen from G, with repetitions allowed. For each g r= G let R(g) = R(g ; gi, $2, ... , gk) denote the number of representations of g in the form
gt,+gi2++g1,=g,
(4.8)
Then if 6 > 0 and 2k >
22 6-2N log N log
(4.9)
we have R(g) > 1 for every g, for all but at most 6Nk exceptional choices of the elements 91, 92, ... , gk.
Erdos and Renyi used the terminology of probability theory in both
the statement and the proof of their theorem; and from this point onward we follow their example. We say that the k elements 91, $2, ... , gk are chosen randomly and independently from G: the probability that gj is any particular group element is 1 IN and that t9l, 92, ... , gk } is any particular ordered k-tuple of elements is 1/Nk. The conclusion becomes that (4.9) implies
Prob(min R(g) > 0) >- 1 - 6.
(4.10)
A useful alternative notation for sums like (4.8) is E1g1+8282+ +Ekgk, where each ei = 0 or 1. This avoids the double suffices in (4.8) but strictly
speaking is an abuse of notation: we mean eg = g if e = 1, cg = Oc if
e=0. It is natural to ask to what extent (4.9) is best possible with regard to the extra factors 6-2 and log N going beyond the obviously neccessary condition 2k > N. We notice that the Erdos-Renyi theorem imposes no condition on the structure of G (other than that it is Abelian), hence a (rather theoretical) approach to this question would be to attempt to determine the most unfavourable structure for G. The best structure would appear to be a direct sum of cyclic groups of order 2. R.J.Miech (1967) observed that in this case, G may be regarded as a vector space over Z2: the elements g which are represented, that is
130
4 Probabilistic group theory
R(g) > 0, comprise a subgroup of G on which R(g) is constant. Using these facts it may be shown that in this case the condition 2k > 8-1N is sufficient to imply (4.10). Erdos and Hall (1978) considered this problem, and imposed a condition on the structure of G which is satisfied by the groups with which we
have to deal in this chapter. This is Condition A For each fixed l E Z+, the number of elements of G of order l is o(N).
Theorem 4.4 (Erdos-Hall) Let G satisfy condition A, and k
k
log
g2
+ O(1),
2 = N.
Then for each fixed r > 0 we have r
card{g E G : R(g) = r} - Ne r.
with probability -> 1 as N -* oo.
This shows that if, almost surely, every element is to be represented, then we must have 2 --* oo. It also suggests but does not prove that we need 2 > log N, just as in (4.9). If condition A is made quantitative, rather more can be said. For example if G is cyclic Erdos and Hall proved that 2 >> log log N is necessary for every element to be represented with probability tending to 1. In this chapter we have to consider two groups: the congruence classes prime to D under multiplication, and (R/Z, +) which we approximate by a cyclic group of large order. In the latter case, condition A is trivial.
In the former, we note that the number of elements of order I does not exceed lw(D)+I (The extra 1 occurs because the powers of 2 from 8 onwards do not have primitive roots.) We shall not apply Theorem 4.4 directly in what follows but draw the reader's attention to the behaviour of R(g) as a Poisson variable: this is the best picture of what is happening that we have. We give some further information on the abstract theory in §4.5.
4.2 The Erdos-Renyi theorem: first variant
In this section we state and prove a variant of Theorem 4.3 which will enable us to settle a conjecture of Erdos (1979).
4.2 The Erdos-Renyi theorem: first variant
131
Theorem 4.5 Let G be an Abelian group of even order N, and H be a subgroup of G of index 2. Let k elements gi, $2, ... , gk be chosen randomly and independently from G. Then the probability that R(g; gl, $2, ... , gk) >- 1 for every g E G, conditional on the event that precisely r of 91, 92, , gk
belong to H, is at least 1 - 6, provided 6 > 0, r < k and
1g 6-2N log N. 2k > 128
(4.11)
This result is due to Erdos and Hall (1976a) (in which the right-hand side of (4.11) is claimed to be smaller; this error does not affect their application). The condition r < k is necessary: if r = k, Eigi + 9292 + always.
+ Ekgk E H
We emphasize at this point that in Theorem 4.3 there is no condition on the structure of G, and in Theorem 4.5 no condition other than N being even as stated, which is necessary and sufficient for the existence of a subgroup of index 2. Proof of Theorem 4.5 We shall write Prob(...) and Exp(...) for the probability and expectation of the event in brackets; Prob(... IA) and Exp(... IA) are conditional on the event A. We call a set of k elements , gk good if R(g; gi, $2, ... , gk) > 1 for every g : it is a (k, r)91, 92, set if exactly r of the elements belong to H. G = (G, x) is the group of characters x acting on G, of which xo is the principal character, (xo(g) = 1 for every g E G) and xi is such that X1(g) = 1 if g E H, = -1 else. is the complex conjugate of x, and the properties of these characters that we need are :
(i) each x is a homomorphism from G into the unit circle, that is Ix(g)I = 1 always, and x(g + g') = x(g)x(g') for every g, g' E G;
(ii) there are N distinct characters, and if we sum over all these we obtain X(g)x(g') x
1
if g = g'
0
else;
(4.12)
(iii) there holds the corresponding orthogonality relation 1
Vg)/ (g) = s
1
0
if x' else=
(4.13)
In fact G is isomorphic to G apart from the change of group operation, but we shall not use this fact here. Notice that x(g) = x(-g).
4 Probabilistic group theory
132
Lemma 4.6 Let l elements 91,92.... , gt be chosen randomly and independently from G, and Mo = card{g : R(g;$1,$2,...,gt) = 0}.
(4.14)
Then for any s < l and j > 0 we have Prob(s)(Mo > gN22-1) <
ii_1
(4.15)
where Prob(s) denotes probability conditional on f91,92, .... gt} being an (l, s)-set.
Proof of lemma
Put R(g) = R(g; gi, $2, ... , gt). From (4.12) we have R(g) =
nr
E x(g)11(1 + x(gt))
(4.16)
i=1
x
and we notice that the principal character Xo contributes 21/N to the right-hand of (4.16). This is the average value of R(g) as g varies and we denote it by a.. If we subtract A from both sides of (4.16) and employ the second orthogonality relation (4.13) we obtain I
E(R(g) - ,)2 =
N E H I 1 + x(gt)I2.
gEG
(4.17)
X*-Xo i=1
For a given set of group elements gl, $2, ... , gt write t
t = E xi (gl)
(4.18)
t=1
so that t = 2s - l for (l, s)-sets. We consider the function of z, defined for
zEC
{0} by
F(z) _
(R(g;91,g2,...,gt)-.1)2
zt
(4.19)
gEG
where the outer sum is over all N' sets of l elements and t is given by (4.18). We employ (4.17) to obtain F(z)
11 zx,(gi)I 1 +x(gt)12
(4.20)
i=1
zxl(g), 1 +X(g)I2}
N X*X0
gE G
(4.21)
4.2 The Erdos-Renyi theorem: first variant
133
Notice that g has taken the place of g;. We evaluate the inner sum. If X * Xi it acts on H as a non-principal group character, and the inner sum is N(z + z-1). If X = Xi the sum is 2Nz. Therefore z-1)1
F(z) _ (1 - N) N1(z +
+ N1-1(2z)1.
(4.22)
Let us write E(s) for summation over all (1, s)-sets. Equating coefficients of z2s-1 in (4.19) and (4.22) yields, provided s < 1,
KS) gEG (R(g) - 2)2 =
(1
\\\
N)
(')N1.
(4.23)
Let us write Exp(s) for expectation conditional on the event that {gl, g2, ... , 911 is an (1, s)-set. There are (s) (z )1 such sets, and so (4.23) implies, for s < 1,
Exp(s) 1:(R(g) - 2)2 = (1
- N) 21.
(4.24)
gEG
By (4.14), the sum on the left is at least Mo22 and we recall that 2 = 21 /N. Hence Exp(s)Mo < (1
- N) N22-1
(4.25)
so that (4.15) follows by Markoff's inequality. This proves the lemma. Let us write ko = I + j where
=
1
log
log2 I
= log N log 2
(loN\1 log 2
+u
'
(4 . 26)
and u is a function of 6 to be determined: it is positive, and such that l E Z. We begin by choosing just l random elements from G, appealing to Lemma 4.6 for an estimate of the probability that a significant proportion of the elements g already have R(g; 91, 92, ... , 91) > 0. We then add j extra elements one at a time, claiming at each stage that the proportion of elements g represented is increased, with probability nearly 1. For this step we employ the following result.
Lemma 4.7 Let H be a (fixed) subgroup of G of index 2 and B be any subset of G of cardinality M. Let an element g' be chosen, either randomly from H, or randomly from the complement of H (in either case the probability that g' is any particular group element is 2/N). Let M' = M'(g')
4 Probabilistic group theory
134
denote the number of elements g such that both g E B, g - g' E B. Then 2Nz
ExpM' <
(4.27)
.
Let S(g) = 0, 1 or 2 according as neither, one or the other, or both of g and g - g' lie in B. For group characters x set Proof of lemma
P(x, B) _ L x(b)
(4.28)
bEB
and let xi be the non-principal character which is identically 1 on H, that is, principal as a character acting on H. We have
S(g) = N
S(g) =
x(g)P(x, B) (l + x(g'))
P(xo, B) (1 + xo(g')) = 2M,
gEG
E S(g)2 = N gEG
IP(x, B)I2
11
+ x(g')12 .
(4.29)
x
Since we also have 1
1 EJP(x,B)I2=M
(4.30)
x
we may subtract the second line of (4.29) from the third to obtain
M'(g') = gEG
(5(g)) = N
I` I P(x, B) I' Re x(g'),
(4.31)
X
where Re denotes the real part. Suppose g' is to be chosen randomly from H. Then ExPRe X(g) =
1 if x = xo
or xi
0
else.
(4.32)
On the other hand if g' is to be chosen from the complement of H we find that ExpRe x(g') _
11 -1
(4.33)
0
In either case we have ExPM' < N(IP(xo,B)I2+IP(xi,B)12)
(4.34)
4.2 The Erdos-Renyi theorem: first variant
135
and this implies (4.27) because IP(x,B)I < M for any character. This proves the lemma. To fix our ideas, let us assume that our final (k, r)-set is to be obtained by choosing the k - r elements from the complement of H first: so when 1 elements have been chosen we must have an (l,s)-set, with s = (I -k + r)+. Note that r < k implies s < 1. Mo is as in (4.14) and AO is the event r1N2_u
Mo <
= r1N22-1.
(4.35)
Let A' denote the negation of the event A. Lemma 4.6 implies Prob(s)(Ao) < r1-1.
(4.36)
Assume that AO occurs and choose a further element $1+1 from H or G \ H as the case may be. For 0 < i < j put
= (l+i-k+r)+
sti
= {g :
B1
R(g;gi,92,...,g1+1) = 0}
M1 = cardB1.
(4.37)
For i >- 1, b; is the set of elements g such that both g and g - g1+1 belong to B;_1, and Lemma 4.7 yields
ExpM; _<
2M 1
(4.38)
N
So we have Exp(,)(M11Ao) <
2r12N2-2u
(4.39)
and we let AI be the event M1 <
4r13N2-2u.
(4.40)
We apply Markoff's inequality to (4.39) to obtain Prob(S1)(A'IAo) <
(211)-1.
(4.41)
Assume both AO and AI occur, denoting this event by AoAI. We pick g1+2 from H or G \ H, and apply Lemma 4.7 again to obtain Exp(sz)(M21AoA1) < 32g6N2_4".
(4.42)
Prob(S2)(AzlAoA1) < (4q)-1
(4.43)
Markoff gives
where A2 is the event
M2 <
128r1lN2-4u.
(4.44)
4 Probabilistic group theory
136
We continue in this way. Thus Ai is the event 2"''12`+'-1N2-2''"
Mi <
(4.45)
where mi = 3(21 - 1) - i. (The recurrence relation is mi = 2mi_1 + i + 1, mo = 0.) Let us set 2 1
(4.46)
S
and fix u in such a way that l in (4.26) is an integer and Mj < 1. Let
- log 2 log C log 2
-j
(4.47)
so that 0 is the fractional part of the first quantity on the right, whence 0< 0 < 1. From (4.45) and the formula for mj, we require u>
3(23 - 1) - j
2j+' - 1
2
log N
+ 23 log 2 log S + 2i log 2
2]
and, bearing in mind that we may have to increase u to make l an integer, we see that we can choose a suitable u satisfying
u<4+
logg 2
log 2 + 2B.
(4.48)
We put (4.26), (4.47) and (4.48) together to obtain
ko
24
(4.49)
The event Aj now includes Mj < 1, that is R(g ; gl, $2, ... , gig) >- 1 for every g E G.
(4.50)
If k satisfies (4.11) then k > ko is also sufficient. We have Prob(,)(A')
< Prob(r)(A jAoAl ... Aj-i ) + Prob(,j-,)(A j-i IAoAl ... Aj-2)
+ Prob(1,)(A'IAo)
+ Prob(,)(A')
(4.51)
4.2 The Erdos-Renyi theorem: first variant
137
and by (4.36), (4.41), (4.43) etc., the right-hand side does not exceed
1+1+1+...+1
d(D, a) _ {d : d = 1(mod D), 1 < d < exp D"},
(4.52)
A(D, a) = d# (d(D, a)).
(4.53)
and
0 as D -> oo if a < 1. Erdos stated
Trivially A(D,a) < D-1(D" + 1) that he could prove that
D--* oo.
(4.54)
He conjectured the existence of a number ao, exceeding 1 and such that both lim A(D, a) = 0,
if a < ao
(4.55)
lim A(D, a) = 1,
if a > ao.
(4.56)
D- co
D-+oc
We shall prove that (4.55) and (4.56) hold, with ao = (log 2)-' = 1.442695..., obtaining slightly stronger results in both cases, in which oo. a - ao - 0, ao + 0 (respectively) as D Theorem 4.8 Let A(D, a) be as defined above, and
a< where c > (lo 2
2
)1'
/logloglogD
1
log-2 c
log D
(4.57)
is fixed. Then A(D, a) -> 0 as D --+ oo.
Theorem 4.9 Let A* (D, a) denote the density of the integers n such that for
every a prime to D, n has a divisor d - a(mod D) in the interval exp exp (
log D) < d < exp D".
(4.58)
Then if + c(D)(log D)-1/2
a=1 1
where (D)
oo as D --> oo, we have A*(D,a)
1.
(4.59)
138
4 Probabilistic group theory
This is a suitable point to state the refinement of Erdos' Theorem 4.1 mentioned earlier. Theorem 4.10 (Erdos and Hall (1976)) Let F(x, D) be as in Theorem 4.1 and D = 2loglogx+(X+o(1))
log1ogx
(4.60)
f
(4 . 61)
Then
F(x , D) = (x + o(x)) 2
Oc
e "2/2 du '
This result had been conjectured by Erdos (1965) and the idea has been a leitmotif in his work in this area. The parameter a in (4.1) has a threshold at log 2, by which we mean that the arithmetical function F(x, D) satisfies different asymptotic formulae on either side of the threshold, in this case
F(x, D) - x a < log 2 = o(x),
a > log 2).
Erdos' idea is to allow a to converge to the threshold and to introduce a further parameter, say X, and a probability distribution function E(X) to give an asymptotic formula on the threshold. E(X) provides a smooth transition or homotopy. Conjecture 4.11 There exists a probability distribution function E(X) such 1/log 2 together in that if A(D, a) is as defined above and D -* oo, a such a way that
a = log 2 + (X +
o(1)) (log
D)-1/2
(4.62)
then
lim A(D, a) = E(X), X E R.
(4.63)
Theorems 4.9 and 4.10 share the property that apparently Theorem 4.3 provides insufficient input from probabilistic group theory to prove them, because of a shortfall in our knowledge about the distribution of primes (mod D), specifically about the possible Siegel zero (mod D). Theorem 4.4 bypasses the difficulty such a zero raises. Proof of Theorem 4.8
Lemma 4.12 (Erdos (1946)) Let s > 0 be fixed, and to -> oo. Then for almost all integers n, and every t, to < t < n, we have IS2(n, t) - log log tI <
(2 + e) loge t logo t.
(4.64)
4.2 The Erdos-Renyi theorem: first variant
139
Erdos indicated that this could be derived from the law of the iterated logarithm in probability theory. There is a number theoretic proof in Hall and Tenenbaum (1988), Chapter 1 together with references to complete probabilistic proofs, e.g. Kubilius (1964). We shall apply Lemma 4.12 with to = D. is the sequence of integers n for which (4.64) holds, and 4(d (D, (z)) n w.
(4.65)
Since d(D, a) is finite, its set of multiples -4 is a finite union of arithmetic progressions and as such has asymptotic density equal to its logarithmic density. So it will be sufficient to show that 80 = o(1) or
E na = o ((a
- 1)-1)
(4.66)
nE°. 8'
uniformly for u > 1; in what follows limit processes are as D -+ co. We set
f (t) =
(2
(4.67)
e) loge t logo t
where e is to be fixed later. We have, for n c
and y < 1,
i(n ) < E ,O(n,d)-log logd-f(d) d]n, de.,1
Z
yo(n'd)(log d)- I °gY
(4.68)
din, dESad
where
Z = exp{(- log y)
(2 + 2e)a log D log log log D,
(4.69)
assuming as we may that D is so large that (2 + E) log log log Da < (2 + 2:;) log log log D. This implies that
E n -a nEa'
nE4
n a'r(n'sl) 00
< Z T n-a E yn(n'd)(log n=1
d)- logy
din,dEsl 00
rriayi2(m,d)
< Z E d-ay°(d)(log d)-logy dE.d
<
m=1
Y d-ayn(d)(logd)-l0gYg(Y,a,d) dc.ad
(4.70)
4 Probabilistic group theory
140
where /
=
g(y,a,d)
I
1 - pa
(1
pa
11 p
« exp{(y-1)1:
p-, }.
p
(4.71)
)))
By the mean value theorem,
I: p-° > Ep 1-(a-1)Epllogp p
p
p
> loglogd-(a-1)logd+0(1),
(4.72)
uniformly for all d and 1 < a < 2. We insert this into (4.71) to obtain g(y, a, d) « da-1(log
d)y-1
(4.73)
and we substitute this into (4.70) to obtain T. n -G
Z
<<
nEg'
1
d-'yn(d)(log E dEd
Z Da(y-1-logy)
<<
_1
d)Y-1-logy
d-lyO(d)
(4.74)
dEsi
provided y - 1 - logy >- 0, because for all d E W we have log d < D. We apply an estimate of Shiu (1980) to the sum on the right. This yields
E d-lyn(d)
«Day-1
(4.75)
1)-1D«(2y-l-tog y)-1
(4.76)
dE.4
so that we now have
ri ° ' Z(a nE4'
We choose y = 2 to minimize the exponent of D (checking that y - 1 log y > 0) and we see that (4.66) will follow from ZDalog2-1 = o(1),
D -> oo.
(4.77)
Let us put a = (log 2)-' - n, and refer to (4.69) for the definition of Z. For (4.77), it will be sufficient to have
rl log D - / C
l0g 2) log D log log log D - +oo
(4.78)
and this holds, by (4.57), if we choose E < Eo(c). This completes the proof of Theorem 4.8.
4.2 The Erdos-Renyi theorem: first variant
141
Proof of Theorem 4.9 Let D > 2 and I (D) denote the interval (u, v] where
u = exp exp( log D),
v = 2 exp
(2a
tog D)
.
(4.79)
By hypothesis a > 1/log 2. We have Dv2a log D < expD'
(4.80)
and if D is sufficiently large, as we may assume, v > u > D. We construct
divisors of n in every congruence class prime to D using only prime factors of n lying in I (D) and possibly one extra prime factor less than D. We show that we are able to do this for integers n belonging to a sequence of asymptotic density -+ 1 as D --+ oo. We begin with the following assumptions about n.
(i) the prime factors E I (D) are distinct, (ii) n possesses a prime factor q < D, q%D. (iii) a log D - 1(D) og D < w(n; u, v) < 2a log D.
where 1(D) -> co as D -* co. The integers n which do not satisfy either (i) or (ii) have density
<
(p( D)
p
p
z=o(1)asD->cc, p>u
p
and those failing to satisfy (iii) have density << 1(D)-2 = o(1) by Tur'n's
familiar variance argument. Notice that a divisor d of n constructed in the manner described above satisfies d < Dvo(u'u'v) < expD'
(4.81)
by (4.80). Next, we strengthen condition (i) above to
(i') the prime factors E I (D) are incongruent (mod D). For each l prime to D define
Z(l)=E{p 1: u
Z « p(D) log Clogu)
(4.82)
142
4 Probabilistic group theory
and the integers n which fail condition (i') have density
z(l)2 <
log V
D) log2
00),
log u
noticing that because u > D the primes in I(D) do not divide D. For a particular n, (we assume henceforth that conditions (i'), (ii), (iii) are satisfied), put w(n; u, v) = t and let Pi, P2, ... , pt be the prime factors
in I (D). Put p, - l,(modD), 1 < i < t so that the congruence classes h are prime to D and distinct. W e say that ,-- ... , lr} is a good set 12
of congruence classes if for every 1 prime to D we can find exponents El, E2, ... , Et such that E, = 0 or 1, 1 < i < t and
l 12 . .
.
l(mod D).
lr'
(4.83)
Else we call {ll,12, ... , lr} a bad set. Let us show that if it is a good set we can construct divisors of n of the shape required. We begin with the case 101 (mod D), when we simply solve (4.83) for the E, which of course
cannot be all zero, and put d = P1'P22 ... A
(4.84)
We have
d1n, d - 1(mod D),
u < d < expD"
(4.85)
using (4.81). When l - 1 (mod D) we have to take care to ensure d > 1. By assumption (ii) above, n has a prime factor q less than, and prime to
D. Clearly q#1 (mod D) so that if l' is such that ql' - 1 (mod D) we have l'#1 (mod D) and we replace l by l' in (4.83) and put d = gP1'P22 ... pt`.
(4.86)
Since l'#1 (mod D) the e, are not all zero and so d > u as before; in fact d satisfies (4.85) as required. It remains to show that the density of the integers n which satisfy conditions (i'), (ii) and (iii) above but are associated with bad sets {11,12, ... , Ill is o(1) as D --> oo. Let ill, 12,...,11} be a bad set of congruence classes (mod D) and pi, p2, ... p, be primes in I (D) such that p, = l,(modD), 1 < i< t. The density of the integers n with precisely these prime factors in I (D) is (P1P2 ... PX 1
11 C1- -
u
(4.87)
4.2 The Erdos-Renyi theorem: first variant
143
and so the density of the integers n associated with this particular bad set is
u
(l_)zl1)
(4.88)
i=1
with Z(l) as in (4.82). Now let E() denote summation over all bad sets of t distinct congruence classes (mod D), counting different orders of the classes multiply. Since each bad set oft distinct classes has t! (plainly bad) permutations,
and we are not concerned about the order of the primes in (4.87), the density of the exceptional integers n associated with these sets is
(fl,
1-
I) I (t)
u
(l,).
(4.89)
i=1
We should explain at this point that we have counted the bad sets in this way because we are going to apply Theorem 4.4. In the probabilistic model adopted in this book, we choose ordered sets of not necessarily distinct group elements {gl, g2, , gk} : if N denotes the order of the group then the number of choices of such a set is Nk and the probability that a particular set is chosen is 1/Nk. It would be possible to work with a model in which unordered sets of distinct group elements were chosen: the formulae would be slightly more complicated and more importantly we should be concerned at every turn with independence. It seems both easier and clearer to proceed in the present manner, accepting a little clumsiness at this stage of the argument. We want an upper bound for the sum C1
u
- -1) p
*1
t! t
iti 71 ,1 ZO 1'
(4.90)
i=1
in which ' denotes that t = w(n; u, v) is restricted by condition (iii) above.
We therefore require an approximation to Z(1) and it is neccessary to take account of the influence of a Siegel zero (mod D) if such exists: in our present state of knowledge of these zeros this may not be negligible and cannot be treated as an error term. We refer the reader to Prachar (1957), Chapter IV and in particular to Satz 7.3 (p.136), from which we deduce that for (1, d) = 1 we have Z(1) =
A
(1 - xi(1)B) + O ((Dlou)
(4.91 )
4 Probabilistic group theory
144
where
/
g A = log I\\\log u
fV I
/
fl
z dw
log w .
AB
,
(4.92)
Here $ E (0, 1) is the (supposed) Siegel zero of the Dirichlet L-function L(s,xl). The Dirichlet character xi must be real and non-principal, i.e. xo; if Q does not exist we put B = 0. Since xl(l) = ±1 X1 = xo, xl taking each value on half the congruence classes prime to D, the effect of the Siegel zero is to increase half the quantities Z(1) at the expense of the other half. These perturbations may not be small and we have to show that their effects cancel. We apply Theorem 4.5. Let G be the group of congruence classes prime to D under multiplication and k the subgroup of index 2 on which the Dirichlet character xi in (4.91) takes the value +1. Now G is isomorphic
to G, with the group operation +, H corresponding to H under the isomorphism. We say that {ll, 1z, ... , it} is a (t, r)-set if exactly r of the
congruence classes li belong to k, and E(t'r) denotes summation over bad (t, r)-sets. The li are ordered and we do not demand from now on that they should be distinct. We deduce from (4.91) that the sum (4.90) is
p(D) to g u
E (t,r) UrV t-r t
(4. 93)
r
where, A and B being as in (4.92),
U=A(1-B)+6, V=A(1+B)+6, 6
(4.94)
Put N = qp(D), k = t, b = b(D) in Theorem 4.5. Here b(D) = o(1) will be spec ifi ed later. Provided condition (4.11) is satisfi ed, that is 128
2t > log 2 b(D)-z (p (D) l og (p (D)
(4 . 95)
we may deduce that the number of bad (t, r)-sets is b(D) ( r ) (` R )t
if r < t , (4.96)
ifr=t,
(Wi)t z whence the sum over r i n (4.93) is
< 6(D) <
C(p(D))t(U+V)t+
2
(fe-f-)tut 2
(4.97)
4.2 The Erdos-Renyi theorem: first variant
145
and the whole sum is log u {b(D)eA+e + e(A+e)/2} log v
«
log U S(D)+(logv)
=0(1),
(4.98)
as D oo, using (4.94) and then (4.92). We have to check that (4.95) holds, with (say) 5(D) = 1/logD. Plainly it would be sufficient to have t
8 + log 2log log D -{ jogD
(4.99)
whereas by condition (iii) at the beginning of our proof we have t > a log D - 1(D)/T, where we must choose 1(D) -> oo. We put 1(D) = with (D) as in (4.59), (the definition of a) and (4.99) holds for sufficiently large D. Thus (4.98) holds, and the density of the integers n without a divisor of the type specified is o(1) as D -> co, that is A* (D, a) --> 1. This completes the proof of Theorem 4.9. The argument presented above to prove Theorem 4.9 may be extended to give a lower bound for A(D, a) when, as in (4.62), we have
a
= log 2 +
(X + o(1)) (log D)-1/2, X E R.
(4.100)
We require u to be smaller, such that log log u = o (IT); a suitable choice is u = expexp((logD)113),
(4.101)
with v as in (4.79). We assume that the integer n satisfies conditions (i') and (ii) as before, the exceptional n having density o(1), and Theorem 4.5 shows that the integers n which also satisfy, (see 4.99), co(n; u, v) > 8 +
1
3 2
g
log log D + log
D
(4.102)
g
but do not have divisors (of the construction described in the proof of Theorem 4.9) in every congruence class prime to D, have density o(1). We strengthen condition (iii) to: (iii') log("" g) + Y
log("g ) < 04n; u, v) < 2a log D,
which implies
w(n;u,v)>-alogD+Yal12(log D)112+O((logD)1/3),
(4.103)
4 Probabilistic group theory
146
and we can choose Y = Y(X,D) in such a way that (4.100) and (4.103) imply (4.102), moreover
Y=
+ o(1),
g2
D -> oo,
(4.104)
By the Erdos-Kac Theorem, (see §4.1, also Elliott (1980) Theorem 12.3) the integers n such that (iii') holds have density re
-
e--' /2 d.
1
2)T Y
whence lim inf A(D, a) > D-.oo
1 2,c
00
e J -X/
W2/2dw.
(4.105)
log 2
This is a partial result in the direction of our conjecture (4.63). The upper bound needs a new idea since the first step would be to remove the iterated logarithm on the right of (4.57), which arises from Lemma 4.12. (This latter result is of course, in itself, sharp.)
4.3 The Erdos-Renyi theorem: second variant
We now state and prove the second variant of Theorem 4.3, which we need for our subsequent proof that d*(t) is a Behrend sequence. Theorem 4.13 Let a E (0,
2]
and 6 > 0 be given and let
p(a) = -a log a - (1 - a) log(1 - a), K(a) = 1 log (a(1- a))
-
2
(a, 6) = 2log b +
log 2 Z
log
µa)
+7.
(4.106)
(4.107) (4.108)
Let N >- max(No(a),6-i°) where No(a) is sufficiently large, and G = (G, +) be an Abelian group of order N. Let k be an integer such that
k>
{log N + K(a) log log N + (a, 6)}, µI and let h be any integer satisfying
min(h, k - h) > ak.
(4.109)
(4.110)
Then if k elements gl, 92, ... , gk are chosen randomly and independently from G, with probability exceeding 1 - 6 every element g E G may be written as a sum of precisely h of these k elements.
4.3 The Erdos-Renyi theorem: second variant
147
It is important that h be specified in advance of the choice of g1,g2,. ,gk. When a = 2, k must be even and h = Z. We then
have µ = log 2, K = 2 so that (4.109) is not much stronger than the corresponding condition (4.8) in Theorem 4.3, and we do get some extra information. The factor k-1 min(h, k - h) is crucial. We can only write down (h) sub-sums of h elements and this must be at least N. It is a straightforward calculation with Stirling's formula, or more swiftly, with the inequality kk Ck/ < min x-h(1 - x)h-k = o<x<1 h/ hh(k - h)k-h
to see that the factor µ(a)-1 in (4.109) is best possible. We need two lemmas corresponding to Lemmas 4.6, 4.7. Lemma 4.14 Let l elements $1, 92, ... , gl be chosen randomly and independently from G, and Mo = card{g : R(j)(g; 91, 92,
,
91) = 0}.
(4.112)
Then for any j < l and rl > 0 we have Prob
2
-
7T < -1.
(Mo>
(4.113)
Proof of lemma Put R(i)(g) = R(i)(g; g1, $2, ... , gl). For any complex number z we have, from (4.12), R(i)(g)zj
=N
X(g)1(1 + X(g+)z)
j=o
(4.114)
i=1
x
whence
E(R(j)(g) h=o
- )l(i))zi = N E X(g) 11(1 + X(g1)z) X *X0
(4.115)
i=1
where
,l(j) =
1
N( j)
(4.116)
is the average value of R(j)(g) as g varies. We put z = re`B and apply
4 Probabilistic group theory
148
Parseval's formula, which yields t
E(R(j)(g) j=0
-
2(i))2r2J
f (1 2rz
=N
x2(S)2
71(S) x,#xo
+x1(gi)z)(1 +x2(gi)Z)dO
i=1
°
x2#xo
(4.117)
and sum over g to obtain A(j))2r2j gEG j=0
2
=N x#xo
1 + x(gi)z12dO. 2- f ff I i=1
(4.118)
°
For non-principal characters x we have, for each i, Exp 11 +x(gi)zI2 = 1 +
(4.119)
r2,
and because the gi are independent, (4.118) and (4.119) imply that i
E(R(j)(g)
Exp
- A(j))2r2j =
gEG j=0
C1 - N (1 + r2)i.
(4.120)
\\\
We equate coefficients of r2j in (4.120) to obtain, for each j< 1,
ExpE(R(j)(g)-A(j))2= C1 gEG
N)
(4.121) (f).
The sum on the left-hand side is at least M°N-2 (x) 2, so that we have ExpMo < N2 I
1
I
(4.122)
and Markoff's inequality gives (4.113). This proves the lemma.
Lemma 4.15 Let B be an arbitrary subset of G of cardinality M. Let f and f be chosen randomly and independently from G, and let M' = M'(B; f, f')
denote the number of elements g e G such that both g - f, g - f belong to B. Then 2
ExpM' = N Proof of lemma
(4.123)
We define, for each group character x, P(x, B) =
X(b), bEB
(4.124)
4.3 The Erdos-Renyi theorem: second variant
149
and for g, g, f' E G,
S(g;f,f') = N
X(g) (x(f)+X(f')) P(X,B)
(4.125)
x
so that S = 2, 1 or 0 according as both, exactly one, or neither of
g-f,g-f'EB. Hence f')) M' =E (S(g;f, 2 J
(4.126)
T S(g; f, f') = 2M
(4.127)
Now gEG
and
iS(g;f,f')2 = gEG
1 N 1: Ix(f)+x(f')I2IP(x,B)I2 X
4N 2
Ix(f)+X(f')I2IP(X,B)12.
+.u
(4.128)
x#xo
For non-principal characters,
Exp2Rex(f)X(f) = 0
(4.129)
so that, from (4.128) and (4.129) we have
ExpES(g;f,f')2 =
4M2
+,2
gEG
EIP(x.B)IZ X*Xo
2M2 IP(X,B)I2
N +2 2M2
N It follows from (4.127) and (4.130) that Exp gcG
(
S(g;f,f') 2
x
+ 2M.
M2
)
N
(4.130)
(4.131)
and together with (4.126), this gives the result stated. Proof of Theorem 4.13 Our argument is based on that of Erdos and Renyi (1965) and this being the case it is similar to the proof of Theorem 4.5. We require an extra mechanism to ensure that the representations
4 Probabilistic group theory
150
of each g E G as sub-sums of the given group elements 91, 92, ... , gk contain precisely h summands. For g, gl, 92, ... , gl, fl, fl, f2, f 2, ... , f v, fv E G the number of solutions of denote by the system of equations (in which every Ei, s' = 0 or 1), g = 6191 + " ' + Elgl + EI+1f 1 + EI+1f i + 81+2f2 + 8'1+2f2' +
+El+vfv +El+vfv, E1+E2+...+El=J,
61+1 + E1+1 = E1+2 + E1+2 =
= El+v + Ei+v = 1.
(4.132)
We have
RO )(g;91,92,...,gi;fl,fi,.. ,fv,fv) i < R(i+v) (g;gl,g2,...,gl,fl,fl,...,fv,fv),
and we put
l=k-2w, j=h-w,
(4.133)
where
_ [loioN]
(4.134)
log 2
and begin by choosing just l elements 91, 92, ... , gi randomly and independently from G : we shall then add w pairs of elements f i, f', one pair at a time in an iterative process. At the start, let Bo = {g E G :
=0},
(4.135)
and put MO = cardB0. Put
(il) = Neu, n > 0
(4.136)
MO < fNe u
(4.137)
and let AO denote the event
so that if Ao is the negation of A0, we have from Lemma 4.14 that
ProbA' < i-1.
(4.138)
We assume AO occurs, and pick the first pair of extra group elements f 1, f i randomly and independently. Let
Bl = {g c G : Roi)(g;g1,...,gl;fl,fi) = 0},
(4.139)
4.3 The Erdos-Renyi theorem: second variant
151
and M1 = cardB1. For g E B1, we have both g - f I E Bo, g - f i E Bo, and we apply Lemma 4.15 which yields Exp(MI IAo) < ri2Ne-2u.
(4.140)
M1 < 2rl3 Ne 2"
(4.141)
Let AI denote the event
so that if A'1 is the negation of A1, Markoff's inequality implies that Prob(A' IAo) < (2q)-1.
(4.142)
We assume that AI occurs, and pick f2, f2 randomly and independently from G. Let
B2={g: Ro)(g;91,...,g1;fl,fi,.f2,f2)=0},
(4.143)
and M2 = cardB2. For g E B2 we have both g - f2, g - f'2 E BI and Lemma 4.15 yields Exp(M2IAoA1) < 4116Ne 4u.
(4.144)
Throughout, the product of events denotes their joint occurrence, and A' denotes the negation of the event A. We let A2 denote the event
M2 < 16nlNe4"
(4.145)
so that Markoff's inequality applied to (4.144) yields Prob(A'2IAoAI) < (4q)-1.
(4.146)
We continue this process. We put, for v < w, B° = {g : R W(
fl, fi,..., f°, fu) = 0},
(4.147)
and M° = cardB We denote by A° the event
M<
2'"°12°+'-1Ne
2'"
(4.148)
where m° = 2(2'- 1) -v. (The recurrence relation is m° = 2mv_1 + v, mo = 0.) We put
rl = 26-1, u >- 2log 6-1 + log 16 + 2
(4.149)
so that, by (4.134),
2'+1 log 2 + 2`1 log rl + log N - 2wu < 0
(4.150)
4 Probabilistic group theory
152
and the event Aw implies Mw < 1. This means that Bw is empty, and for every g
R(3+w)(g;gi,g2,...,9l,fi,fi,...,.fw,fw) > 0
(4.151)
as required. As in the proof of Theorem 4.5, we have Prob(Aw) < Prob(A%,jAoA1...Aw_1) +Prob(Aw_1 AoAI ... Aw_2)
+Prob(A' IAo)
+Prob(A' )
<
- = 8.
(4.152)
h
It remains to calculate how large we need k to be, given the restriction min(h, k - h) >- ak. Substituting from (4.149) into (4.136), we find that we need
G')
> 16e26-2N.
(4.153)
kk(4.154)
For k >- 2, 0 < h < k, we have the (sharp) inequality
(h) > hh(k - h)k-h
8(-h
which may be derived from Stirling's formula in the form: for n >- 1,
n! =
)n
27rn
(e
exp
In
8n E (0,1)
(4.155)
in all the cases in which k > 4, and which checks in the remaining cases. Hence (h) >- (2k)-1/2 exp(y(a)k)
(4.156)
with p(a) as in (4.106). Next, we have, on recalling that I = k - 2w, j = It - w, (4.157)
By (4.110) we have h(k - h)/k2 >- a(l - a). In view of the inequality 1 - x >- e2x, valid for 0 < x < 2, we also have
kv
-h)
v2
-2kv >- exp (h(k -h))
4.4 The Behrend sequences c*(t)
153
if we assume that 2w < all - a)k. Thus (4.157) implies
() >
(') h(a(1 - a))- exp
((2k) .
(4.158)
Since 2k > N and by (4.134), w log 2 < log log N, we shall ceratinly have 2w < a(1- a)k for such N > No(a), indeed we may assume that for such
N the exponential factor on the right of (4.158) is close to 1, say > 1/J for convenience. We combine (4.156) with (4.158) to obtain 2k-1/2 (a(1
C,)
>
- a))" exp (µ(a)k)
2k-1/2(logN)z-"exp(µ(a)k)
(4.159)
with K = K(a) as in (4.107), using w log 2 < log log N again. The right-
hand side of (4.159) is an increasing function of k - the condition p(oc)k > 12 follows from (4.109) - and so we may replace k by the smaller number ko (which need not be an integer) satisfying µ(a)ko = log N + K(a) log log N + c(a, S). Hence 1
(1) >
(---)
so that (4.153) will follow from C log N ko
z 32e26-2
)
(4.160)
which, by the definition (4.108) of (a,S) is equivalent to log N
10
Z Ce
z(a)
(4.161)
for which a sufficient condition is 20.5 log N >- K(a) log log N + (a, S).
(4.162)
By hypothesis N >- max(No(a), 5-10), and this implies (4.162) if No(a) is sufficiently large. This completes the proof of Theorem 4.13.
4.4 The Behrend sequences d*(t) The main result in this section is Theorem 4.17, from which it will be a short step to Theorem 4.23 that ,sV*(t) is Behrend. We begin by stating a result of Erdos and Hall (1974) which was a precursor of Theorem 4.17 and has a similar but technically easier proof.
4 Probabilistic group theory
154
Theorem 4.16 Let c be a fixed real number and 0 < 0 < 1. Then the integers n having a divisor d such that
0 < Illogd - 011 <
1oglogn
2-1oglogn-c
(4.163)
have asymptotic density 1
Go
(4.164)
e )-2//2dy;
moreover if there exists a (necessarily unique) integer m(0) such that 0 - log m(0) (modl) and we allow equality on the left in (4.163) then the density is increased to 1
m(0) 2
I
e y'12dy +
e Y2/2dy.
1
2,n
(4.165)
c
Furthermore we may replace c in (4.163) by c(n), or -fi(n), where fi(n) --> oo as n -> oo. In the former case the asymptotic densities in (4.164) and (4.165) become 0 and 1/m(0) respectively; in the latter case both densities become 1.
This result should be compared with Theorem 4.10. The underlying group, namely (R/Z,+) is infinite in Theorems 4.16 and 4.17 and we have to replace it by a finite cyclic group of suitable order. As explained in §4.1, this involves an approximation (which will be made clear in the
proof of Theorem 4.17) for which there is a price: property (i) of the mapping p no longer holds exactly, and our application of probabilistic group theory is therefore impure. It will turn out not to present any serious difficulty.
Some explanation of Theorem 4.16 may be helpful in preparation for the statement of Theorem 4.17. Recall that by the Erdos-Kac theorem (1940), stated in §4.1, each of the sequences {n : co(n) > log log n + c
log log n},
(4.166)
{n : 0(n) > log log n + clog log n},
(4.167)
has asymptotic density given by (4.164). Since 2w(n) < i(n) < 2°(n)
(4.168)
the same is therefore true of the sequence {n : r(n) > 2lo9 logn+c
1og1ogn}.
(4.169)
Thus the theorem is in accord with a very simple heuristic model in which the fractional part of log d is uniformly distributed on [0, 1) when d runs
4.4 The Behrend sequences d*(t)
155
through the divisors of a large random integer n. That is, these fractional
parts are spaced 1/i(n) apart. This model appears again in the next chapter but the reader should beware of interpreting such arguments too literally. For example, Hall (1975a) proved that if g(n) := min {II log(d/d') II
:
d, d'I n, d
d'}
(4.170)
then
g(n) = (log
n)-1og3+o(1)p.p.,
(4.171)
a result which should be compared with the Maier-Tenenbaum theorem ((1984), Divisors, Chapter 5) in which II II in (4.170) is replaced by I I and
the right-hand side of (4.171) is multiplied by log n. An equally simple, but different, heuristic model involving the distinct ratios of the divisors (cf. Divisors §3.3) predicts the behaviour of g(n). A corollary of Theorem 4.16 is that the sequence -5/'
2
_ {a > 1
:
II log a II < (log a)-log z
log log a)} (4.172)
is Behrend, provided (a) -* co as a -+ co; moreover it is necessary for the conclusion that (a) be unbounded. We proceed to Theorem 4.17. This depends on Theorem 4.13 in probabilistic group theory. In a parallel fashion, Theorem 4.16 depends on Theorem 4.3. Theorem 4.17 Let c be a fixed real number and 0< 0 < 1, t E (0,12)U(12,1). Then the integers n having a divisor d such that both 0 < II log d -ell < T-109 109 n-c
log log n
(4.173)
and
< tfl(n), (if t < z ) (if t > 1),
i2(d) { > tn(n),
(4.174)
where T = exp (µ(t)) and µ(t) = -t log t - (1- t) log(1- t) as in (4.106), have asymptotic density 00
e YZlzdy. 27C
(4.175)
c
Furthermore we may replace c in (4.173) by fi(n) --> co or -co when the density becomes respectively 0 or 1, and we may if we wish require that d, in addition to satisfying (4.173) and (4.174), is a multiple of any particular prime factor of n.
4 Probabilistic group theory
156
The last part of the theorem provides a device, useful in some applications, to ensure that d is fairly large. For example we may choose P+(n) as the particular prime and since P+(n) > n°(1) p.p. we then have log log d > log log n - co(n) p.p.,
(4.176)
where io(n) is any function tending to infinity with n. We use this device later when we show that q*(t), (t < 11) is a Behrend sequence. Proof of Theorem 4.17 The proofs for t < 11, t > 12 are similar and we 2 treat these cases together, pointing out the slight differences as we meet them.
We split the integers n < x into four main classes, which we specify in turn as we encounter them. The zero class comprises those integers for which S2(n) > log log x + (log log x)23 : it contains o(x) integers by the
Hardy-Ramanujan theorem. The first class comprises those integers n < x not in the zero class for which 0(n) >_ log log x + c
log log x + (jo(t) + 1) (log log x) 3 ,
(4.177)
where jo(t) is an integer depending on t only, to be specified later. By the Erdos-Kac theorem, the first class has cardinality
1f
00
x
2n
e Y2 /2dy + o(x)
(4.178)
c
and we shall prove that all but o(x) of these integers have a divisor of the required kind. We split the first class into sub-classes Cj, jo(t) < j < j, (x), assigning n to Cj if
(fl(n) - log log x - c
og og x
(log log x)3
L
I = J + 1.
(4.179)
Here ji (x) = [(log log x)113 - c(log log x)116].
(4.180)
We put I (x) = (u, v] where
u = exp ((loglogx)1/3)
,
v = x1/(1oglogx)2
(4.181)
and we have
E (n(n) - fl(n; u, v)) - 5x log log log x. n<X
(4.182)
4.4 The Behrend sequences d*(t)
157
We assume henceforth that first class n satisfy S2(n) - S2(n; u, v) < (log log X)113;
(4.183)
by (4.182) the number of exceptional n is o(x). We denote by Cj' the set of integers n in Cj for which (4.183) is satisfied. For n c Cj we deduce from (4.179) and (4.183) that S2(n; u, v) > k(j) where k(j) = [log log x + c
log log x + j(log log x) 3 ].
(4.184)
Let the k(j) smallest prime factors of n which lie in I(x) be denoted by pi, p2, ... , pk(j). We consider divisors d of n which are products of exactly h(j) of these primes, where h(j) is specified below. We are going to apply Theorem 4.13 and this involves a parameter a, at our disposal, and requires in condition (4.110) that min (h(j), k(j) - h(j)) > ak(j). We consider the cases t < 1, t > 12 separately. If t < 12 we put 2
h(j) = [ak(j)] + 1, a = t.
(4.185)
We shall then have fl(d) = h(j) < tc2(n) if k(j)+t-1 < S2(n). By (4.179) and (4.184) a sufficient condition for this is t-i < (log log x)1/3 or x > xi(t).
We may assume this holds, so that any divisor d of the kind described above satisfies (4.174). If t > 1 we put
h(j) = [(1 - a)k(j)], a = 1 - t -
(4.186)
where q = 3(log log x)-2/3. From (4.179) and (4.184) we have )(n)-k(j) < 2(log log x)1/3 + 1, and qk(j) > 2(log log x)1!3 + 2 provided x > x2(c, t). Hence qk(j) >- t(TZ(n) -k(j))+1 which implies [(t+q)k(j)] > tf2(n). Thus any divisor d with 92(d) = h(j) satisfies (4.174) when t > 2. We assume henceforth that x > max(xl, x2).
For t < 2, p(a) = p(t), and we shall need a relation between p(a) and p(t) when t > I'. Recall from (4.106) that p(t) = -t log t - (1- t) log(1- t),
so that p'(1 - t) = - log(1 - t) + log t < - log(1 - t) because t < 1, and p' < - log(1 - t) in the interval to the left of 1 - t. It follows from the mean value theorem and (4.186) that there exists 0t) such that q < 113(t) implies p(a) > p(1-t)+q log(1-t). Hence for x > x3(t) and all t, we have p(a) > p(t) + 3 log(1 - t)(log log x)-2/3.
(4.187)
Next we set, for each j, jo < j <- jl, Nj = [exp{p(a) (log log x + c log log x +(j_ 1)(Iog log X)1/3 (4.188)
and we define Gj to be the cyclic group of congruence classes (mod
4 Probabilistic group theory
158
Nj) under addition. We are going to apply Theorem 4.13 with G = Gj,
N = Nj, k = k(j), h = h(j) and 8 = (log log x)-1, and we note at this point that there exists an x4 = x4(c, t) such that for x > x4, the conditions Nj > max(N°(a), 8-10), (4.109) and (4.110) are all satisfied. We assume henceforth that x > max(x1, x2, x3, x4). We define the mapping pj : Z+ --> Gj by the relation [Nj log m] - pj(m) (mod Nj);
(4.189)
this does not satisfy p(mm') = p(m) + p(m') for all m, m' but does so approximately, and this is sufficient for our purpose. Indeed, if g(O) E G, g(O) - [NCO] (mod Nj) and p(pi,) + p(pi2) +
+ p(pi,,) ° g(O) (mod Nj)
(4.190)
then 11 log(pi, pi, ... pih) - O 11
< hN) .
(4.191)
Nj
We are now in a position to specify the integer jo(t) which appears in the definition (4.177) of the first class. From (4.184), (4.187) and (4.188) we have, for sufficiently large j,
k() < T-109109'loglogx
(4.192)
Nj
where T = exp (p(t)) as in the statement of the theorem. We define jo(t) to be the smallest integer such that j >- jo(t) implies (4.192). We have to refine the class C further. We say that n E Cj" if n E C and the congruence classes p(p) (mod Nj) are distinct, where p runs through all the KI(n; u, v) prime factors of n belonging to 1(x). Of course this also implies that these prime factors are distinct.
Lemma 4.18 For each fixed /3 > 0 and every subinterval E of [0, 1) we have
p- (log
logll
1
EI
log u
u
1
+O
log u
+ O#(e
log u),
(4.193)
where JEJ is the length of E.
This is a straightforward deduction from the Prime Number Theorem in the form
7r(z) = liz +Op(ze2# and we omit the details.
logi)
(4.194)
4.4 The Behrend sequences sd*(t)
159
For each 1, 0 < l < Nj, let Pa(l) denote the set of primes p E I (x) such that p(p) = 1, that is [Nj log p] = l (mod Nj).
(4.195)
An equivalent condition is log p c El(l) (mod 1),
E3(l) _
l + [,!__!). Nj Nj
(4.196)
We deduce from Lemma 4.18 that
E p 1<< NT 1 log log x
(4.197)
PEPj(l)
uniformly for jo(t) < j < j, (x), whence 2
card {Cj' \ Cj" }
<
xY(Ep1 I
<<
pEl)
xNJ-.'(log log x)2,
(4.198)
and
card{Cj' \ C } K
log x)3 = o(x).
(4.199)
Let n E Cj" and write pj(pi) = gi for 1 < i < k(j): the gi are therefore distinct congruence classes (mod Nj). We say that n is good if {gi, g2, ... , gk(j)} is a good set of congruence classes in the sense that every congrunce class g (and so in particular g(O)) has at least one representation
g = gii + giz +
in which i1 < i2 <
....+ gihv> (mod Nj)
(4.200)
< ih(j). A good integer n has a divisor d satisfying
the requirements of the theorem and we have to estimate the number of members of Cj" which are not good, let us say bad. Accordingly let 91,92-_g. be distinct congruence classes (mod Nj) where m >_ k(j), such that no sub-set of k(j) of these classes is good in the sense described above, and moreover m < log log x + (log log x) 3 .
(4.201)
By Theorem 4.13, the number of choices of $1, 92, ... , gm is <6Njm.
(4.202)
If n is a bad integer associated with this set of congruence classes, we have p,,,q where pi E I(x), log pi E Ej(gi) (mod 1) for 1 < i < m n = PI P2
4 Probabilistic group theory
160
and q has no prime factor in I(x). If n is to be in the first class (4.201) holds.
Lemma 4.19 Let I be a set of primes of which none exceeds x. Then
card{q:
q<x,pjq-p01}<<xfl(1-p).
(4.203)
pEI
For a proof see van Lint and Richert (1965), Hall (1974a) or Chapter 3 of Halberstam and Richert (1974). The number of integers n as above is
card{q : q <
, pjq - p I(x)}
x
(4.204)
p1"'Pm and we note that by the definition (4.181) of v and (4.201) we have v < xlvm so that Lemma 4.19 is applicable and the cardinality in (4.204) is
«
x
1- 1
P1P2"'Pm
11
(4.205)
p
pEI(x)
Hence the number of bad n corresponding to the set
« «
x
x
to gv ) 11 i=1 09u M log u ( log v ) (N
p
Iog pEEi(gi) 1 l og
gm is ( 4. 206 )
1
u
I
l og u Y
I
m
(4 . 207)
by (4.193) and (4.201). Hence the number of bad integers in C such that S2(n; u, v) = m is, combining (4.202) and (4.207), m
K Sx
( log v) 019 -U
m!
(log ( log 10911
(4.208)
and we sum over m, and then over j to deduce that there are << bx(log log x) 3 = o(x)
(4.209)
exceptional integers in the first class. Our treatment of this class is complete. The second class comprises the integers n < x such that (4.177) is false,
but Q(n) > log log x + c log log x - (log log x) 3 .
(4.210)
4.4 The Behrend sequences d*(t)
161
Since )(n) = log log x + (c + o(1)) og log x for all these integers, the cardinality of the class is o(x) by the Erdos-Kac theorem. The third class comprises the integers n < x for which (4.210) is false, and we show that the number of these integers having a divisor of the type specified in the theorem is o(x). We retain u as in (4.181) and we begin by disposing of the integers with such a divisor d satisfying the extra condition P+(d) < u. For this purpose we replace condition (4.173) by 0 < Il log d - 011 < (log x)-µ(t)/2
(4.211)
which is weaker except for small integers n whose number is negligible. We are going to show that
d-1 +' d-' = o(1), P+(d)
P+(d)do
(4.212)
P+(d)
where we have spilt the sum at do = exp((log log x)7), the * denoting that (4.211) holds. The first sum on the right does not exceed d-1
<
d-1
(log do)-1
log d
P+(d)
P+(d)Su d>do
1
< <<
(log
do)-1
fJ (- P Y P
log e PSu
P
(log log x)-7(log u)2 = o(1)
(4.213)
because log u = (log log x)3. Notice we have not used (4.211) : obviously
we must do so in estimating the remaining sum in (4.212). Let m = 0,1,2,3 ....; the sum of the reciprocals of the d's in the range m - (log x)-µ(t)12 < log d - 0 < m + (log x)-µ(t)/2
(4.214)
is
(4.215)
if indeed there exists an integer d satisfying (4.214): recall that in Theorem 4.17 we do not allow logd = 0 in the case 0 e logZ+(mod 1). Let d(x, 0) denote the smallest integer d for which (4.211) holds, and let d(x, 0) oo as oo, m(x, 0) satisfy (4.214) when m = m(x, 0). We have d(x, 0)
x - co, and we sum (4.215) over the range m(x, 0) < m < log do + 0(1)
4 Probabilistic group theory
162
(if this is non-empty) to obtain d_1 << P+(d)
(loglogx)7 (log x)1(t)/2
+ d(x, 8)-1 = 0(1).
(4.216)
d<do
Notice that in this part of the sum (4.212) we have not used the condition P+(d) < u. Together with (4.213) this establishes (4.212).
Next we consider the integers int the third class with a divisor d satisfying the requirements of the theorem but none such with P+(d) < u. We refer to these integers as being in the fourth class. We want to show that the cardinality if this class is o(x). Let us exclude from the class integers n for which
P+(n) < w = xl/loglogx
(4.217)
By Lemma 4.19, the number of such integers is
x J (1 - 1) << x(log log p
x)_1
= o(x).
(4.218)
w
For the remaining integers n in the fourth class we write n = mp, p > w, and we consider whether such a decomposition, in which m also belongs to the fourth class is possible or not. The number of cases when this is possible is (-X
<
*
m<x/w
7r
m
) <<
x log log x * 1 log x
m
(4.219)
where * denotes that m is in the fourth class. Let us write m = qpl p2 ... Pr
where P+(q) < u and u < pl < P2 <
< pr. Of course P+(m) > u because d%q, and we may assume that the prime factors of m which exceed u are distinct: the m's with a repeated prime factor > u contribute
K
x l og log x p>u -
s= o(x).
(4.220)
s5x
Next, m has a divisor d of the form d = api p22 " . pr', al q, E; = 0 or 1 Vi,
(4.221)
which satisfies (4.173) and (4.174). We may restrict our attention to integers n > _IIX so that (4.173) implies 0 < Il log d - 011 <
2
(4.222)
4.4 The Behrend sequences d*(t)
163
where
Iv
T-109109x-cloglogx
(4.223)
By definition of the fourth class, P+(d) > u and hence there exists j, 1 < j < r, such that sj = 1. But then (4.222) restricts the fractional part of log pj to lie within the union of M sub-intervals of [0, 1) each of length 1. Here M is the number of divisors of gplP2 ' ' ' Pj-lPj+l ... Pr which
satisfy (4.174), and plainly M does not exceed the number of divisors of n itself with this property, which we denote by M(n, t). It follows that if E' denotes the summation over the primes pj as above then by Lemma 4.18 (in which we substitute x for v), we have
EI « M(n, t)l log (
log u
(4.224)
.
Lemma 4.20 Let s be a positive integer and X, Y be real numbers such that 0 < X < Zs < Y < s. Then the number of s-tuples {81, 62, ... , bs} in which each bi = 0 or 1 and, respectively 6
X
(4.225)
6
Y
(4.226)
does not exceed
exp
sp (X) I , exp {sy
(4.227)
},
(s)
where p is defined by (4.106).
The two results are equivalent: if we replace bi by 1-6; and X by s - Y in (4.225) it becomes (4.226). We restrict our attention Proof of lemma to (4.225).
Let 0 < z < 1. The number of s-tuples for which (4.225) is satisfied does not exceed 1
1
6 1 - 06 2 = 0
1
...
z31+az+...+a,-X
= (1 +
z)sz-X
(4.228)
s,,=O
and we put z = X/(s - X) which does not exceed 1 by the hypothesis about X. This gives the result stated.
4 Probabilistic group theory
164
Lemma 4.21 Let M(n, t) denote the number of divisors of n such that (4.174) holds. Then M(n, t) < exp{f2(n)y(t)},
(4.229)
where 11 is defined by (4.106).
Let n be squarefree, 92(n) = s, n = P1P2 Proof of lemma typical divisor be
d=pigp2z...pS
each bi = 0 or 1.
ps and a (4.230)
We apply Lemma 4.20 with X = M(n) if t < 2 and Y = 2(n) if t > z. This gives the result stated in this case. In the general case let n = p` p22 ... pkk where a1 + a2 + and let n' = p1l)P12) ... p1
P2 )P22) ...
P(2a2)
... Pkl)Pk2) ...
Pkak)
+ ak = s, (4.231)
be any squarefree number with s prime factors. We define a multiplicative, injective mapping 0 from the set of divisors of n into the set of divisors of n' by setting Y'(PYj) = P(l)P(2) ... p(Pi)
Ri < ai
(4.232)
and we have S2 (O(d)) = 0(d) for every d. Hence M(n, t) < M(n', t) < exp (sy(t)) because n' is squarefree. This completes the proof.
We return to (4.224) and we recall that for integers n in the fourth class, (4.209) is false, that is S2 (n) < log log x + c
log log x - (log log x)13.
(4.233)
Hence (4.223) and (4.229) imply
M(n, t)l < A
(4.234)
A = exp{-(Iog log X)1131.
(4.235)
where
We may now estimate from above the sum on the right of (4.219). This does not exceed
E q-1E E P+(9)<_u
r
E"(P1P2...Pj-1Pj+1...Pr)-1 Y'P71
(4.236)
4.4 The Behrend sequences c*(t)
165
where the sum E" runs over all products of r - 1 distinct primes p; E (u, x], E' is as in (4.224) and ro is the maximum value of r. By (4.233), ro < log log x and so by (4.224) and (4.234) \th\e sum in (4.236) is <<
r (log
(log u)r2A
I
1
I
r
g /// / u
ro
< roAlogx.
(4.237)
It follows from (4.219), (4.220), (4.235) and (4.237) that the number of integers n in the fourth class of the form mp, p > w, where m is also in the fourth class is < Ax(log log x)3 + o(x) = o(x).
(4.238)
It remains to consider the integers n in the fourth class for which such a decomposition is impossible, that is if p is any prime factor of n exceeding w then m = n/p is not in the fourth class.
Lemma 4.22 There exists an absolute constant B such that if y >-
2,
1 < w < y, and E is any sub-interval of [0, 1) then 1 < B I E) y(log y)-1 + Ofl(ye
B
log')
(4.239)
w
where JE l is the length of E and fi is arbitrary.
This is a straightforward deduction from the Prime Number Theorem (4.194).
If n is one of the above integers it has a divisor d, necessarily a multiple of p satisfying (4.174) and (4.222). This places the fractional part of log p within the union of not more than M(n, t) sub-intervals of [0, 1) each of length I (the position of these sub-intervals depending on m = n/p), and so the number of such n does not exceed
EE
(m) 1
(4.240)
m<x/w w
where (m) denotes that log p lies in the relevant set (mod 1). We apply Lemma 4.22, with w = xl/1oglogx, y = x/m, IEI = 1. By (4.223) the error term is negligible, fixing /3 = 1 say, and so (4.240) becomes, using (4.234),
«
A m (log m m<x/w
< Ax(log w)-1 log x = o(x).
(4.241)
4 Probabilistic group theory
166
This completes our treatment of the fourth class integers and our proof of Theorem 4.17 except for the last part which involves a small change in the treatment of the first class integers only, and which we leave to the reader. Theorem 4.23 Let 0 < t < 1, and (a) -+ oo as a -> oo. Then the sequence d'(t) is a Behrend sequence, where .W ('-z) is defined by (4.172), and for t < 1, t > z respectively
d`(t) =
{a > 1 : Q(a) < t log log a + (a)Vllog log a, a}
Illogall <
d* (t)
= {a > 1 fl(a) >- t log log a - (a) :
(4.242)
log log a,
Illogall <
T being as in Theorem 4.17, that is T = exp{-t log t - (1- t) log(1- t)}. Proof of Theorem 4.23 The case t = z is a corollary of Theorem 4.16 and is left to the reader. We consider the case t > 1. Put
1(n) = min din
(lolon_lolod og log n
og log n
(4.244)
Then 1(n) -+ oo as n -> oo, for either we have log log d < log log n when the first term on the right of (4.244) is at least 4 og loga n or else the second term exceeds '-ze(d) and d --> oo. By (4.244) we have, for every divisor d of n,
log log d - (d) log log t l< log log n - 2;1(n) log log n.
(4.245)
We apply Theorem 4.17 with 0 = 0 and c =
For almost all n, there exists a divisor d satisfying both (4.173) and (4.174). We call this divisor a and by (4.173) and (4.245) a fulfils the second condition in (4.243). We may assume Q(n) > log log n - 1(n) og og n by the Hardy-Ramanujan theorem and we see from (4.174) and (4.245) that a also fulfils the first condition in (4.243), whence a E d*(t). Thus almost all integers n have a divisor in .sd`(t) and d*(t) is a Behrend sequence. It remains to deal with the case t < 1, when (4.173) is again stronger than we need for the second part of (4.242) but (4.174) is not sufficient for the first part without some extra information. Let c1(n) be as in (4.244).
We apply Theorem 4.17 with 0 = 0 and c = as before, so that for almost all n we can find a divisor d satisfying both (4.173), (4.174) which is, in addition, a multiple of the greatest prime factor P+(n) of n.
4.5 A conjecture of Erdos and Renyi
We choose io(n) = i
167
og log n in (4.176) so that if n is large enough, we
have log log n < log log d +
log log d p.p.
whence
log log n < log log d + 2 (d) log log d p.p.
(4.246)
because (d) -> oo as d oo by hypothesis. As in the case t > 2, (4.173) and (4.245) imply that a (= d) fulfils the second condition in (4.242). Next, by the Hardy-Ramanujan theorem we have i2(n) < log log n + i (n)
log log n p.p. and we combine this with (4.245) to obtain
f2(n) < 2 log log n - log log d + (d)
log log d p.p.,
whence by (4.246) tS2(n) < t log log d +
log log d p.p.
(4.247)
We substitute (4.247) into (4.174), noticing that in the present case t < Z, so that a (= d) also fulfils the first condition in (4.242) p.p., whence d*(t) is a Behrend sequence as required.
4.5 A conjecture of Erdos and Renyi
In this final section we discuss the accuracy of the Poisson model for R(g) provided by Theorem 4.4 when A = 2k/N is large. We recall that one of the hypotheses of this theorem is that log). = 0(1). If we ignore this condition and allow A, to be large, then we are led to the conjecture that almost surely card{g E G : R(g) = r} = 0
(4.248)
when A is so large that -, Ar
e r!
1
(4.249)
Thus we expect to require 2 > log N for every element to be represented. If we put
A= (I+ 0) log N, r = s log N
(4.250)
then (4.249) reduces, approximately, to
0 > s log(1 + 0) -slogs + s.
(4.251)
168
4 Probabilistic group theory
For positive 0, (4.251) holds if s < sl(0) or s > s2(0) where 0 < sl(0) < 1 + 0 < s2(0) < oo
(4.252)
and so we expect the values of R(g) to cluster about their average A, with approximately normal distribution of variance A. It is not difficult to show that as 0 grows, the ratio s2(0)/sl(0) -+ 1, whence R(g) - 2 for all g.
The above argument is speculative. In their original paper, Erdos and Renyi proved (Theorem 1) that if 2k > 8-2 6-1N 2
(4.253)
then with probability >- 1 - 6 we have for all g, (1 - s)2. < R(g) < (1 + e)2.
(4.254)
In fact this is a straightforward deduction from the formula
Exp E(R(g) - 2)2 = (1
- N )2 k
(4.255)
gEG
which follows from (4.17). Notice that k in (4.253) is very large, indeed 2 > N. Nevertheless, Erdos and Renyi conjectured that the exponent of N in (4.253) cannot be reduced if (4.254) is to hold, without extra conditions on the group structure.
Although it disagrees with the heuristic model derived from Theorem 4.4 explained above, the conjecture seems reasonable. Various authors including Miech (1967), Bognar (1970), Hall (1972), and Hall and Sudbery (1972) studied this conjecture, in each case reducing the
exponent of N in (4.253) at the expense of restrictions on G, more precisely on the orders of the elements. These arise because the high moments Nm(G) = Exp
R(g)m
(4.256)
gEG
depend on the group structure : Bognar (1970) showed that this is the case if m >_ 4.
Erdos and Hall (1976b) obtained the following result. This disproves the conjecture of Erdos and Renyi, but there is still a large gap between what we know to be true and the heuristic model.
4.5 A conjecture of Erdds and Renyi
169
Theorem 4.24 Let A be fixed and NA/IoglogN
klog2 > (1+BlogloglogN)logN loglogN J)
(4.257) (4.258)
where B = B(A) depends on A only. Then with probability -+ 1 we have, for all g, that (4.254) holds.
Although there is some way to go before such results are applicable to, say, the discrepancy estimates in Chapter 6, the author believes that this is one of the promising avenues of further study.
5
Divisor density
5.1 Introduction The idea of divisor density is due to Hall (1978).
Definition 5.1 Let d be a (positive) integer sequence and 0 < z < 1. We say that d possesses divisor density z, and write Dd = z in this circumstance, if
i(n, a) = card{a : a E d, aIn} = (z + 0(1)) i(n) p.p.,
(5.1)
that is there exists a further sequence of asymptotic density 1 on which the
ratio i(n, d)/i(n) converges to z.
Thus a random divisor of a large random integer has probability z of belonging to d. This only makes sense if i(n) is large (indeed if z is irrational (5.1) requires i(n) -+ oo) and so we only require (5.1) to hold We stress that the o(1) term in (5.1) tends to zero as n tends to infinity (through a suitable sequence of asymptotic density 1), not as i(n) -+ oo. This latter condition would be stronger but too rigid for our purpose. We may of course assume if we wish that on the sequence of asymptotic density 1 referred to in our definition, we have r(n) -+ o0 or even 'c(n) > (log n)iog2-E
(5.2)
and we leave as an exercise for the reader to check that if z is not a dyadic rational then the weaker of these assumptions is obligatory. The definition is unusual for two main reasons: firstly the involvement of another, supplementary sequence of asymptotic density 1 and secondly
because it is not at all clear from the start that any sequence 4 exists having divisor density z, z * 0 or 1. Indeed, Erdos was initially foxed and 170
5.1 Introduction
171
wrote in a letter dated January 1976 that it should be possible to prove that whenever it exists, divisor density must be equal to 0 or 1. Erdos observed that no arithmetic progression other than Z+ (or Z+ \ [1,no]) has divisor density and nor does the sequence of squarefree numbers. (The reader should verify these observations.) However, the sequence
slo(z) = {a : loga < z(mod 1)}
(5.3)
(i.e. the fractional part of log a lies in [0, z]) satisfies Ddo(z) = z : this follows from Theorem 5.14 below. This was the first example of such a sequence, but is not logically the most straightforward. We construct a fairly basic sequence d below such that Dd = z, from first principles. (The reader may not care to take an example such as (5.3) on trust, fearing that the remainder of the chapter may be nugatory.) Once convinced by these examples Erdos immediately made a far-sighted conjecture about divisor density which we shall meet later.
Clearly if Dd = z > 0 then d is a Behrend sequence. Even when Dd = 0, it may be possible to show that there exists a sequence {zn} such that zn -* 0, and Zn >
i(n)
i(n d) ' znt(n) P.P.
which again implies c is Behrend. Thus we might have zn = (log n)-fl, n > 2, $ < log 2. This idea will be taken up in the next chapter.
We note two straightforward consequences of our definition. If d1 and Q2 are disjoint sequences such that Dd1 = zl and Dd2 = z2 then D(sll ud2) = Z1 +z2. If d' is the complement of d1 then Dc/' = 1-zl. We proceed to the construction of a sequence d1 (z) such that Dd1(z) = z. This cannot be too easy in view of Erdos' remarks quoted above. We shall see later that there is quite a simple result (Theorem 5.2) about the structure of an integer sequence with divisor density which immediately excludes sequences of the wrong form; and this will help to explain the `counterexamples' mentioned above. Let a E Z, b e Z+. We begin by showing that the sequence
d = {d : 1Z(d) = a(mod b)} has Dd = 1/b. We have (with e(O) = exp(2niO)), rc(n,d)
I
b-1
J:e r=0
r)
dln
Cr b(d)/// \\\
(5.5)
5 Divisor density
172
The inner sum is multiplicative and when n is prime it has absolute value 2 Icos(7rr/b)I. Let m(n) be the product of the prime factors p of n such that p2%n. Then from (5.6) we have T(n, d) -bi(n)
n-1
1
(n
b
m(n)
2 cos r=1
7rr w(M)
h
7C
Si(n) (cos )
w(M)
(5.7)
(We assume b > 2 else there is nothing to prove.) We have co(m(n)) = co(n) - w(mn)) and since the average order of o(men)) is a constant, for oo as n -> oo we have w(mn)) < (n)p.p., whence w(m(n)) has fi(n) normal order log log n. Hence the right-hand side of (5.7) is o ('c(n)), and Dd = 1/b as required. Now set, for i = 1, 2,3,..., ,cli
= {d
S2(d)
= {d :
2i-' JI52(d)}.
2'-'(mod 2i)} (5.8)
From the above, D,4i = 2'. Let the binary expansion of z be z = .£16283 ... (6i = 0 or 1).
(5.9)
We claim that
d = U{d :
Si = 1}
(5.10)
possesses divisor density Dd = z. This requires a little care since it is not true in general that an infinite disjoint union of sequences possessing
divisor density has divisor density equal to the sum of the individual densities; a gegenbeispiel is Rj = {j}, (j = 1,2,3,...) in which D-4j = 0
for every j but U-Vj = Z+. We therefore give a detailed argument employing the selection principle. Fore > 0 let R(6) denote the sequence of integers n such that i(n' ` ') T(n)
_Z >9
(5.11)
where sd and z are given by (5.9) and (5.10). We claim that if a(6) has zero asymptotic density for every e then Dd = z. To see this let X(n, 6) be the characteristic function of 4(6) so that by hypothesis
E
x(n,8) = o(x).
(5.12)
nSx
Let j e Z+. By (5.12) there exists xj such that x > xj implies
Ex(n,j-') <j-'x. nSx
(5.13)
5.1 Introduction
173
Put x(n) = x(n, j-1) for xj < x < xj+1. For these values of x we have E X(n) <- 1: x(n, J 1) < j-1x, n<x
(5.14)
n<x
because x(n, j-1) is an increasing function of j, whence
E X(n) = o(x),
x --> oo.
(5.15)
n<x
The sequence on which x(n) = 0 therefore has asymptotic density 1 and from the defintion of x(n) and (5.11) we have (C(n,d)/i(n)) -> z on this sequence, that is Dd = z. It remains to show that dR(c) = 0 always. There exists h so that -919283 ... Eh > Z -
1
2
s
(5.16)
and we truncate the union (5.10) at h. We can add divisor densities in a finite disjoint union whence r(n, d) > (z - 29 + 0(1)) i(n) p.p. Next, put
salt = U dj = {d :
S2(d) = 0(mod 2i)}.
(5.17)
j>i
Then Dd; = 2-i. Also there exists k such that (5.18)
and since k
.acU{S/i:9i=11Us4k
(5.19)
i=1
we have i(n, d) < (z + 1-29 + o(1)) p.p. - together with the lower bound already obtained this establishes that dd(c) = 0, whence DR = z. Thus for each z E [0, 1] there exists an integer sequence d which possesses divisor density z. We now state our first theorem. This easy result, already referred to above, is very useful for showing that a given sequence cannot have divisor density; indeed even in such cases as the squarefree numbers it provides a negative test which is as quick as a direct argument. Moreover we gain some insight into the structure of sequences with divisor density. Surprisingly this theorem made its first appearance rather late in Hall and Tenenbaum (1986).
174
5 Divisor density
Theorem 5.2 Let sd be an integer sequence. In order that d should possess divisor density Dd equal to z, it is necessary that for each fixed integer q we have Dd(q) = z, where
d(q) := {m E Z+ : mq E 21}.
(5.20)
As an example, let us dismiss the possibility that the sequence of squarefree numbers should possess divisor density, by taking q to be any square exceeding 1. d(q) is empty, so that z would equal zero, which is clearly false.
Proof We have sl(gig2) = d(gi)(g2) for every q1 and q2, hence we may assume that q is prime. Then
i(n, d(q)) = card{d : din, dq E c} = card{d : dqlnq, dq E sd}
= i(nq,d)-card{t : tInq, t E c, q%t} = r(nq, d) - i(nq-fl, d ),
(5.21)
where of IIn. Let us fix /3. By hypothesis, Dd = z whence the right-hand side of (5.21) is
(z + o(1)) i(nq) - (z + o(l))
(z + o(1)) i(n) p.p.,
for the n such that q# 11 n, which have asymptotic density q-#(1 - q-1). Hence if fi < fib where fib - oc sufficiently slowly, we have i (n, d(q)) _ (z + o(1)) i(n), that is Dd(q) = z as required. There is no sufficiency version of this theorem in the literature, and it is difficult to envisage a situation in which it would be easier to establish that c(q) possesses divisor density than that 4 itself does so. However it would be of some interest from a theoretical standpoint to determine minimal conditions on a sequence .2 c Z+ \ { 1 } such that if Dd(q) = z for every q e .2 then necessarily Dd = z. By Theorem 5.2 we may restrict our attention to primitive .2. (There is a mistake in Hall and Tenenbaum
(1986) at this point, where we stated that it would be sufficient to consider the case when .2 contains primes only.) An alternative proof of Theorem 5.2 follows from Theorem 5.9 below; Theorem 5.9 is also likely to be relevant to this problem about J. Let
Ed-' = oo. dES/
(5.22)
5.1 Introduction
175
Then
xEd-1 n<x
(5.23)
d<x dEsd
and the right-hand side of (5.23) is (z + o(1))xlog x = (z + o(1))E i(n)
(5.24)
n<x
if and only if 4 has logarithmic density &V = z. This might suggest some causal connection between divisor density and logarithmic densities, although of course we know from the examples already considered that
8sl = z cannot, in general, imply Dd = z. The truth is quite clear-cut. Theorem 5.3 Divisor and logarithmic density are independent. More pre-
cisely, given (z, w) E [0,1]2 there exists a sequence d such that bd = z, Dd = w, or if we wish, either one of these equations holds and the other density is not defined.
We shall deduce this from the second part (concerning -9±) of the following theorem. We include the first, negative, part of this theorem as a precautionary note: it might at first sight be imagined that these sequences should possess divisor density.
Theorem 5.4 Let z be a fixed real number. Then the sequence -9(z) = (d >- 3 : o(d) < 2 log log d + z
log log d}
does not possess divisor density. On the other hand the sequence -9+(z) = {d >- 3 : w(d) < 2 log log d + 0(d)
log log d}
in which 0(d) --> oo, has D-9+ = 1. If instead 0(d) -* -oo, the corresponding sequence being denoted by 2-, then D-9- = 0. We begin with some lemmas.
Lemma 5.5 Let 0< a < y < 2. Then
n n<x
log n
y0(n) In
(logd)
a
<"Y x(log x)2y-1,
where tha dash indicates that d = 1 is omitted from the sum.
(5.25)
5 Divisor density
176
Proof of lemma Denote the sum above by S. We invert summations, put n = md, and employ the inequalities
(log md)a « (log m)a + (log d)«,
(5.26)
E y)(')
(5.27)
w(log
<<
2w)3'-',
m<w
of which (5.27) is a special case of (0.38). We obtain S «y S(x, a) + S(x, 0)
(5.28)
where yn(d)
2x )Y_l+a
x
(log d)« d
d
E (log
d
r
(log 2r)Y-'+a
d5x
r<x/d
E(log 2r)Y-'+a ` ,
L (log d)
r5x
YS2(d)
d<x/r
« E(log
2r)Y-'+«
r Clog
r<x
2r l Y
1
a .
(5.29)
At this point we note that from our hypotheses both exponents exceed -1, whence we obtain S(x, a) «a,Y x(log
x)2y-1
which we insert into (5.28). This proves the lemma.
Lemma 5.6 Let fi(n) -- oo as n card{d > 1
Proof
:
oo. Then we have
d1 n, log log d < log log n - fi(n)} = o (i(n)) p.p.
Denote this quantity by 1(n, ). For these divisors of n we have log n > eo(n) iog a
and we apply Lemma 5.5 with y = 1 and a = 3, which implies that
E _(n, )d.)13 K X 2°(n)
n<x
whence for all but o(x) integers n < x we have L(n, ) < e fi(n)/420(n)
(5.30)
5.1 Introduction
177
We have i(n) > 20n), and for almost all n we also have f2(n) - co(n) < fi(n)/20log 2. Hence for x + o(x) integers n < x, 0(n, ) < e (n)l5i(n). This completes the proof. Lemma 5.7 The number of divisors d of n such that co(d) < 2co(n) + t
(5.31)
co(n)
is
(Go(2t)+o(1)+O where
Go(u) =
K2(n)
1
27 I-.
w(n))) wn
-
(5.32)
r(n)
e2/2dv
and the o(1) in (5.32) is a function of co(n). This result is uniform in t.
Perhaps the quickest proof is to treat w(d) as the result of Bernoulli trials on the squarefree kernel of n, and apply the Central Limit Theorem. We leave it to the reader to construct a proof to his taste. We begin by showing that for fixed z, the sequence
Proof of Theorem 5.4
9(z) does not possess divisor density. Let n > no(z) and co(n) = log log n + w
log log n.
(5.33)
Then if din and d E 9(z) we have co(d) < 2 log log n + z 2(w(n) - w
log log n
co(n)) + z VG -(n) + O ((1 + IzI (Iwi
+
w2
(5.34)
that is w(d) < Ico(n) +
(z - w + 0(1))
w(n).
Alternatively let din and ow(d) <
2
log log n + z
log log n - (log log 07'.
(5.35)
5 Divisor density
178
This is again equivalent to (5.35), and either d e 2(z) or if not, then log log d < log log n - fi(n)
where fi(n) -+ co. By Lemma 5.6 the number of such exceptional d is o (i(n)) p.p., and so negligible. For all the remaining divisors the condition d E -9(z) is equivalent to (5.35), and by Lemma 5.7 the number of these divisors is (Ga(2z - w) + o(1)) i(n) p.p.,
(5.36)
since we clearly have Q (n) - w(n) = of co(n)) p.p. The variable w defined
by (5.33) is normally distributed, whence from (5.36) D-9(z) does not exist.
We may replace z throughout this argument by a function 0(d) tending slowly towards +oo. The z appearing in (5.36) becomes 0(n). We require
that the error term in (5.34) is o
w(n)). We deduce that D-9+ = 1,
D-q- = 0 as claimed. This completes the proof. Proof of Theorem 5.3
We set
M = {b > 3 : co(b) < S log log b}.
(5.37)
By the Hardy-Ramanujan theorem we have d-4 = 0, and by Theorem 5.4,
D- = 1. Let 0 be the complement of . so that we have 6.1J = 0, SB' = 1. We can find a sequence c1 such that 5sl1 = z, or Sd1 does not exist, and a sequence 42 such that D cd2 = w, or Dd2 does not exist. We put
d=(d1na')u(d2na) which has the properties required. This proves the theorem.
We have now shown that for each z E [0, 1] there exists a sequence . such that Da = z, moreover this definition of density is quite distinct
from the densities with which we are more familiar. A sequence d possessing positive divisor density is necessarily Behrend; the converse is false (witness the sequence of primes). This provides a method (sometimes
the only method known) for establishing that certain sequences are Behrend.
In the next section we shall give several necessary and/or sufficient conditions for Dd to exist.
5.2 Necessary and sufficient conditions
179
5.2 Necessary and sufficient conditions
In the previous section we introduced the idea of divisor density and showed that it is non-vacuous. We now seek conditions which are either necessary or sufficient, and in one important case both, for a given integer sequence to have divisor density. We recall that an essential part of the definition requires the existence of a supplementary sequence of asymptotic density 1 on which z(n, d) =
(z + o(1)) i(n). This is unavoidable, but the condition can be eased in a rather surprising and useful way. We shall find that it is sufficient to have i(n, d) = (z + o(1)) i(n) on a sequence of logarithmic density 1. From a practical point of view this is rather important. If we want to employ Dirichlet series, with or without complex variable methods, to establish that a given sequence has divisor density, the series employed are almost certain to have abscissae of convergence uO = 1. We then need a Tauberian theorem to infer back from properties of the infinite series in the half-plane a > 1 to properties of the finite set of integers not exceeding
x. The most natural tool at this point will be the Hardy-LittlewoodKaramata theorem, which will leave a residual weight n'1 on the integer n, n < x. Therefore our conclusion will be about logarithmic density. We may compare what happens when we consider sets of multiples rather than divisor density. Here the classical theorem is that of Davenport and Erdos (1937), (1951), (Theorem 0.2) that .W(s) possesses logarithmic density. The pre-war proof of this theorem reproduced in Chapter 0 depends on the Hardy-Littlewood-Karamata theorem in just the manner described above. However in this case there is no possibility of a second step to asymptotic density since by Besicovitch's theorem (Theorem 0.1) may not exist.
These observations notwithstanding, we recall that the DavenportErdos theorem contains a second conclusion, that the lower asymptotic density of a set of multiples is equal to the logarithmic density 8t(sd). As we pointed out in Chapter 0, this means that &#(d) = 1, that is d is Behrend, already implies d#(sd) = 1. It was this fact which originally suggested the truth of the next theorem, (although it is not actually used directly in the proof). We shall employ the notation p.p.l. to mean `on a sequence of logarithmic density 1'.
Theorem 5.8 Let d be an integer sequence such that i(n, d) = (z + o(1)) i(n) p.p.l. Then 4 has divisor density D W = z.
(5.38)
5 Divisor density
180
Proof We deduce from (5.38) that 1
/ i(n,_)
2
= o(log x)
z
nsx n (` i(n)
(5.39)
and the essence of the proof is to deduce from (5.39) that 2
n<x
((d)
(5.40)
However the factor z(n)-2 is technically rather inconvenient and we consider the simpler sums ('C(n, d) - zi(n))2
L(x) n<x
n40(n)
(5.41)
and S(x)
4I n)n) (i(n,
d) - zz(n))2.
(5.42)
n<x
Since z(n) < 2n(n) (5.39) implies
L(x) = o(log x).
(5.43)
Suppose that we can prove that S(x) = o(x).
(5.44)
This implies (5.40). To see this, notice that since the average order of c2(n) - co(n) is a constant, for any fi(x) -* oo we have 2n(n)
-
< 2a(n)-w(n) < fi(x)
(5.45)
for all but o(x) integers n < x. Hence the sum on the left of (5.40) does not exceed (x)2S(x) + o(x) = o(x)
(5.46)
provided fi(x) -> oo sufficiently slowly. Of course (5.40) implies that
DsV=z. We therefore set out to deduce (5.44) from (5.43), and, denoting the characteristic function of a by x, we begin by defining
r(m) := E{(x(d) - z)(x(d) - z) : [d, d] = m}
(5.47)
5.2 Necessary and sufficient conditions
181
so that we have both L(x)
_
r(m)
= m<x m.40(m)
(5.48)
h.412(h)
h5x/m
and
r(m)
S(x) m<x
1
E
4s2(m)
(5.49)
40(h)
h<x/m
For each fixed y E (0, 2) we have 1
E yO(n) = C(Y) + O
(j-))og
n<x
x(1og
x
where C(Y) =
r(Y)
, H(Y) - -
\
y
1
1-P
x)y-1
(5.50)
1
(5.51)
I
P
(1
-p///
We put y = 4, and deduce from (5.48) and (5.50) that L(x) = 4C 14
f
J
x (log )' dR(t) + 0(1).
(5.52)
t
We truncate the sum (5.49) at xn and xl-n, (0 < 11 <
Since
jr(m)I < 3n(m) we have I r(m)I 40(m) m5xn
1
40(h)
3
K
14)
m
m<xn
h<x/m
x -3/4
0(m) x
(log m)
K x(log x)-3/4(llog x) 4
< ;1
3
X.
Also I r(m)I 40(m)
x' i<m<x
1
h
40(h) h<x/m
(1
(3)0(m)
n(h)
mh 0(h)
(4)
x
X)-1/4
h (log h
Therefore
S(x) = C
(I4)
xlog t)x x' n
Xn
3/4
i
dR(t)+O(11ax+x(logx)-1/4) (5.53)
5 Divisor density
182
where
R(t) '_
m
(5.54)
.'in( M)'
Let L(s) denote the Laplace transform of the function L(e°), moreover set
k(s) :=
J0
"0 e-SDdR(e°), (s > 0).
(5.55)
From (5.52) we have
L(s) =
00
e-S'L(e')dv
J0 = H (1)4R(s)s
514
+ O(s 1).
(5.56)
By hypothesis, L(e') = o(v) whence L(s) = o(s2) and, from (5.56), k(s) = o(s 3/4) as s -+ 0+. Put rl (m) = r(m) + P(m) so that rl (m) > 0 and rl(m)
R1 (t)
(5.57)
m.40(m)
m
is increasing, moreover from (5.50) with y = 43, we have
R1(t) = R(t) + 3 C (3) (log t) 4 + 0(1).
(5.58)
We also have, directly from (5.57),
k, (s) =R(s)+
1 -
(1 + s)4
1
p++s
E
1
3
;WI-+s
(4)
0(m)
(5.59)
3- H(4)
a
1 - 4pl+s
s4
(5.60)
P
as s -> 0+. Since k(S) = o(s3/4), we deduce from (5.59) and (5.60) that
k1(s) - H ( 4) s314, (s -> 0+)
(5.61)
and the Hardy-Littlewood-Karamata theorem (Lemma 0.4) yields R1(e°) -
H(4)va ,
IF(42)
(v --+ co).
(5.62)
5.2 Necessary and sufficient conditions
183
Comparison of (5.58) with (5.62) gives R(e°) = o(v4).
(5.63)
Now consider the integral
I (u, q) := j
0-n)n
(u - v)-3/4dR(e°).
(5.64)
ryn
Provided riu -+ oo, we obtain from (5.63), after integration by parts, that I(u, ) = 0(1W3/4.
(5.65)
We substitute x = e", t = e° in (5.53) and provided q log x -+ oo, we have
S(x) = C (4) xI(u,q)+O(g4x) x{o(1)q-3/4 + O(q 4 )}
(5.66)
by (5.65). If q -> 0 sufficiently slowly as x -> oo, the condition q log x -> oo
is satisfied and the right-hand side of (5.66) is o(x). This is (5.44), and the result follows.
The next result is a necessary and sufficient condition for a given sequence to have divisor density. The sufficiency part is particularly useful and will be the basis of several proofs in this chapter. The result is due to Tenenbaum (1982) and pre-dates Theorem 5.8, which appeared in Hall and Tenenbaum (1986). With hindsight, we may now use Theorem 5.8 to simplify the sufficiency proof given by Tenenbaum because we need only establish the (apparently) weaker condition (5.38)
to obtain Dd = z. Theorem 5.9 Let sad be an integer sequence and x be its characteristic function. Then a necessary and sufficient condition for Dsa1 = z is that, as x tends to infinity, we have
x(n) - z = of k<x
n4n(n)
log x).
(5.67)
n<x n-O(mod k)
We remark that Theorem 5.2 is an immediate corollary of Theorem 5.9.
If we restrict k to be a multiple of q, and put k = hq, n = mq, x = yq in (5.67) we obtain, for fixed q and y -> oo, X(mq) - z .'10(m)
= of
h
so that Dd = z . Dd(q) = z as required.
log Y)
(5.68)
5 Divisor density
184
We begin by defining, for a > 0,
Proof of Theorem 5.9
00
.
Z
.
sl) - zi(n)) (5.69)
nl+a4S!(n)
n=1
and we claim that Dd = z if and only if we have
f(a) = o(a-1) as a -+ 0+.
(5.70)
First, if Dsd = z then the sum S(x) defined in (5.42) satisfies S(x) = o(x), whence (5.70) holds, by integration by parts. Second, if (5.70) holds then, on setting a = 1/log x we have as x -* oo,
L(x) =
(T(n, -4) - zi(n))2 n4f)(n)
n<x
= o(log x)
(5.71)
from which we deduce (5.38), that is
i(n, d) = (z + o(1)) i(n) p.p.l.,
(5.72)
and D,sd = z follows from this by Theorem 5.8. (We proceed from (5.71) to (5.72) via (5.39). The step from (5.71) to (5.39) is analogous to that from (5.44) to (5.40) in the proof of Theorem 5.8.)
We require an alternative formula for f (a) and we introduce the functions g(a)
:_
fk(a)
:=
E nl+ 4n(n) 00
(4) a-114, (a
0+),
X(kt) - z
(5.73)
(5.74)
t1+Q4'1(t) ' t=1
where H is as in (5.51). We also notice that ml+a4i2(m)
=
1 - 4p1+a 1
kl+o4n(k) k Im
(5.75)
pI k
and we denote the product on the right by A (k, a). We have
f(a) _
1
n=1
nl+Q4f'(n)
= g(a) "0
J>(d) - z)(X(d') - z) d,d'In
(X(d) - z)(X(d) - z)
EL.
00
"0
"0
[d d'11+a40([d,d'])
d=1 d'=1
= g(a)
X(d) - z)(X(d') - z) d=1 d'=1
(dd')1+a4i2(dd')
k'+Q411(k)2(k, a) kl(d,d')
(5.76)
5.2 Necessary and sufficient conditions
185
by (5.73) and (5.75). We change the order of summation, writing d = kt,
d = kt' to obtain odn(
.f(a) = g(a)
U l+ 7
k)
fk(a)2.
(5.77)
k=1
We are now in a position to prove the sufficiency part of our theorem. We write X(n) - z (5.78) Ak(U) n4°(n)
n<e°
n=_O(mod k)
so that (5.68) reads 00
Ak(u)I = o(,/U), (u --+ oo).
(5.79)
k=1
We have -
CIO
-'
Ifk(a)I k1+640(k)
x
_
e audAk(u)
=
k=1
0
k=1 00
e °uAk(u)du
a T, k=1
0
w
e ° u E IAk(u)I du 0
=
k=1
(5.80)
O(a-1/2)
as a - 0+. From (5.73) and (5.74) we have I fk(a)I < g(a); moreover 0 < ,(k, a) < 1 whence (5.77) yields f (G) <_
g(a)2
-EIMO = o(a-1) k1+Q40(k) -
(5.81)
k=1
by (5.73) and (5.80). This is (5.70), and we have seen that Dsd = z follows.
Next, we remark on the proof of necessity: we may assume that (5.70) holds, whence from (5.73) and (5.77) we have 00
1(k, a) .fk(a)Z
= 0(a-314), (a -> 0+).
(5.82)
k=1
It is not difficult to check that we have 00
A(k, u)-' k1+a40(k)
k=1
0+).
(5.83)
5 Divisor density
186
and (5.82), (5.83) and the Cauchy-Schwarz inequality yield I.fk(a)I
o(U-1/2), (6
]r1+adO(k)
-' 0+),
(5.84)
k=1
that is, see (5.80), QudAk(u)
f o00 e
= o(a-1/2), (a
--,. 0+).
(5.85)
We are going to show that this implies (5.79) and our argument follows
the main lines of Karamata's proof of the Hardy-Littlewood-Karamata theorem. The details are not quite straightforward and we procced in stages. Firstly, we deduce from (5.85) that for each fixed polynomial P we have eQUP (eou)dAk(u) = 0(6-1/2).
(5.86)
0
We introduce the familiar function
0
0,
Q0(v)
v-1
(5.87)
e-1 < v< 1
and we approximate Qo in a certain sense by polynomials. More specifically, for each e > 0 we construct, by Weierstrass' approximation theorem, a polynomial Q(E)(v) such that
(0 < v < 1) if n (e ",
0< Qo(v) - Q(E)(v) < 1, Qo(v) - Q(')(v) < e
(5.88)
e-).
(5.89)
From (5.86) and (5.87) we have 0
00
I e°uQo(e°u)dAk(u) < where Ji(a,e)
k=1
J1(a,e) + J2(a,e)
f
eouQ(e)(eQU)dAk(u)
o(a-1/2)
(5.91)
00
J2(Q, e)
= 1: 1 o
<
J'
(5.90)
eou{Qo(e au) "0
- Q(E)(e °u)}dAk(u)
eau{Qo(e'u) - Q(E)(e Qu)}dA(u),
(5.92)
5.2 Necessary and sufficient conditions
187
where A(u)
(5.93)
ri4u(n) n<e°
From (5.88), (5.89) and (5.92) we deduce that J2(a,£) < E
E
in) ( a,,f,(n)
+ J3(0,£)
(5.94)
k=1
where 1+£
dA(u) = A
J3 (9, £)
fill,
(
1
a
-AC
1)
.
(5.95)
\ a
The first term on the right of (5.94) is « £a-112. Also A(u) - A-Ifu for a suitable constant A, whence from (5.95), J0, E) « £a-1/2 for 0 < a < ao(£). From (5.90), (5.91), (5.94) and (5.95) we now have Ak 11
1
«£a-1/2,
0 < a< a1(£)
(5.96)
k=1
that is, (5.79) holds - and this is just (5.68) as required. This completes the proof. One of the striking differences between the definitions of asymptotic and logarithmic densities on the one hand, and divisor density on the
other, is that in the former cases the existence of the density dd = z or 8d = z as it may be, is expressed immediately and simply by an asymptotic relation of the form
1: x(n)f(n) = (z + o(1)) E f(n)
(5.97)
n<x
n<x
in which x is the characteristic function of d and f is a simple weight, 1 for asymptotic, and n-' for logarithmic density; whereas in the latter case, no such relation appears in the definition of divisor density, and it is not obvious that (5.97), with some suitable f, either implies or is implied by the proposition D.sd = z. Returning to Theorem 5.9, certainly condition (5.67) has some resemblance to (5.97) but of course the imposition that we must add the absolute values of the separate sums arising from different k makes it much stronger: we cannot disentangle the z as in (5.97). However, an immediately corollary of Theorem 5.9 and the triangle inequality is that whenever Dd = z we must have x(n)i(n)
n<x
n4n(n)
_ (z + 0(1)) n<x
i(n)
00(n)'
(5.98)
5 Divisor density
188
so that we have a necessary condition with the same shape as (5.97). There is no loss in the triangle inequality when z = 0 or 1 and so in these cases (5.98) is also sufficient for Dd = z. For 0 < z < 1 this condition cannot be sufficient as it would assign a divisor density to every arithmetic progression, and to the sequence of squarefree numbers. In the original paper (Hall (1978)) on divisor density it was shown in Theorem 3 that Dd = z implies (5.97) for a certain positive multiplicative function f, different from that which appears in (5.98). At first sight this
discrepancy seems surprising, because we cannot readily interchange distinct multiplicative functions in a relation like (5.97); and although the explanation is simple enough we are led to an interesting general question about multiplicative functions which we shall discuss later. For the moment we concentrate on our next result which provides a necessary
condition for a sequence to have divisor density: if Dc = z then (5.97) holds for a large class of functions f ; and the two functions mentioned above are just particular examples. Such a class first appeared in Hall (1981), and a wider class was given in Tenenbaum (1982). The following theorem includes the latter result. Theorem 5.10 Let g : Z+ -> R+ U {0} be multiplicative, and such that
(i) E {Jg(p) - 1 I /p : all primes p} < co,
(ii) there exist fixed 01 and 02, with 01 > 0, 2 > 02 > 0, such that 0 < g(p°) < 0101, for all prime powers p°.
Then if d is an integer sequence having divisor density Dd = z, we have
E X(n)g(n) = (z + o(1)) n5x
nr(n)
n<x
g(n) nr(n)
(5.99)
where x is the characteristic function of d. The condition (5.99) is certainly not sufficient to imply Dsd = z when 0 < z < 1, for any g in the class. In general it is also insufficient when z = 0 or 1, an exception being the g implicit in (5.98). To justify our first remark, notice that for every g in the class there exists a constant co(g) such that
V n
n (n)
co(g)(log x) l .
(5.100)
5.2 Necessary and sufficient conditions
189
To see this, we observe that
fo
e-sud
f E g(n) 1 n<e°
1:
ni(n) J
00
g(n) nl+sti(n)
jl7
+...J1
g(p) g(p2) 1 + 2pl+s + 3p2+2s
P
2
P
C1+S(p)+S(p2)
\
2p
2p
3P2
as s -* 0+, with H as in (5.51). We apply the Hardy-LittlewoodKaramata theorem to deduce (5.100). Now let .1 be a set of primes for which
E{p l
: p E °Y} < ao
(5.102)
and let 5 be the sequence of integers with no prime factor in Y. Then x(n)g(n) n<x
ni(n)
wco ( g)(l ogx )2
(5 . 103)
where
w=
fJ 11++ /
g(p)
g(P2)
P
\
2p
3p2
+
\
l
I
(5.104)
/
and we can arrange 9 so that w takes any prescribed value z c (0, 1). Thus (5.99) holds, but Dd does not exist. Our second remark is easier to justify. We put g(n) = p(n)2 and Q the sequence of squarefree numbers, so that (5.99) holds with z = 1, or z = 0 if we take the complement of c. In Hall (1978) there appears the conjecture that there is no function f, multiplicative or otherwise, such that (5.97) standing alone implies that d possesses divisor density z. This conjecture has not received
much attention - it is after all negative - but it quickly leads into questions about multiplicative functions and their mean values which are of independent interest (indeed would seem to be fairly basic). We return to this question and its ramifications after the following proof. Proof of Theorem 5.10
n<x
In view of (5.100) we have to show that (x(n) - z)g(n) = of ni(n)
log x).
(5.105)
5 Divisor density
190
We are going to derive this from Theorem 5.9, i.e. (5.67), and to this end we write
-
4n(n)g(n)
h(k)
i(n)
(5.106)
kin
which defines the multiplicative function h via the Mobius Inversion Formula. From (5.106) we have h(p) = 2g(p) - 1 so that lh(p) - 11 = 2
Ig(P)-11 p < oo
(5.107)
and we also have, for v > 2 and every prime p, h(p')I <- 01(403)', 03 := max(1, 02).
(5.108)
We fix r > 1 such that (403)' < 8 and put s = r/(r - 1). Also we write X(n) -z nn(n)
Ak
(5.109)
nSx n=_O(mod k)
and we note that, uniformly for k < x,
Ak <
(log x) a
(5.110)
k4n(k)
The sum on the left of (5.105) is (5.111)
h(k)Ak k<x
and we apply Holder's inequality, with exponents r and s (as specified above) to obtain
1/r(E
E h(k)Ak
(Ih(k)IflAkl)
k5x
1/s.
IAk1)
k<x
(5.112)
k<x
By hypothesis, Dd = z and so by Theorem 5.9, (5.67) holds. Hence the second factor on the right of (5.112) is o((log x)1/2s). We have, by (5.108) and (5.110), I h(k)I r IAkI k5x
(log x) a E k<x
I h(k)I r
W(k)
1+
(log x)4 P
Ih(P)I'
4P
+
16 p2
+
« (logx)I f 1 + r l h(P) -11 + r(r -1)I h(P) -112 p5x
4P
8P
(5.113)
5.2 Necessary and sufficient conditions
since r < 2 (automatically) and so for u
191
0,
( 1 +u)' < 1 +ru+ 2r(r- 1)u2. The product on the right of (5.113) is bounded by (5.107) and so the first factor on the right of (5.112) is << (log x)1/2r, whence the sum on the left-hand side is o(,/l). This establishes (5.105) and completes the proof of the theorem. We return to the question whether or not the relation E x(n)f(n) = (z + o(1))E f(n)
(5.114)
n<x
n<x
in which x is the characteristic function of the integer sequence d can imply Dd = z. For simplicity we restrict f to be non-negative, but we do not suppose at this point that it is multiplicative. We must have 00
J (n) = oo,
(5.115)
n=1
for if the infinite series were convergent we could set sl = Z+ \ {no} for some no such that f (no) > 0: (5.114) would then hold with z < 1, whereas
Dd = 1. Next, we have seen that (5.98) implies Dd = z when z = 0 or 1, and so our question becomes nugatory unless (5.114) can hold for some z E (0, 1). The necessary condition for this is f (n) = o (T, f (m)JJJI, as n
oc,
(5.116)
m
moreover (5.116) implies that for every z, 0 < z < 1, there exists x or, what is the same thing, a sequence d for which (5.114) holds. Definition 5.11 We denote by F* the class of non-negative functions f satisfying (5.115) and (5.116), and by .W1.F' the sub-class of multiplicative
functions belonging to F*. For each f E F' and z E (0, 1), K(f,z) denotes the class of integer sequences d such that (5.114) holds.
Notice that if g satisfies the hypotheses (i), (ii) of Theorem 5.10 and f 1(n) = g(n)/ (nz(n)) then f 1 E JL *. Suppose now that there exists an f E .F' with the property that (5.114) implies Dd = z. By Theorem 5.10, (5.114) holds with f replaced by any of the functions f 1 described above, that is
K(f,z) c n{K(f1,z), all f1}.
(5.117)
192
5 Divisor density
The function f must be very oscillatory, because K(f,z) may not contain any sequence, such as the sequence of squarefree numbers or an arithmetic progression, which does not possess divisor density. It seems likely that this must admit some bizarre elements into K(f,z) which do not belong to the class on the right of (5.117).
Conjecture 5.12 There is no function f E F' such that (5.114) implies
Ds/ =z. Failing this, we make the following conjecture, which is a question purely concerning multiplicative functions.
Conjecture 5.13 Let e, f E .F' and K(f,z) 9 K(e,z) for all z E (0,1). Then there exists a constant y such that e(n) __ nv f (n).
(5.118)
Clearly if Conjecture 5.13 is true then any gegenbeispiel to Conjecture 5.12 lies outside
5.3 Slowly switching sequences
Initially Erdos believed that the relation r(n, Q1) = (z + 0(1)) i(n) p.p.,
(5.119)
could not hold for 0 < z < 1, but he was soon convinced that (5.119) was possible, by the example (5.7) and, more significantly, by the example
sad = {d : logd < z(mod 1)}.
(5.120)
At this point Erdos made the following conjecture, perhaps the most important in the development of this subject: let (i) uj+l > cvj, vj > cud for j E Z+ and some fixed c > 1,
(ii) d = U{Z+ n (uj, vJ : j = 1, 2, 3.... } (iii) Sd = Z. Then sad has divisor density D W = z.
Of course, if true, this includes (5.120) as a special case. At first sight the conjecture flies in the face of Theorem 5.3 which states that divisor density and logarithmic densities are independent: however we recall that Theorem 5.3 depends on Theorem 5.4, and we cannot impose this sort of restriction on ow(d) for the integers d contained in a relatively
5.3 Slowly switching sequences
193
long block such as (uj, vj] ; it is essentially a Poisson variable. Blocks of consecutive integers occur elsewhere in this book, e.g. in §1.3, where we are concerned with certain kinds of Behrend sequences: the point is that the multiplicative structure of the integers in a long block is in
effect random, or random enough; depending on the precise context the blocks can sometimes have quite moderate length. For example the blocks which appear in Corollary 1.10 are of type (u, u + u"] where a is arbitrarily small. A proof of the above conjecture of Erdos may be found in Hall (1978). Since then it has been realized that it is not so much the length of the blocks but their frequency which matters for a result of this sort, in other words we have to limit the number of switches in and out of c which can occur up to x by some function of x. If we extend the characteristic function of d onto R+ by setting 1
w xO
0
if w E U(uj, vj]
if w < ul or w E U(vj, uj+i]
(5.121)
then our condition becomes an upper bound on the total variation of x on [0, x]. The following is a result of this type. Theorem 5.14 (Hall & Tenenbaum (1986)) Let
(iii) d _ U{Z+ n (uj, vJ : j = 1, 2, 3, ...}
(iii) 8d = z, (iii) card{j : uj < x} < R(xn)
where q(x) -> 0 as x - oo, and R(x) is an increasing function such that R(x) < x and such that there exists a corresponding Stieltjes measure A(t) on [0, satisfying both 2]
4Pn) =
(iv) n<x
f
1/2
xi-`d2(t) + O(x/R(x)),
O
and
(v) 'Id),(t)l «t-1/4dt, (0 < t < z Then 4 has divisor density Dd = z. Notice that (iii) implies uj/j -+ co. Since
fJdw
Ea-i =J u;
u;
W
+
5 Divisor density
194
condition (ii) translates immediately into
fx (ii')
x(w) - z dw = o(log x).
At first sight the theorem looks rather technical, and it may be helpful for us to see immediately how the truth of Erdos' conjecture follows from it. We set H(4)t-1/4dt (5.122)
),(4)T(4)
with H as in (5.51). By (5.50), condition (iv) is satisfied with R(x) = (log x)7/4. We choose j(x) such that j(x) -+ 0, log x fG (q (X) log x)7/4, and
conditions (i)-(v) are all satisfied. By the theorem, Dd = z. The formula (5.122) is far from optimal, and we have Theorem 5.15 (Hall & Tenenbaum (1986)) There exists a constant c > 0 and a Stieltjes measure .1 such that conditions (iv) and (v) of Theorem 5.14 are satisfied with cl (log x) s
(5.123)
R(x) = exp (log log
Proof of Theorem 5.14
We write x(n) (') z n4
Ak
=
n<x n-o(mod k)
k4
(5.124)
°(k) Bk
where z
x Bk
m<x/k
m40(m)
<< (log k)
.
(5.125)
We assume that q(x) -> 0 as x -,. oo so slowly that condition (iii) above holds; and we set
K = K(x) =
xi-n(x).
(5.126)
For k > K, the trivial bound on the right of (5.125) yields
Bk «rl(x)'(logx)4.
(5.127)
We write 4(m)
M(x) :_
1
C4)
f xi-`do(t) + E(x) 0
(5.128)
5.3 Slowly switching sequences
195
say, and we have for k < K,
+ x/k x(uk) -
0(,(X) Bk =
a (log x)
z dM(u).
(5.129)
U
xn
We split the integral on the right into two parts Ji and J2 arising from the two terms on the right of (5.128). The first part is xlk
Ji
x(uk) - z
L
u
f
(1
-
0
j(1 - t)k`d(t)
z
(v) -IT -t
x Jk.1
d(5.130)
on substituting u = v/k. By hypothesis (ii'), there is a function e(x) 10 such that
Ix
x(w) - z dw « e(x) log x
(5.131)
w
and, by (5.131) and the Second Mean Value Theorem, the innermost integral on the extreme right of (5.130) is << (kxn)-`E(x) log x if, as we may assume, e(x) log x is an increasing /function of x. Hence
Ji < E(x) log x
J0
21
X -qt Id2(t)I
< e(x)q(x)-3/4(logx)4
(5.132)
by condition (v) above. Next, J2
x(uk) - z
Jx/k
u
dE(u)
/ x/k uk z x/k x( uk ) - E(u)1 + J x( z
u
xn
z
u
xn
E(u)du
x/k 1
+ I,
(5.133)
-E(u)dx(uk) u
after an integration by parts. By condition (iv), E(u) < u/R(u), where R is increasing, whence J2
du uR(u)
x/k
1
+
R(xn)
0
dx(uk).
(5.134)
We may assume R(u) >- (log u)7/4, so that the integral is convergent. Hence by hypothesis (iii), we have J2 = 0(1). We deduce from (5.127), (5.129) and (5.132) that Bk < q(x)1/4(log x)1/4 + E(x°)q(x)-3/4(log x)1/4 + 1,
(5.135)
5 Divisor density
196
and we may assume that q(x) - 0 as x -+ co so slowly that the right-hand side is o((log x) 4 ). Hence 1
JAk1
k<x
= o ((logx)4)kSxE k4°(k) o ((log x) z)
(5.136)
,
and we apply Theorem 5.9 to obtain our result. This proof is shorter and easier than the one appearing in Hall (1978),
which was limited to establishing the truth of Erdos' conjecture. The simplification comes from Tenenbaum's theorem (Theorem 5.9) and arises essentially because the sums Ak are linear in X. If we were to estimate the sum
(t(nd) r(n)
2
(5.137)
z
n<x
or its `logarithmic' variant (5.39) directly we should not be able to avoid awkward and lengthy calculations involving l.c.m. [a, a'] for pairs a, a' E d. Of course an upper bound for (5.137) is implicit somewhere within the proof - in the presentation here, Tenenbaum's method 'linearizes' the sum (5.39) and the step up to (5.137) is the content of Theorem 5.8. Proof of Theorem 5.15 We obtain the following more general result: for complex y, such that yJ < 1, there exists a Stieltjes measure AY on [0, z ] such that Id),Y(t)I KY t ReY,
0
2
(5.138)
and
yu(n) = J0 2 xl-`d2y(t) + DY (R(Xx))
(5.139)
nSx where R is given by (5.123).
We begin with Perron's formula, (Titchmarsh (1951) Lemma 3.12), with
x restricted to have fractional part 1: this restriction may be dropped in the final result. Because IyI < 1 this gives yn(n) n<x
=
1
+iT
,cc-iT
s
(s, y) s ds + 0
x 1T x)
(5.140)
5.3 Slowly switching sequences
197
where c = 1 + (I /log x), T is at our disposal, and
/
4(n)
1
(s,Y)=V'yns
(5.141)
n= 1
P
Let H(s, y) := exp j
l
ym EE mp m
(5.142)
P
Then H is analytic in the half-plane Re s > 2, and if connected sub-domain of this half-plane in which then for s E we have
is a simply has no zeros or pole,
C (s, y) = H(s,
(5.143)
Richert (1967) deduced from Vinogradov's method (Vinogradov (1958))
that
((a
+ it) <
tlloo(1-,,)311 (log
(5.144)
ItI)
uniformly for a > 2, Iti > 2. Theorem 3.10 of Titchmarsh (1951) then gives the zero-free region {a + it : a > 1 - 29(t)} where we may take 2))-2/3 (It, +3)) -113 9(t) = co (log(ItI + (log log with a suitable co > 0. (We assume 9(0) < 4.) We obtain by inserting a cut in the real axis from a = 1 to 1 - 29(0). In we have I I (s) I << (log(Itl + 2))' ,
Is - 1I >> 1
(5.145)
by (5.144), and we deduce that in the smaller region {a+it : a > 1-9(t)} we have Ilog ((s)I << log log(Iti + 3)
Is -11 » 1.
(5.146)
For this we apply the Borel-Caratheodory inequality (Titchmarsh (1939), p.174) to the function g(z) := log ( (1 + it + z), with concentric circles of radii r1 = 1 - a, r2 = c3r1/2 centred at z = 0. Let us assume Iti > 3. We have M(r1, g) <
2r1
r2-r1
A(r2, g) + r2 + r1
r2-r1
1(0)1
where M(r,g) := sup{Ig(z)I : IzI = r} , A(r,g) := sup{Reg(z) : IzI = r}. We have Ig(0)I K loglogltl, A(r2,g) < loglogltl by (5.145) and (5.146) follows.
Now we move the contour of integration in (5.140) to the contour
G comprising the curves a = 1 - 9(t), -T < t < 0 and 0 < t < T, horizontal segments 1 - 9(T) < a < c at t = ±T and a lacet from
198
5 Divisor density
so := 1 - 0(0) winding positively around s = 1. We have assumed (adjusting co if necessary) that 0(to) < 1, and we (approximately) optimize T by requiring that T = x&(T) or T = exp{c(log x)315(log log
x)-1f5},
(5.147)
say. The contribution to (5.140) from the error term O(T-lx log x) and all of G except the lacet is absorbed by the error term given in (5.139). We collapse the lacet to two real line segments joining so to a small circle of radius e, centre 1. Near s = 1 we may write C(s, y) =: h(s, y)(s - 1)-Y
(5.148)
where arg(s - 1) := 0 at s = 1 + E and, for Is-11 < 21 h(s, y) = H(s,y){1 + y(s - 1) +
}Y.
(5.149)
(The radius of convergence of the series for (s - 1)1'(s) is infinite but H is only analytic for Re s > 1; (s - 1)((s) 0 for Is - 11 < 14.13...). Since Re y < 1 the contribution to the integral from the small circle -> 0 as E -* 0 and so we may finally collapse the lacet to two line segments, on which we substitute s = 1 - t, to obtain a main term f 02 xi-`d.1Y(t)
(5.150)
in which d2y(t) =
sin
7ry
7r(1 - t)tY
h(1 - t, y)
(5.151)
for 0 < t < 9(to), d2y(t) = 0 for 0(to) < t < z. The factor sin iry arises in the usual way when we evaluate arg(s - 1) on the two edges of the lacet. The reader should compare (5.151) with (5.122), in which we may replace 1 by y, noting of course that f(y)F(1 - y) = it/sin icy. The error term in (5.139) is effectively the same as that in the Prime
Number Theorem. However the problems are different in that in the Prime Number Theorem we are committed to a main term lix whereas here we may choose, and theoretically optimize, the measure d)y(t). We are unable to make any tangible advantage of this apparent freedom at the time of writing. 5.4 Slowly switching sequences: a reformulation
We recall that a slowly switching sequence has the form
d = U{(uj, v;] n Z+ : j = 1, 2,3,...J
(5.152)
5.4 Slowly switching sequences: a reformulation
199
uj < x} has limited growth as a function of x. Let z E (0, 1) be arbitrary. Then provided uj < vj < uj+l for all j, as we where card{ j
:
assume, there exists an infinite class of continuous, strictly increasing functions f such that
f(ui)
= j,
j = 1,2,3,...
f(v3) = j+z, j=0,1,2,...,
(5.153)
where we have set vo := 0 for convenience. We may then write
d = {d : f(d) E (0,z](mod 1)}.
(5.154)
Often we write the condition on d as f(d) < z (mod 1): notice that we exclude f (d) - 0 (mod 1).
The natural choice for z is 8d if the logarithmic density exists. If it does not exist, or we choose z * 8d, f will be too oscillatory to serve any useful purpose. Evidently we can employ (5.154) to describe a given sequence or we can choose any strictly increasing f and regard (5.154) as defining 4. An example of this latter procedure is the sequence (5.120). If we do this, with the end in view that we shall be able to deduce that Dd = z from Theorem 5.14, it is clear that we need sufficient conditions on f for the sequence (5.154) to have Sd = z. Since the sequence {uj} will have broadly exponential growth it will be convenient to write f (w) = F(log w), where F is defined on R.
Theorem 5.16 Let F : R+ -+ R, F
oo, and there exist To such that
F'(t) > 0, t >_ To, F E C2 ([To, oo)). Let IF,(ttzI
dt = o(T),
T -+ oo
(5.155)
ITO
and F(t) = o(e`). Then the sequence
d = {d : 0 < F(log d) < z(mod 1)},
(5.156)
has logarithmic density z, for each z E (0, 1]. If in addition t3/5
F(t) = exp Co(l). (log t)1/5)
(5.157)
then d has divisor density z.
The second part of the theorem is a corollary of the first part together with Theorems 5.14 and 5.15. We may set F(t) = t", a > 0 or F(t) = (log t)fl, $ > 1 for example, and have Dd = z. Thus the sequence (5.120)
has Dd = z.
5 Divisor density
200
In the proof, it will be convenient to have F E C2(R), F'(t) > 0 for all t, with F increasing from z to oo on (-oo, oo). We may alter F, say for t < Tl, to achieve this without upsetting either conclusion. Proof of Theorem 5.16 We have to prove the first part. Let f (w) _ F(log w), and g : (z, oc) -> R+ be the inverse of f. For given x, put f (x), N = [fl. Then
Ed-1 = Elo (g(n+z) g \ g(n) n
+lo g
min(x,g(N+z)))
\
/
g(N)
dEd
+ O (1:
g(n)-1).
(5.158)
n
Since f (w) = o(w) we have g(n)/n -+ oo as n -> oo and the error term above is o(log x). We put h(u) = log g(u), 'Y(u) = [u] - u + 1 and we have N
h(n) =
f
h(u)du + h(N) -
W(u)h'(u)du.
(5.159)
2
There is a similar formula with h(u) replaced by H(u+z) and N by N-1. We subtract these, and deduce that the main term on the right of (5.158) is Nf
min{h(C), h(N + z)} -
h(u)du + 2 {h(N -1 + z) - h(N)}
J N-1+z
N
N-1
-
JT(u){h'(u + z) - h'(u)}du + JT(u)h'(u)du + 0(1). N-1
1
(5.160)
We approximate h(u), wherever it occurs, by h(C) = log x, and note that N-1
N
f(u){h'(u + z) - h'(u)}du < 2 f I h"(u)l du.
(5.161)
So (5.160) becomes Nf
C
/
z log x+ O
Ih"(u)I du + \I J 1
J N-1
Ih'(u)I du + 1
I
(5.162)
5.4 Slowly switching sequences: a reformulation
201
and since
Ih'(u)I < f Ih"(v)j dv + 0(1), (u < ) 1
this gives (z + o(1)) log x in (5.158) provided
J
h"(u)I du = o(log x).
(5.163)
1
We substitute u = f (w) so that the left-hand side becomes x
x
1 F"(log w)
d (w f,(w))-1
Jdg,(f(w))
w F'(log w)2
g (f (w))
dw
(5.164)
Uj
U1
since f (w) = F(log w). Finally we substitute w = et, x = eT to deduce from (5.155) that the integral in (5.164) is o(log x). Hence (5.163) holds and 8d = z as required. This completes the proof of Theorem 5.16. The sequence
sl = {d : (log log d) < z(mod 1)}
(5.165)
has logarithmic density if and only if /3 > 1, so that in this sense condition (5.155) is best possible; and the theorem shows that this sequence also has divisor density equal to z when /3 > 1. It is conceivable that when /3 < 1 the sequence manages to have divisor density: of course this would not necessarily be equal to z. We may rule this possibility out, for z E (0, 1),
by reference to Theorem 5.10. The proof is by contradiction. Suppose then that s (as in (5.165)) has Dc/ = w, and set g(n) = 1 identically in the theorem, so that by (5.99),
n<x nEd
ni(n)
(w + 0(1)) n<x
ni(n)
(cow + 0(1))
log x
(5.166)
for a suitable (positive) constant co. Let m - oo through Z+ and associate with each m, the numbers x, u, v such that (log log x)fl
=m+z
(log log u)fl
=m = m+1.
(log log v)fl
(5.167)
By hypothesis, /3 < 1 and 0 < z < 1 whence
log (x) >> log x, u
log(') >> log x, x
(5.168)
5 Divisor density
202
and we deduce from (5.166), substituting x = u, x = v, that both 1
u
,(n )
= (cow + 0(1)) (
log x -
log u)
(5.169)
= (cow + 0(1)) (
log v -
log x).
(5.170)
nE.d
and
2
ni(n)
x
For u < n < x we have (log log n)# < z (mod 1) by (5.168), and (5.169) implies that w = 1. For x < n < v we have (log log n)$ > z (mod 1), and (5.170) implies that w = 0. This is the required contradiction and the proof is complete. It therefore seems that the first condition, (5.155) of Theorem 5.16, is realistic, and the theorem is a useful generator of sequences with divisor density. The status of condition (5.157) is less clear.
6
Divisor uniform distribution
6.1 Introduction Let f be a real-valued function defined on the positive integers. We shall be concerned in this chapter with the distribution (mod 1) of f(d) as d runs through the divisors of a large integer n. We are therefore interested in the counting function card{d :
d1n.
0 < f(d) < z(mod 1)},
(6.1)
where 0 < z < 1. We confine our attention to uniform distribution, and we ask whether the cardinality in (6.1) is approximated by zT(n), and if so then how good is the approximation. We put
d(z) = d(z; f) = {a : 0 < f (a) < z(mod 1)}
(6.2)
so that we may write this cardinality more simply as r (n, d(z)) and we define the discrepancy
A(n; f) =
sup
IT (n, d(z)) - T (n, d(w)) - (z - w)T(n)I .
(6.3)
O<w
Definition 6.1 (Hall (1976)) The function f is divisor uniformly distributed if and only if for every s > 0, we have 0(n;f) < eT(n)p.p.
(6.4)
An equivalent definition is that on a suitable sequence of asymptotic density 1 we have 0(n; f) = o (T(n)).
(6.5)
Although the formal definition of divisor uniform distribution was given in 1976 it had been shown previously in Hall (1974) that the 203
6 Divisor uniform distribution
204
function f (d) = log d has this property, moreover that for this particular function we have A(n;f) < i(n)z+E p.p. (6.6) There is a warning note for the reader at this point. The function log n (n E Z+) is not uniformly distributed (mod 1) in the traditional sense; and we shall soon have other examples of functions f such that f (n) is not uniformly distributed (mod 1) for n E Z+ but nevertheless f is divisor uniformly distributed. These functions present themselves naturally. In the opposite direction we shall construct a function which does possess the former property and not the latter. Uniform distribution and divisor uniform distribution are therefore independent, but related, properties. Let us set z = 1 in (6.3) and apply (6.5). We deduce that if f is divisor uniformly distributed then for every w e (0, 1) we have i (n, d(w)) = (w + 0(1)) i(n) p.p.,
(6.7)
equivalently Dd(w) = w. That is, a necessary condition for f to be divisor uniformly distributed is that the sequence d(w; f) should have divisor density w for every w E (0, 1). This condition (which also holds, trivially, for w = 1) is also sufficient. Let Dd(w) = w, 0 < w < 1 and let q E Z+. Then for 1 < 1< q we have Dd(l/q) = l/q, whence IT
((-))
qi(n) < !i(n), 1 < l < q, p.p.
(6.8)
Because (r(n, d(w)) is a non-decreasing function of w, we may deduce from (6.8) that IT (n, d(w)) - wz(n)I <
2
z(n),
0 < w < 1, p.p.,
(6.9)
q
and so (6.4) holds, with s = 4/q. Since q is arbitrary, we conclude that f is divisor uniformly distributed. The ideas of divisor uniform distribution and divisor density are closely connected: we note that divisor density was defined later (Hall (1978)), so that examples of sequences possessing divisor density, in particular that in (5.3), were already available when the definition was made. In studying divisor uniform distribution we may distinguish, broadly, two
approaches: the application of results about divisor density, and the direct method, employing classical techniques from the theory of uniform distribution. The first approach yields, very quickly, a wide class of divisor uniformly distributed functions but does not readily give explicit upper bounds for the discrepancy 0(n; f ). For this task the second approach is
6.2 Applications of divisor density
205
almost obligatory; the reason for this will be explained at the beginning of §6.4.
Before proceeding further, let us notice a conclusion which may be drawn from (6.6). By the Hardy-Ramanujan theorem, (6.6) implies that for f (d) = log d, we have A(n;f) < (log n)log2+E P.P.
(6.10)
whence we may assert that, provided p < log 2, almost all integers n i possess a divisor d such that 0 < log d < (log n)-/(mod 1).
(6.11)
Hence for the same values of p the sequence
IC// = {a : 0 < log a < (log a)-/(mod 1)}
(6.12)
is Behrend. This result should be compared with Theorem 4.13, which
shows that c/ is Behrend if and only if µ < log 2 (and is even more precise). Now (6.10) is not best possible, and we give a better result later. However it is certainly not known whether or not we have A(n; log) < (log n)E p.p.,
(6.13)
indeed this is one of the longstanding open problems in the area; and we should need (6.13) to hold to be able to infer that d/ is Behrend for every y < log 2 by this method. Of course a bound for the discrepancy of any distribution contains more information than the existence of points in short intervals; we may be able to construct Behrend sequences from bounds for discrepancy but such constructions should be viewed as corollaries rather than the main objectives of this theory.
6.2 Applications of divisor density
In this section we show that various functions are divisor uniformly distributed, as a consequence of results in the last chapter. Theorem 6.2 Let F E C2(R+) be increasing from some point onward, and for a suitable To let T
J
if
!F'(t)2 dt = o(T), 0
T , oo
(6.14)
6 Divisor uniform distribution
206
moreover let t3/5
F(t) < exp
(6.15)
(O(1)(logt)1/5)
Then f (d) = F(log d) is divisor uniformly distributed.
This is a corollary of Theorem 5.16, which establishes that the sequence d(z) in (6.2) has divisor density z for each z. It follows that the functions (i) (log d)", a > 0 (ii) (log log d)#, /3 > 1
are divisor uniformly distributed. Moreover the condition on /3 is sharp, by the argument following (5.165). We may also prove this directly as follows. Let 1 < u < 2. By Dickman's theorem, the integers n for which
P+(n) > nl/n have asymptotic density log u. Half the divisors of such an integer have the form d = P+(n)m, and for these d, and 0 < /3 < 1, we have
(log log d) =
log log P+(n)) B + O
log m log n(log log
(log log P+(n)) fl + O C1 -u1)
,
(6.16)
because m < n/P+(n) < nl-u. If u is sufficiently close to 1, we may deduce that (log log d)O E In for at least r(n)/2 divisors of n, where In is an interval of length < 3. Hence (log log d)# is not divisor uniformly distributed.
We shall see later in this chapter that if f is divisor uniformly distributed but has a suitably small derivative, this argument will give a useful lower bound for the discrepancy.
There has been a great deal of work on uniform distribution of sequences {F(n) : n E Z+} (see e.g. Kuipers and Niederreiter (1974)), and we take advantage of this in the next theorem. Hall (1978) (Theorem 1, Cor.) proved that if F E C'(R+) satisfies the conditions (i)
{F(n) : n E Z+} is uniformly distributed (mod 1),
(ii)
F'(x) -> 0 as x -> oo,
(6.17)
then the sequence
d(z) = {a : F(loga) < z(mod 1)}
(6.18)
has divisor density Dd(z) = z, 0 < z <- 1. Hence the function f (d) = F(log d) is divisor uniformly distributed. Hall and Tenenbaum (1986)
6.2 Applications of divisor density
207
(Theoreme 6, Cor. 1) showed that we may take, more generally, f(d) = F (0(d)) where 0 satisfies suitable conditions (not quite stated precisely). We develop this idea a little further here. It is more convenient to work with a function of log d.
Theorem 6.3 Let F satisfy the two conditions in (6.17) and f (d) _ F (g(log d)), where g is such that
(i) g'(u)/g(u) >> 1/u, (this condition need hold for u > uo only),
du « U, (ii) I u I g(u)gif(u)I g(u)
U
oc,
o
and
u3/5
(iii) g(u) < exp
u
co.
Then f is divisor uniformly distributed.
Thus we may take g(u) = u", a > 0 or g(u) = exp(u#), 0 < /3 < 3/5. Condition (iii), like condition (6.15) in Theorem 6.2, goes back to Theorem 5.15, and ultimately depends on the zero-free region for the Riemann (-function. Condition (ii) is a quasi-convexity condition: if g" is ultimately of fixed sign it is weaker than condition (i). Proof of Theorem 6.3 For u > uo, g has an inverse which we denote by h. We put vo = g(uo) and we note that conditions (i) and (ii) translate into 01)
hv)
«v,
v>vo
(ii') f Ivh"(v)I dv «h(V), V
V
oo.
vo
We shall have to consider the second forward difference of the function h, and we note that m+2
h(m + 2) - 2h(m + 1) + h(m) = I
(1 - I v - m -1I )h"(v)dv.
(6.19)
m
It follows that we have, for integers M,
mlh(m+2)-2h(m+1)-h(m)I vo<m<M
<J
M+1
vIh"(v)Idv
vo
<<
h(M + 1)
<<
h(M).
(6.20)
6 Divisor uniform distribution
208
The first inequality follows from (6.19), the second from condition (ii'), and the third from condition (i'). After these preliminaries, we are able
to follow the proof in Hall (1978) fairly closely. Let z E (0, 1) and < min(z,1 - z). We consider the sequences
sl+ = {a : sad-
F([g(log a)] < z + (mod 1)} F([g(log a)] < z - (mod 1)}.
= {a :
(6.21)
Our first objective is to prove that z + g, and we restrict our attention to sad+, writing z + = w for convenience. We denote by x(n) the characteristic function of the sequence {n F(n) < w(mod 1)} and :
we put
=
gy(m)
1 : m < g(log a) < m + 1 I ,
M=
(6.22)
[g(log x)].
(6.23)
We have from (6.21)-(6.23) that
E
1
a5x aE W+
= E x(m)2(m) + O (2(M))
a
(6.24)
m<M
and, by hypothesis that F is uniformly distributed, we also have 1: x(n) = wm + Emm,
(6.25)
n<m
where Em -* 0 as m -> oo. By partial summation, we deduce from (6.24)-(6.25) that 1
= w log x + O (2(M) + EMM2(M))
a<x aEd+
+O(
(6.26) m<M
From (6.22), for m >_ vo, 1
2(m) =
{a
:
eh(m) < a < eh(m+t) }
= h(m + 1) - h(m) + O(e him)).
(6.27)
Condition (i') implies that h(m + 1) - h(m)
_
m+1
«
m+' dv
h'(v)dv << h(m + 1)
m
m
h(m+ 1) m
K
h(m) m
v
(6.28)
6.2 Applications of divisor density
209
Condition (iii) implies that h(m) >> (log m)513 and together with (6.27) and (6.28) this yields m2(m) << h(m),
m > vo.
(6.29)
Hence the first error term on the right of (6.26) is (M-1 + EM)h(M) = o(log x)
(6.30)
because h(M) < log x by (6.23). As for the second error term, we have from (6.27),
2(m + 1) - 2(m) = h(m + 2) - 2h(m + 1) + h(m) + 0 (-),
(6.31)
using the lower bound h(m) >> (log m) 3 , whence by (6.20),
E m I2(m) - 2(m + 1)I << h(M) + 1.
(6.32)
vo<m<M
It follows that the second error term in (6.26) is o (h(M)) = o(log x), and
ad+ = w = z + as required. Clearly Sd- = z -
as well.
Next, we notice that Q+ and d are slowly switching sequences, in the sense of §5.3: the switches can only take place when g(log a) increases
through an integer value and by condition (iii), Theorem 5.14 applies with the R given in Theorem 5.15. Therefore DJ-! = z ± , that is both
i(n, d ) = (z + + o(1)) i(n) p.p. By hypothesis, F' --> 0. Hence there exists ao = all a > ao we have
(6.33)
F, g) such that for
IF (g(log a)) - F([g(log a)])I <.
(6.34)
Therefore for all n, we have
i(n, d-) - ao <,r (n, d (z)) < i(n, d+) + ao
(6.35)
d(z) = {a : 0 < F (g(log a)) < z(mod 1)}.
(6.36)
where
Now consider the integers n < x. We can let = fi(x) -> 0 as x -> oo sufficiently slowly so that ao, as above, does not exceed og x. For all but o(x) exceptional integers n < x we have i(n) > (log x)3/5 by the Hardy-Ramanujan theorem, whence ao < r(n)5/6. From (6.35), we have, with o(x) exceptions, 5/6 IT (n, d (z)) - zt(n)I < ( + 0(1)) T(n) + t(n),
(6.37)
6 Divisor uniform distribution
210
whence Dd(z) = z. Since this holds for all z e (0, 1) (and trivially for z = 1), we conclude that f (d) = F (g(log d)) is divisor uniformly distributed. This completes the proof of Theorem 6.3.
Next, we construct an example to show that the sequence If (n) n c Z+} may be uniformly distributed (mod 1) and yet f is not divisor uniformly distributed. This is modelled on the proof of Theorem 5.3 :
(that divisor and logarithmic density are independent). Put f (n) =
I
.,/n,
if w(n) > 2 log log n
1
else.
(6.38)
n c Z+} is uniformly distributed, so is {f(n) n E Z+} p.p. On the other hand, by Theorem 5.4 we have because f (n) = Dd(z; f) = 0, with Q(z; f) as in (6.2), for every z E (0,1). Hence f is not divisor uniformly distributed. Since { i i
:
:
The theorem which follows is a rather brisk corollary of Theorem 5.8. It is potentially very useful because logarithmic density is often technically easier to cope with than asymptotic density.
Theorem 6.4 In order that f should be divisor uniformly distributed it is sufficient that we have 0(n; f) = o(r(n)) on a sequence of logarithmic density 1.
Proof On this hypothesis, (5.38) holds for si = c(z; f ), (0 < z < 1), whence by Theorem 5.8, Dd(z ; f) = z. Hence f is divisor uniformly distributed.
The last theorem in this section is a necessary and sufficient condition for a function f to be divisor uniformly distributed. It is due to Tenenbaum, and appeared in the same paper, Tenenbaum (1982), as Theorem 5.9. It is motivated by Weyl's criterion for the uniform distribution (mod 1) of an integer sequence, in that it involves the functions e (v f (n)), for each fixed v c Z+. We employ the familiar notation e(x) = exp(2xix). Theorem 6.5 A necessary and sufficient condition that f should be divisor uniformly distributed (mod 1) is that for each fixed v E Z+ we should have
E k<x
e (v f (n)) n<x n-0(modk)
n4°(n)
= of log x).
(6.39)
6.2 Applications of divisor density
211
Proof We begin with necessity. Let f be divisor uniformly distributed, so that for each z e (0, 1] we have D{a : 0 < f (a) < z(mod 1)} = z.
(6.40)
For k e Z+ and each z, put (Dx,k(z) _
ox { n41(n) g n<x
_
: 0 < f(n) S z(mod 1)}
.
(6.41)
n=_o(modk)
Then by Theorem 5.9 and (6.40) we have, for each z 'xk(z) - ztDx,k(1)I = o(1).
(6.42)
k<x
For each v E Z+ we have JO1
e(vz)d'x,k(z) = 2itiv J 0/
e(vz){'x,k(z) - z(1x,k(1)) dz.
(6.43)
The left-hand side of (6.42) is << 1, whence we may apply Lebesgue's dominated convergence theorem to (6.42) and (6.43) to obtain, for each v, 1
e(vz)d I ,k(z) = o(1),
:
--+
(6.44)
k
which is (6.39).
Next, we prove sufficiency. Let (6.39) hold for each fixed v e Z+. We are going to show that for each zl and z2 such that 0 < zl < z2 < 1 we have, for almost all n, that card{d : dl n, zi < f (d) < z2(mod 1)} < (z2 - zi + 0(1)) i(n).
(6.45)
The opposite inequality will then follow by considering the complementary set. We employ Fejer means, defined as follows. Let p(t) be real, integrable and periodic on R with period 1, and its Fourier series be
ap(t) - >
,
cve(vt).
(6.46)
Then the Nth Fejer mean CO, gyp) of T is the (real, c_v = c,,) trigonometric polynomial N
aN(t, (p)
=
(i - N
cve(vt)
v=-N
N(u){Ot + u) + T(t - u)}du, 0
(6.47)
6 Divisor uniform distribution
212
in which FN(u) is the familiar Fejer kernel
.F NW _
sin 2 Nnu
(6.48)
N sine 7tu
Let qp be the characteristic function of an interval I (mod 1). From (6.47)-(6.48), CO, (p) > 0 for all t, moreover if t c I and the distance from t to 01 is at least , then we have 6N(t, gyp) > 1 - 2 f N(u)du > 1 I
1
2Ni; .
(6.49)
We may assume that z2 - z1 < 1 in (6.45), else there is nothing to prove, and we suppose 2 < 1 (z2 - zl) and let I be the interval (z1 - , z2 + ) on the torus T = R/Z. Then the cardinality in (6.45) does not exceed
AE6N(f(d),(p), A= I
(6.50)
d in
We require a (straightforward) generalization of the sufficiency part of Theorem 5.9, in which x, the characteristic function of S, is replaced by a weight function w : Z+ - [0, 1], in (5.68), and the conclusion is that we have
E w(d) = (z + o(l)) i(n) p.p.
(6.51)
dIn
We shall apply this with w(d) = UN (f (d), cp) and z = z2 - z1 + 2 = co in (6.47). We have to show that the hypothesis of the modified Theorem 5.9 holds, viz.
E k<x
6N(f (n), (p) - Co
= of log x).
n4fl(n)
n<x n=_O(mod k)
By (6.47), the left-hand side of (6.52) is
( (c N
k<x
n<x n=_O(mod k)
4Sn)
(i_I) ///
v#0
N
e (vf(n))
<2E(1-N)IcvIE v=1
e (vf (n))
v=-N
k<x
n<x n=O(mod k)
n.411(n)
= of log x)
(6.52)
6.3 Weyl sums and additive functions
213
by the hypothesis (6.39). Hence (6.52) holds, and implies (6.51), that is
E orv(.f (d), (p) = (z2 - zi + 2 + o(1)) i(n) p.p.,
(6.53)
din
from which we deduce that card{d : d1n, zl < f(d)
/
1
2ND
I
(z2 - zl + 2 + 0(1)) i(n) p.p.
(6.54)
The Fourier coefficients cv are uniformly bounded, indeed acv < 1 from (6.46) since 0 < cp < 1. We may therefore let N = N(x) -* oo sufficiently
slowly so that (6.52) holds, and put c = 1//. Hence (6.45) holds, and this is all we need. By means of Theorem 6.5, it was shown in Dupain, Hall and Tenenbaum (1982) and Hall and Tenenbaum (1986) that the functions f(d) =
0 irrational, and f (d) = d", a > 0, a Z are divisor uniformly distributed. These results do not follow from Theorems 6.2, 6.3 or 6.9 (below), because f increases too fast: Theorem 5.15 is insufficient for us to cope with the weights yn(n) by partial summation. No estimates for the discrepancy were given in the above proofs, but recently Tenenbaum has obtained rather strong results of this type, which are to be published elsewhere. For this reason we do not pursue these quick f here, and refer the reader to the literature and Tenenbaum's forthcoming paper (199y). Od,
6.3 Weyl sums and additive functions
A natural approach to questions about uniform distribution (mod 1) is to consider the associated Weyl sums. Accordingly for real-valued arithmetic functions f and v E Z \ {0} we define S-, (n; f) = E e (v f (d)).
(6.55)
dIn
Since S_v(n; f) = S,(n; f) we shall for the most part be concerned with v E Z+.
In this and the next section it is to be understood that when we write card{d : din, w < f (d) < z(modl)}, we count divisors for which f (d) - w
or z(modl) with weight z each in the cardinality. There is a well-known quantitive upper bound for the discrepancy in terms of Weyl sums due to Erdos and Turan (1948). In the present context this reads as follows.
6 Divisor uniform distribution
214
Lemma 6.6 Let L(n; f) be as in (6.3) but with the end-point rule above. Then we have, uniformly for N E Z+,
0(n;f) << , n) +
y IS,(n;.f)l.
(6.56)
v=1
This provides us with a standard method to show that f is divisor uniformly distributed, and moreover to estimate the discrepancy, provided we can find p.p. bounds for the Weyl sums. The easiest case, which we study in this section, is when the arithmetical function f is additive, for then S.(n; f) is multiplicative. We begin with a theorem of Katai (1976). Theorem 6.7 A necessary and sufficient condition that the additive function f should be divisor uniformly distributed is that we should have
P
n
(6.57)
Ilvf(p)Il2 = 00
for every v c Z+, where Ilxll = min(Ix - ml, m E Z.
We note that there is a mistake in Katai's statement of his theorem, in that he has 2v instead of v in (6.57). (The redundant 2 makes its appearance in the formula for gm(p) on p.210 of his paper.) Proof We begin with sufficiency, deducing from (6.56) that
o(n;f)
<x
x +E 1 N
ti(n)
v==1 v
E IS,(n;f)l
(6.58)
T(n)
n<x
We apply the Halberstam-Richert inequality (Divisors, Theorem 01) to the inner sum, which is therefore
«xH1 1-p 1
p <x
/\\
1+
IS"(p;f)I 2p
+
(Sv(p2;f)l + .. . 3p2
//
< x f l+ Icon Irvpf (p)I p<x
\\\
-
estimating IS,(pr; f )l < r + 1 trivially, for r > 2,
K x ex:)
1- ICosnvf(p)I p
P<x
xexp -41: 1 Ilvf(p)II' p<x
p
.
6.3 Weyl sums and additive functions
215
By hypothesis (6.57) holds, whence each inner sum on the right of (6.58) is o(x), and we may let N = N(x) -- co in such a way that the sum over v is also o(x). Hence 0(n; f) = o (i(n)) p.p. by Markoff's inequality. We give a slightly longer and more complicated proof of neccesity than Katai's; this gives a little extra information. We begin with the identity 2
1.1 0
Csin7rvt)21S,(n;f)12,
du =
1 - ti(n)
(6.59)
J
7rv
*O
dIn
u-z
of which the proof may be left to the reader. We denote the integral on the left by A2(n; f, t), and we observe that for all t, A2(n; f, t) < 0(n;
f)2.
(6.60)
For a > 1, we have °°
7-l
T IS,.(n;f)12 n°i(n)2
1 + S"(P;f)12/ 4pv
P
n=1
p
(a) 77
1+
IS"(p;f)12 -4
1
)
(a) r1+( ISv(p;f)1 - 2 x(26) 11
X26)) r 1 Since p-' < 2,
1
2PU
P°
p C1- 2pQ
lvf(p)12)
l /
.
(6.61)
the factors in this product exceed 1 -
11 of (p) II
37r2/32 > 0. Now suppose that for some v, say v = µ, the series in (6.57) is convergent. Then from (6.61) we have, uniformly for a > 1,
0 n-1
IS,u(n;f)12
n°i(n)2
»µ (6), yy
(6.62)
whence from (6.59), we have
0 A2(nr,fzt)>t"' mcjnj 00
c(a),
a>1
(6.63)
n=1
for any t such that pt is not an integer. By hypothesis f is divisor uniformly distributed and 0(n; f)/i(n) -> 0 on a sequence of asymptotic,
6 Divisor uniform distribution
216
and hence logarithmic density 1. (Of course there is no distinction, by Theorem 6.4.) Therefore 0(n; f)2 A1i(n)2 = o G (a)),
a - 1+
(6.64)
n=1
and (6.63)-(6.64) are in contradiction by (6.60). Hence the series (6.57) diverges for every v as required. This proof raises a question about the status of the proposition A2(n; f, t) = o (ti(n))2 P.P.,
(6.65)
whether or not f be additive. If t is rational then (6.65) does not imply that f is divisor uniformly distributed. For example if t = h/k then the distribution of f(d) (mod 1) might be, approximately card{d : d1 n, f (d) < z(mod 1)} = (z + 2 sin 2nkz)i(n),
(6.66)
where 2nk IAI < 1. If t is irrational, then (6.59) and (6.65) imply that for every v E Z+, 00
ISv(n;f)IZ
na,r(n)2 = o(4(a)), (a -> 1+).
(6.67)
n=1
We put a = 1 + (I /log x), and apply Cauchy's inequality which yields, for each fixed v, (n; f E I S,,ni(n)
)I
= o(log x),
x -> oo.
(6.68)
n<x
Hence f is divisor uniformly distributed, by the Erdos-Turin inequality (Lemma 6.6) and Theorem 6.4. Perhaps the simplest additive function is f (d) = log d. Either of Theorems 6.2,6.3 show that it is divisor uniformly distributed (the latter slightly less obviously). To apply Katai's theorem we need that
Epivlogp112=o0, vEZ+
(6.69)
P
which is an easy deduction from the Prime Number Theorem: the more precise result E 1 11 v log p 112
= 11) log log x + O (log log(3 + v))
(6.70)
p5x P
follows from Lemma 6.15, see (6.153). If we inserted (6.70) into the proof
6.3 Weyl sums and additive functions
217
of Theorem 6.7 we should obtain a p.p. upper bound for the discrepancy from (6.58), which in this case reads 0(n; log) n<x
T(n)
« x(log
(6.71)
log log X.
This leads to a weaker result than (6.6) because 3 < bypass the inequality
1- Icositvf(p)I >-
i
log 2, but if we (6.72)
4IIvf(P)II2
which is wasteful when II v f (P) II < 2, we may replace the 3 in (6.71) by The next theorem, proved independently by Hall (1975) 1-n> 1log2.
and Katai (1976), gives a stronger result, obtained by using a weight different to 1/i(n) in (6.58) or (6.71). Theorem 6.8 Let c(n) -> oo as n -+ oo. Then for almost all n, log log n}.
A(n; log) < (log n)b0g(4/n) exp{ fi(n)
(6.73)
The right-hand factor arises from the Hardy-Ramanujan theorem. We shall actually prove that A(n; log) <
(44
w(n)
)
(log log n)A p.p.
(6.74)
for a suitable A.
Proof We write i(n, 9) _
d`0,
6ER
(6.75)
dIn
as in Hall (1975); Divisors, Chapter 3. Then S, (n; log) = i(n, 2irv).
(6.76)
By Lemma 30.3 of Divisors we have, uniformly for 0 < y < yo and v E Z+, E li(n, 2xv)I YC(n) << x(log
x)4(y/n)-1 logB(2
+ v)
(6.77)
n5x
where B = B(yo). We put y = yo = iv/4 and deduce from (6.77) and Lemma 6.6 that, uniformly for N >_ 2, we have TC
A(n; log) (4)
w(n)
< N-' x log x + x logB+ N.
(6.78)
nSx
(The sum involving the factor N-1 is estimated trivially as if we had y = 1. If this term were critical it would be better to insert a p.p. upper
218
6 Divisor uniform distribution
bound for i(n) at the outset.) We put N = [log x], and the result, that is (6.74), follows from Markoff's inequality, for any A > B + 1.
6.4 Weyl sums and discrepancy
We made no attempt in the previous chapter to obtain an explicit, quantitive error term in the formula i(n, sd) = (z + 0(1)) i(n) p.p.
(6.79)
in any of the cases except (5.5) in which we established that Dd = z. Much of the theory of divisor density depends on the Hardy-Littlewood-
Karamata theorem, and although a best possible quantitative form of this has been obtained by Freud (1952-4) the remainder term is, even in the best circumstances, very weak. For this among other reasons it is usually better, if we want explicit upper bounds for the discrepancy 0(n; f ), to tackle the problem directly. The Erdos-Turan inequality, Lemma 6.6, employed in §6.3 is a natural approach. However, unless f is additive, so that the Weyl sums S,,(n; f) are multiplicative, it is difficult to make any headway with (weighted) average orders of IS,,(n; f )I. This means that an almost obligatory initial step in any problem is to apply Cauchy's inequality to that of Erdos-Turan, to obtain 2
A(n; f )2 «
-
N
+ log(N + 1) E y
1 Sv(n; f )12 .
(6.80)
v=1
This undoubtedly introduces a weakness into the method, and the reader may convince himself of this by deriving a p.p. upper bound for A(n; log) from (6.80) as starting point. (He will arrive at (6.6), or (6.72) with instead of 4/n.) But, often, (6.80) is forced; and we now give a general result based upon it. This is a slight refinement of Theoreme 5 of Hall and Tenenbaum (1986). Theorem 6.9 Let f E C1(R+) be such that uf'(u) is ultimately monotonic. Let F, R1 : R+ -+ R+ be such that both F(t) and Fl (t) = t/F(t) tend monotonically to oo, moreover F(t) >> log2 t, and let
Fl og u)
<< I u f'(u) I << Rl (F(log u))
(6.81)
where R1 is such that the function R(t) = Ri(logt) satisfies (5.138) and
6.4 Weyl sums and discrepancy
219
(5.139) and in addition
R1 (1t) <
R1(t)1_k
(6.82)
for some fixed K > 0. In particular we may take Ct3/5
Ri(t) = exp
(6.83)
(log(2 + t))1/'
with a suitably small positive c. Then f is divisor uniformly distributed, and
for each fixed y, 0 < y < 1, we have Y 0(n)
n<x
n
0(n;
10 F f)2 < (log x)4Y (log
l0 x))2
(6.84)
Fl (log X)2y
In fact the conclusion that f is divisor uniformly distributed follows from (6.84) if we put y = 1. For we may deduce that
0(n;f)2 <
411(n)F1(logn)-1/3
on a sequence of logarithmic density 1, and the upper bound on the right is o(i(n)2) p.p. The conclusion is then implied by Theorem 6.4. The optimal y may not be 1, and depends on the circumstances. We discuss this point and give corollaries after the proof.
We remark that for any reasonable choice of R1, such as (6.83), condition (6.82) is easily satisfied. Proof of Theorem 6.9
We consider the sum Y
Z(a,v) = 1=:
4(n)
n
a>1
ISv(n;.f)I2,
(6.85)
where Sv(n; f) is the Weyl sum introduced in (6.55). We put
y)
C(s,Y)
cp(n;a,Y) = n fi vl
(6.86)
(1- p°
and we have 00
00
Z(a,v) = 4(a,Y)
0([dt])
e(vf(d)-vf(t)) d=1 t=1
[d t]
n(dt)
=
(a, Y)
d=1 t=1
(P(yQ(r)
(dt)° e (vf (d) - of (t)) rI(d,t)
y)
6 Divisor uniform distribution
220
2
C(6>Y)E 0"
y)
Yn(r)(P(r; r2a
r=1
E
Y'(m)e (vf (mr)) ma
m=1
2
<
Ya(r) [ Yc(m)e(vf(mr))
(6>Y)
ra
r=1
(6.87)
ma
IL m=1
by (6.86). We want an upper bound for the inner sum, which we denote by T. We put g(u) = e (v f (ur)). By partial summation and (5.139) we have, for any M >_ 2, Tv = O((log MY) 00
+JM
d
(u
2
Cgua)) {L u1 °d2 (v)+O
du
(1(0g))}dU (6.88)
At this point we fix M = M(r) in such a way that F(log Mr) < 2 log M.
(6.89)
This is in order because we have F(t) = (t/Fl(t)) = o(t) by hypothesis. Notice that for u M we have log ur < log ur log ur F(log ur) = = F(log Mr) log Mr
F1 (log Mr)
F1 (log ur)
log u + log r <
I log M. 2 logM+logr
1
- 2 log U.
(6.90)
The error term arising from the integral in (6.88) is
J
I ug'(u)I + I g(u)I
du
ua
R1(log u)
M
<
fby R1(F(log ur)) du
v
«v
uaR1(log u)
fm
R1(I logu)du uaR1(logu)
(6.91)
(6.81) and (6.90). By condition (6.82) on R1, this is R1(t)-"dt.
<< v
(6.92)
log M
It is a straightforward consequence of (6.82) that for every e > 0, the integral
f R1(t)-Edt J0
6.4 Weyl sums and discrepancy
221
is convergent. Hence the integral (6.92) is
«
Ri(t)-"/2dt
Jlog M vR,(log M)-K12,
<<
and we have, after an integration by parts, that 1
00
TV = 0 ((log M)y + vRi (log M)-K12) + J 2 (1 - v)d.1y(v) J 0
u (+) du.
M
(6.93)
We substitute u = Met in the inner integral here, which becomes M-(G+v-t)
J
e(Q+v-i)te (vf(rMet)) dt
0
= M-(U+v-I) o e (Q+v-i)G'(t)dt
(6.94)
0
where
G(t) _ f
e (v f (rMew)) dw.
(6.95)
By hypothesis, the function
/-{vf(rMew)} = vrMe'f'(rMew) is monotonic, moreover (6.81) implies that dw {v f (rMe" )} »
F (w + log(rM))
(6.96)
whence Lemma 4.2 of Titchmarsh (1951) yields 1
G(t) < -F (t + log(rM)),
(6.97)
V
and the right-hand member of (6.94) is therefore
« 1 tT
e v_1)tF t + log(rM)) A
(6.98)
M°+v-1
Now F is subadditive, because for l, 2 > 0, 2) =
+
<
2) F1(I F1( i) + F(ez) whence the integral in (6.98) is
«
F (log(rM)) VMV
1
+
VMV
0°
(6 0
v - 1)e (°+v i)tF(t)dt.
(6.99)
(6.100)
6 Divisor uniform distribution
222
Substitute t = s/(a + v - 1). We have f°°
J
s
e sF
ds
a+v-1
s esF \a+v-1) Ids
1
<
F(
<
1
)
\a+v-1
{
1 + f e SsdsI 1
I.
1
a+v-1
(6.101)
By (5.138) we have Jd2y(v)I < v-Ydv, and we insert this into (6.93) and employ (6.98), (6.100-6.101). We obtain M)-x/2
Tv « (log M)Y + vR1(log
+v
/z
J vM
{F(log(rM))
F (Q + v - ) } dv
o
«
(logM)Y+vR1(logM)-"/2+I(logM)Y-1F (_---), (6.102)
because F (log(rM)) < i log M by (6.89). This is a lower bound constraint on M, and we now determine M = M(r) by setting
log M = 3F(log r) + F I
a
1)
(6.103)
.
Notice that if a is sufficiently close to 1, say a < co, M is so large that F(log M) < 1 log M. Then F (log(rM)) < F(log r) + F(log M) < (3 + 1-!) log M, that is (6.89) holds. We deduce from (6.102)-(6.103) that
for a < co, Y
Tr <
-K/2
1
(_L1) +vR1
(F Ca
1))
(6.104)
and we insert this into (6.87), which yields
Z(a,V)
«
a 00
+ (a, y)
2Y
+v2R1 CF Ca ! 1))-K}
S2(r)
yrQ Flog r)2Y.
(6.105)
r=1
This last term on the right is absorbed by the first one. This is clearly
6.4 Weyl sums and discrepancy
223
true of that part in which logr < 1/(a-1). The remainder does not exceed C(a,
0(r)
yrO (log r)Zy
)2y
F1(i
4(a, y) 1 F1(Q11)2y (a - 1)3y
2y
1
<<
Y)ZF
U-1
because l;(a, 1)_y for 1 < a < 2. Now consider the sum on the left of (6.84). We put a = 1 + (1 /log x), and employ (6.80). The sum is therefore
<E Oc'
n=1
yn(n)
0(n; f)2
n° o
N
0(n)
< N2 E ynQ i(n)2 + log(N + 1) Y Z(a, v) n=1
v=1
V
< NZ (a - 1)-4y + log2(N + 1)(a -1)-2yF
(_! 1
+ (a - 1)-2yN2R1 (F
1
(6.106)
a-1
We put N = [F1(log x)y]. Since F(logx) >> (log log x)2 and we may
take Ri(u) = exp(b/) for any b, the term involving R1 in (6.106) is negligible. This gives (6.84) and completes the proof.
Corollary 6.10 Let f be as in the statement of the theorem, and fi(n)
oo.
Then C
0( n; f) < c(n)i(n) 1 -
log F1(log n)
fl(n)/2
log Fl (log n)
2 log log n
(6.107)
on a sequence of logarithmic density 1.
We note that, by the Hardy-Ramanujan theorem, the right-hand side does not exceed i(n)F1(log
n)(-'/4)+E
P.P.
(6.108)
so that (6.107) is always non-trivial.
Corollary 6.11 For a > 0, we have A(n; log") < i(n)(log n)-q(OE)+B
(6.109)
6 Divisor uniform distribution
224
on a sequence of logarithmic density 1, where
rl(a)=--log 1- mina, 1)
(6.110)
For J3 > 1 we have 0(n; (log log)#) < i(n)(log log
n)((1-fl)/4)+6
(6.111)
on a sequence of logarithmic density 1. Regarding log", see also Hall and Tenenbaum (1986), p. 135. We prove Corollary 6.10 and leave 6.11 as an exercise. Proof of Corollary 6.10 We consider the integers n < x, and we show that (6.107) is satisfied unless n belongs to an exceptional set E = E(x) such that 1
n<x,nEE
n
= o(log x).
(6.112)
We put 1
y
2
1+
log F(log x) log log x )
-1
=
1
4
1
log F1(log x) 2 log log x
1
(6.113)
and, with this value of y, (6.84) becomes y n5x
0(n)
n
0(n; f)2 « (log F1(log x))2 log X.
(6.114)
Hence if 1(x) -> oo, we have 0(n;.f) < 1(x)2n(n)
tog F1(log x)
C1- 2 log log x
n(n)/2
logFi(logx),
(6.115)
for all integers n < x not belonging to E. We now require that n has all the following properties: (i)
(ii)
(iii) (iv)
n > x4x), s(x) --> 0. 1 log log x < Q (n) < 2 log log x. 8 < log F, (log n) < log log n. 20(n)
< 1(x)T(n)
We include the exceptional n in E. This is clearly permissable in case (i),
and in case (ii) by the Hardy-Ramanujan theorem; also (iii) holds for n > no because F1 -+ oo, F1(log n) = log n/F(log n), and F --+ oo. Finally
6.4 Weyl sums and discrepancy
225
we have, for fixed A < 1,
n<_x
1
2
n
z(n)
«A log x
(6.116)
whence (iv) is in order as well.
Since fl(n) > 1 log log x by (ii), the right-hand side of (6.115) is a decreasing function of log Fl for log Fl > 8. By (iii), we may therefore replace F1(log x) by F1(log n). We apply (iv) to obtain
0(n; f) <
(x)zi(n) Cl
S2(n)/2
n) - log21ogFl (log log x
log Fi (log n).
(6.117)
log s(x) log log x
(6.118)
We have, by (i) and (iii), log Fi (log n) 1
2 log log x -
l
log Fi (log n) 2 log log n
1
and we insert this into (6.117) and apply the right-hand inequality in (ii), which yields A( n; f)
<
(x) )
)
2
i (n ) 1
(log 21og log n )
fl(n)/2
l og Fs (l og n ) .
(6 . 119)
We set
fi(x) = c(x)min{" fi(n)
:
n > x'(x)}
(6.120)
and let E(x) -+ 0 sufficiently slowly that fi(x) --+ co. Hence (6.107) holds for the n not counted in (6.112). This completes the proof.
As we indicated in the introduction to this chapter, we can employ upper bounds for 0(n; f) to construct Behrend sequences. Let {zn n E Z+} be a positive decreasing sequence such that zn -> 0 and, on a subsequence of Z+ of logarithmic density 1, znr(n) > 0(n; f) + 1. For :
such n there exists a divisor d > 1 such that 0 < f (d) < Zn < Zd (mod 1),
(6.121)
whence the sequence
,Q/= {a > 1
:
0 < f(a) < za (mod 1)}
(6.122)
is Behrend. As an example, we apply this idea to the functions f (d) _ (log d)", (log log d)1 using the results contained in Corollary 6.11.
6 Divisor uniform distribution
226
We begin with f (d) = (log d)", and we may take z = (log n)-n for any 11 < ri(a). We put A = 1/a and we conclude that 00
= U {a :
k < (log a)" < k + (log a)-" }
k=1
W
c
U{a :
V < log a < k2 + 2(k + 1)'
1k-n'}
(6.123)
k=1
is Behrend. This is always non-trivial, and it includes Erdos' conjecture concerning the sequence d2 defined in (1.110) if
(k + 1)"k-n' < log 2,
k > k1;
(6.124)
For Erdos' conjecture we are interested in the case A. > 1 only, and (6.124) reduces to
we need only consider the tail of sl by Corollary 0.14.
1 + 2 log (1
- 2) < a
(6.125)
or a > .76070..., or A < 1.31457...; this is the result of Hall and Tenenbaum (1986) mentioned in Chapter 1. As we pointed out there, it has been superseded by the sharp condition A < 1/(1 - log 2). We examine the case f (d) = (log log d)#, /J > 1. We may take z, _ (log log n)«1 and we conclude that = U{Z+ n (Tk, HkTkl }
(6.126)
k
is Behrend, where
log Tk = exp(kl/8)
(6.127)
log Hk
(6.128)
=
k(5( 1-#)/4#)+e log Tk.
This comes within the ambit of Theorem 1.15. If fl < 5, condition (1.159) is satisfied and the theorem applies to show that d is Behrend. As /3 increases, the exponent of k in (6.128) decreases towards -5/4, so that condition (1.160) is easily satisfied. However (1.161) fails: the Tk do not increase sufficiently rapidly (cf. (1.175)). Thus sd is a new Behrend sequence whenever /3 > 5.
6.5 Lower bounds for discrepancy
a-theorems are useful in that they put a limit on what we can expect to achieve with upper bounds. Perhaps of more interest is that they usually
6.5 Lower bounds for discrepancy
227
involve completely different methods; we may be able to see further. We begin with a simple extension of the proof given at the start of this chapter that for $ < 1, (log log d)# is not divisor uniformly distributed. Theorem 6.12 Let f (d) = F(log d) where F is differentiable and I F'(u)I << u for fixed K > log 2. Then
4(n; .f) >
i(n)µ-E
p
(6.129)
where µ = (K - log 2)/(1 - log 2).
Proof Let 2 E (0, ic) be at our disposal. Let m = m(2) be the largest divisor of n such that P+ (m(2)) < exp ((log and fi(n) -+ oo as n
(6.130)
oo, Then we have log P+(n) > fi(n)-1 log n p.p.;
(6.131)
moreover by Theorem 07 of Divisors also log m(2) < (n)(log n)2 p.p.
(6.132)
Now consider the divisors d = P+(n)t where tim. We have
lg F(logd) = F (log P+(n)) + O ((100p7fl)y) = F (log P+(n)) + O (fi(n)'+K(log
(6.133)
that is f (d) E E where JEl << (n)i+K(logn)2-".
(6.134)
Let 1; (n) = (logn)00). Then we deduce from (6.134) and the HardyRamanujan theorem that i(n) I E I < (log n)b0g
2+A-K+0(1)
(6.135)
By Turan's method, co (m(2)) has normal order 2 log log n, whence the number of divisors d of n such that f (d) E E is at least i(m) > 2'(m) > (log n)Aiog2+o(i).
(6.136)
We obtain a lower bound for the discrepancy, for example 0(n; f) > Zr(m), if gy(m) > 2i(n) IE I and so we choose 2 so that 2 log 2 > log 2+2-K,
that is 2 < (K - log 2)/(1 - log 2). Our result follows from (6.136).
6 Divisor uniform distribution
228
We
could
if
wished
we
replace
throughout
(log n)°(1)
by
oglogn), 1(n) -+ co. Of greater interest would be a precise lower bound for A in the case IF'(u)j << u-1`Y(u) when 'Y is slowly increasing. Theorem 6.12 shows that 0(n; log") - oc p.p. for every a E (0,1-log 2). For larger values of a we do not have a lower bound for the discrepancy, and this includes the case a = 1. The function f (d) = log d is of particular
interest, as it is both additive and the restriction to Z+ of an analytic function, and it has been an open problem since 1974 to prove that oo p.p.; in the opposite direction the problem of improving Theorem 6.8 is almost as old. The last result in this chapter comes quite close to, but does not succeed in showing that 0(n; log) is large. A(n; log)
We modify the problem slightly by introducing a weight on the divisors, rn(d), where r < 1. That is we consider the distribution function Y.{rn(d) : din, w < logd < z(mod 1)}
(6.137)
with the usual rule about the endpoints: any divisors d such that log d - w or z (mod 1) count 1-zrn(d). We can if we wish replace )(d) by 0-)(d) with no material difference. We write T(r)(n) = E rn(d)
(6.138)
dIn
and we have the following result.
Theorem 6.13 Let s > 0, and e(x) -+ 0 as x -> oo. Let r and t be such that 2
(g ) g g log x x log logg log
< r < 1 - ex p
(6.139)
and
0 < t < (log log x)-2-E.
(6.140)
Then for all but at most o(x) exceptional integers n < x we have 2
rn(d) 0
- t2(r)(n)
du >- t2 exp{E(x)
log log x}.
(6.141)
dIn
u-f' dog d
There is an alternative `for almost all n' formulation in which we change x to n in each of (6.139-6.141). Although r is very close to 1 it
6.5 Lower bounds for discrepancy
229
does not seem easy to remove it altogether. Perhaps a fairly simple extra idea has been overlooked.
Proof of Theorem 6.13 The Weyl sums, which we write as divisor functions, take the modified form rn(d)e(v log d)
T(n, 27rv ; r) =
(6.142)
dl n
and we seek a lower bound for the sum N
Ii(n,21rv;r)12
(6.143)
V=1
to hold for x + o(x) integers n < x. We put
N = N(x) _ [(loglogx)2].
(6.144)
Our argument is based on that used in Hall (1994) (proof of Theorem 2) concerning the function max{jv(n,0)I : a < 0 < b} and we set out to show that, normally N
E log+ 1,r(n, 27rv ; r)l
(6.145)
V=1
is large (in which log+ u := max(log u, 0)). Initially we have to exclude small prime factors and so we introduce w = w(x) -> oo as x -* co and write
?(n, 0; r, w) = 11 (1 + rp`B).
(6.146)
p[n, p>w
We may assume that n has no repeated prime factor > w since the number of exceptions is << x/w log w. We put N
L(n) _
(6.147)
log Ir(n, 27rv ; r, w) l v=1
and N
Mk(n)
flog It(n, 27rv; r, w)j Ik ,
k E Z+,
(6.148)
V=1
and we show that, normally, L(n) is much smaller than M1(n). We begin with an upper bound for L(n).
6 Divisor uniform distribution
230
Lemma 6.14 Let N
fN(t) _
log I1 + re(vt)I
(6.149)
v=1
and E(T) = It E [0, 1)
:
A(t)1 > T}. Then E(T) comprises at most
N(N + 1) disjoint intervals and, uniformly for r < 1, we have IE(T) I = measE(T) << T2.
(6.150)
This holds for all N E Z+, independently of (6.144).
Proof
We may write fN(t) = log I P(e2iit)I where P is a polynomial of degree 1 N(N + 1), whence the equation I f N(t) I = T has at most N(N + 1)
roots in [0, 1). The first part follows from this. Next we have (we leave the calculation to the reader),
J
log I 1 + re(µt)I log 11 + re(vt)I dt < 12. (µµV)2
(6.151)
whence i
.f IfN(t)I2dt « E
(µµy)z
«N,
p
and (6.150) follows.
Lemma 6.15 Let g be a periodic function of period 1, having bounded variation over the period. Let
g(u)du, I (g) = f
t I g(u) - 81 du, V (g) = f I dg(u)I .
(6.152)
0
Then for real 0 * 0, 2 < w < z and every c, we have
T, P-1 g(O log P) =Slog
lgw
(--)
+D
(W)
+ oc {(M(g) + (1 + I0I)V(g))
e./}
, (6.153)
where M(g) = sup IgI.
This is Lemma 30.1 of Divisors, except that the period is 1 and we have replaced V(g) by I(g) in the first error term (cf. Divisors (3.6)). Lemma 6.16 Let fi(x) -+ oc as x -> oo and w = exp ((log log X)20).
(6.154)
6.5 Lower bounds for discrepancy
231
Then for all but o(x) integers n < x we have
I L(n)l < N'I2(log log x)3/2 + (x)N log (1).
(6.155)
Proof
Set T = (N log log x)1/2 in Lemma 6.14 so that we have IE(T )I 1 /log log x by (6.150). By Lemma 6.15, with g the characteristic function of E, we have 1
w
: logp E E(mod 1)
p
= IEI
f log (log-) +O (lo1 w I +O(N2e Ogw) K 1 g
g
by (6.144), (6.154) and the bound for JEl. Let co'(n,E) denote the number of prime factors of n such that p > w, log p E E (mod 1). Then we have
1: w*(n, E) < x T, n<x
w
1 : logp E E(mod 1)} << x
p
(6.156)
whence co*(n, E) < fi(x) with o(x) exceptions. By (6.147) and (6.149) we have
L(n) = E fN(logp)
(6.157)
p1 n, p>w
and we infer that for the unexceptional n there holds IL(n)t < w(n)T +(o*(n,E)Nlog C1
1
r)
(6.158)
inserting the trivial upper bound for I fN I on the right. We may assume w(n) < 2log log x by Hardy-Ramanujan, and (6.155) follows. Lemma 6.17 Let r, N and w be as in (6.139), (6.144) and (6.154). Then for x + o(x) integers n < x we have IL(n)I < fi(x)-1N provided c(x)
loglogx
(6.159)
oo sufficiently slowly.
This is an immediate deduction from the previous lemma.
Next, we want a lower bound for Mi (n). For this we employ the moment inequality used in Hall (1994) which is of course a familiar application of the Cauchy-Holder inequality, viz. M2(n) < M1(n)213M4(n)113.
(6.160)
6 Divisor uniform distribution
232
Thus we require a lower bound for M2(n) and an upper bound for M4(n). We select the easier task first. Lemma 6.18 Let r, N, w and c be as above. Then for each fixed k we have, with o(x) exceptions,
M2k(n) < c(x)N(loglogx)k.
(6.161)
Proof We consider the sum H(x,A) _
Ii(n,0;r,w)I2
(6.162)
n5x
where 0 = 2irv for 1 < v < N and 2 = +(log log x)-1/2. Thus 2log(1 - r) I << 1 and the Halberstam-Richert inequality (Theorem 01, Divisors) yields
H(x,.l) <<
log x
exp
i 11 + rp`B I1 1: w
.
(6.163)
p We apply Lemma 6.15, with q = v, c = 1 and g(u) =I 1 + re(u)I2 ,
M(g), I (g), V W « 1.
Hence the sum in the exponential in (6.163) equals F(A, r) log I log x \ logg w
I+ O I v to w+ ve log w \ g
(6.164)
2J
(6.165)
where F(2., r) =
J
1
11 + re(u)I' du =
2R
I 1 + re" I A
The error term in (6.164) is negligible by (6.154). A classical theorem of Hardy (1915) on integral means implies that F(2, r) increases with r, whence
F(2, r) < F(2,1) _
1 + O(2)
(6.166)
for JAI < 2. (For F is meromorphic, with poles at -1, -3,-5,..., moreover 2n
1
a
F(A,1)
A=O
27r
logl1+e"Idt=0,
(6.167)
o
whence the right-hand inequality in (6.166) follows from Taylor's theorem.)
6.5 Lower bounds for discrepancy
233
We deduce from (6.163-6.164) and (6.166) that for JAI < 1/ og ogx and x large enough, we have IH(x,A)I < x. We put A = ±1/ oglogx, to obtain
n cosh
log IT(n, 0; r, w)I
n<_x
og log x
<< x
(6.168)
whence N
E loge` I i(n, 0; r, w) I <
(6.169)
n<x v=1
and (6.161) follows.
It is quite likely that the factor fi(x) in (6.161) is actually redundant, provided we replace < by
IQI,andforyEQandn<xlet f(n,µ) = Ef(P,µ),
(6.170)
pin
where If (p, µ)I < R always, and f (p, µ)= 0 when p > x1/4. Let
F(n) = E f (n,µ)2
(6.171)
µEQ
and put
E f(p,µ) P
P
E f(p,µ)f(p,v) P
ps
Ef(p,P)2f(p,v) P
ps
(6.172)
6 Divisor uniform distribution
234
W (µ,v) = E f(p,µ)2f(p,V)2 PS p
ZS(µ) = XS(µ, µ) and
F = E{Z1(µ) +
V (P)2
(6.173)
- Z2(u)}.
pEQ
Then we have, for R >_ 1,
E(F(n)-F)2=xJ+O
CIQI2R4x
(6.174)
log4 x
n<x
where
J = Y E{Wi(µ,v)-7W2(µ,v)+12W3(µ,v)-6W4(µ,v) pEQ vEQ
+2 (X1(µ, v) - X2(µ, v))2
+2V(µ){Yi(v, µ) - 3Y2(v, µ) + 2Y3(v, µ)} +2V(v){Y1(µ, v) - 3Y2(µ, v) + 2Y3(µ, v)}
+4V(µ)V(v) (X1(µ, v) - X2(µ, v)) }.
(6.175)
Proof We expand the square in (6.174) so that the sum becomes F(n)2
- 2F
n<x
F(n) + [x]F2.
(6.176)
n<x
We have
>F(n) _ E1: E1] f(p,µ)f(q,µ) n<x
pEQ n<x pin
qin
EE1: f(p,µ)f(q,µ) q] +0(1) pEQ p
q
and we note that the error term here is, trivially, « IQI main term is x
{ EE pEQ
p
(6.177) 111111
R2n(x1/4)2. The
f(p,,uff _ (q, _
f (P, l1)2
p*q
p
=x
pq
{Z1(µ) + V(µ)2 - Z2(µ)} = xF pEQ
(6.178)
6.5 Lower bounds for discrepancy
235
by (6.172). Trivially, IFI < IQIR2(loglogx)2 so that the error term arising from this sum is absorbed by that given in (6.174). Next, we have
E F(n)2 n<x
n<x pEQ vEQ
f(n,p)Zf(n,v)2
f(p,p)f(q,p)f(r,v)f(s,v) x pEQvEQ p
x
{
r
q
sl
+0(1) }
[p, q, r, s]
(6.179)
111
and again we make a trivial estimate of the error term, which does not exceed that given in (6.174). We write the main term in the form
xF2+x>>K(p,v)
(6.180)
pEQ vEQ
so that
K(p,v) =
1
1
[p, q, r, s]
[pq] [rs]
EEEEf(p,p)...f(s v)
p
q r (pq,rs)> 1
s
(6.181)
The proof will be complete if we show that the inner sum in (6.175) is K(p,v), and we leave this to the reader. We note that the condition f (p, p) = 0 for p > x4 is a luxury which we could forego if the application demanded it. To apply the lemma we put
f(p p)
={
Q=
log I 1 + re(p log p) I 0
if w < p < x14
{1, 2,3,...,N}
R = log (i
(6.182)
else
)
(6.183)
(6 . 1 84)
We observe that for n < x, n may have at most three prime factors > x14 whence
log Ii(n,2np;r,w)I = f(n,p)+O(R),
(6.185)
if as we have assumed, n has no repeated prime factor > w. The inequality (a + b)2 < 2a2 + 2b2 now yields
F(n) = E f (n, p)2 < 2M2(n) + O(NR2). pEQ
(6.186)
6 Divisor uniform distribution
236
We need an approximate formula for F, defined in (6.173), and an upper bound for J, defined in (6.175). We begin with the observation that all the sums in (6.172) in which s > 2 are negligible: they do not exceed 4
p>w
4
P << w <<
lo
(6.187)
-.
Next, we consider the sums V(p) to whichgwe apply Lemma 6.15. We have
f
i
log! + re(u)I du = 0,
r < 1,
(6.188)
0
so that we obtain, uniformly for p < N, that 1
I VGI) I «
µ log w
+ pRe 1Og w « (log log x)-20
(6.189)
estimating I(g) by Cauchy's inequality and (6.151). We see from this that all the terms in (6.175) which involve a factor V(p) are negligible: for example Yl (p, v) << R3 log log x « (log log x)5/2 etc. Hence we are only concerned with the terms Z1(p) in (6.173), Wi(p,v) and Xi(p,v)2 in (6.175).
Let us define i
Ak(r) _
(6.190)
Jlogk I1 + re(u)I du.
We have, with F as in (6.166), 00
4h(2h) i A2h(r) = 2 { F
C 2r) + F
hE =O
\-
2
r) }
whence A2h(r)
Zi(p) = A2(r) log °g W + 0 ((log log x)-20)
(6.191)
Gg
and
W1(p, p) = A4(r)log
Cloggw
) +O ((loglogx)-20
(6.192)
uniformly for p < N, and then Cauchy's inequality and (6.192) imply that W1 (It, v) << log log x, 1 < p, v < N. (6.193) To cope with Xi(p, v) we have to consider the function g(u) = log 1 + re(pu)I log I1 + re(vu)j,
(6.194)
6.5 Lower bounds for discrepancy
237
and we treat this as having period 1. (There would be a slight saving in Lemma 6.15 if we replaced g(u) by g(u/(µ, v)) and took 0 = (µ, v), but this is unimportant in this application.) The mean value g has been estimated already in (6.151): we have (y, V)2 191
(6.195) µv
uniformly for r < 1. The total variation may be estimated by Leibniz' rule and is
f IdgI
(6.196)
whence we have IXi(µ,v)I <<
(µ,v)z log log X. µv
(6.197)
We note that
E
(P, v )4 zvz
KN
,
(6 . 98)
p
and so we obtain from (6.193), (6.198) that J << Nz log log x + N(log log x)z.
(6.199)
Next, from (6.190), we have
Az(r) =
1
2
00
zm
1
mz > g,
m=1
I
2
< r < 1.
(6.200)
Hence from (6.191) we have
F> cN log log x,
c>0
(6.201)
subject to our constraints on r, N and w. We have J = o(N2 (log log x)z) from (6.199), whence we deduce from Lemma 6.19, (6.174), that
F(n) > cN log log x
(6.202)
2 for all but o(x) integers n < x. Hence (6.186) implies
M2(n) > 5 cN log log x
(6.203)
for these same integers. Together with Lemma 6.18, this gives, via (6.160),
Mi(n) > c(x)-112N
loglogx
(6.204)
6 Divisor uniform distribution
238
for x + o(x) integers n < x. Together with (6.159) this implies for large x, N
E log+ (i(n, 27cv; r, w)I > 3
I
x
(6.205)
V=1
with at most o(x) exceptions, and we may strike out the w at the cost of an error term < 2N log log w. log
< 40e(x)N
(j_!:_) r
log log x
(6.206)
by the definitions (6.139), (6.154) of r and w. Thus if fi(x) ---+ oo sufficiently
slowly, we have from (6.205-6.206) that, with o(x) exceptions, N
log log x
(6.207)
EIi(n,2xv;r)12>Nexp{a(x) loglogx}
(6.208)
log+ I i(n, 27rv ; r) I > 2 s(x)N V=1
whence by Jensen's inequality, N V=1
Finally, we have an inequality, in the weighted case, analogous to (6.59). We restrict the sum on the right to the range 1 < v < N: for these v and the values of t specified in (6.140) we have sin 7rvt
>
7rv
2
t
(6.209)
7r
and since we may assume that N > 7z2/4, we obtain (6.141) as stated. A result of Hall (1975b) is that for each fixed real 0 * 0 we have I t(n, 0)I < exp { fi(n)
log log n} p.p.
(6.210)
provided fi(n) -+ cc. No direct method of combining this with the ErdosTuran inequality to bound 0(n; log) has been found, nevertheless, in view
of (6.210) and Theorem 6.13 it seems reasonable to suppose that the order of magnitude of 0(n; log) is something like exp{
og log n}, p.p.
H(x, y, z)
7.1 Introduction
We consider the important special case
d = 5(y, z) = (y, z] f1 Z.
(7.1)
We denote the counting function of -# (d(y, z)) by H(x, y, z), following Tenenbaum (1984). Thus H(x, y, z) = card{n : n < x, 3d : din, y < d < z}.
(7.2)
This function may be studied independently of sets of multiples. In
a general setting, y and z may be functions of x, perhaps the most interesting case being when they are powers of x. Tenenbaum (1980) proved the following theorem: let
y=x(i-")l`, z=xi/`, 0_1.
(7.3)
lim x 1H(x, y, z) =: h(u, t)
(7.4)
Then
(We have used the notation employed in Divisors §2.4 which contains a proof; extra information is contained in Divisors, and in the original paper.) Within the framework of sets of multiples y and z are necessarily independent of x. 4 is finite and is therefore a Besicovitch sequence, and we are interested in the density d.& (d(y,z)). We may write this down explicitly, using either the inclusion-exclusion principle or total exists. Properties of the function h(u, t) were given.
decomposition sets; however what we usually really want is an asymptotic
oo and z is a suitable formula for the function d.# (d(y, z)) when y function of y. Failing this we seek good upper and lower bounds. 239
7 H(x, y, z)
240
H(x, y, z) is the subject of Divisors, Chapter 2 and the work presented here is a sequel, but with a different point of view. Our main purpose in this short chapter is to settle the conjecture which appears in Divisors §2.2.
This requires some introduction in order that our presentation
should be self-contained and comprehensible.
7.2 Short intervals We refer to the interval (y, z] as short, in the present context, if z < 2y. We define /3 = f3(y, z) implicitly by the equation
z=y+ylog fly, 0
(7.5)
We think of z as a function of y and /3 as a parameter, and we seek an asymptotic formula for d./# (d(y, z)). Of course we are interested as well in the question of how large x must be for the formula H(x, y, z) - d./# (d(y, z)) x
(7.6)
to hold as x, y, z --> oo.
Tenenbaum (1984) discovered a threshold in the behaviour of d. '(d(y,z)) at the point /3 = log 4 - 1. The term threshold was introduced in Hall and Tenenbaum (1989) and applies to the circumstance
in which we have an asymptotic formula for an arithmetical function containing a parameter (such as /3), where the main term discontinuously changes shape as the parameter passes through a critical value (the
threshold). Quite often it happens that the formula on one side of the threshold has a simple shape, for which a natural or heuristic explanation is available, while on the other side the formula is more complicated and mysterious or may even be unknown. It may be possible to remove the discontinuity by introducing a further parameter, and this applies in the case under discussion. Let = (y) and
fl=log4-1+
(7.7)
og logy
Then we deduce from Divisors, Theorem 21(i) that if and z is given by (7.5) and (7.7), we have
(d(y, z)) - log -# y;
00 as y
oo,
(7.8)
moreover provided x > z2, we also have
H(x,y,z)-xlogBy
i(n,d(y,z)), n<x
(7.9)
7.2 Short intervals
241
because the sum on the right is, for all x, y, z, equal to x(log(z/y) + O(1 /y) + O(z - y). We can write i (n, ((y, z)) in the alternative form i(n; y, z) used in Divisors. Notice that (7.9) implies that the sum on the right is dominated by the contribution of integers for which i(n; y, z) = 1. Let
Q(2)_Alog2-2+1, 2>0
(7.10)
and define G(fl)
l
Q
0
/3,
/3>log4-1.
(7.11)
Theorem 21(iii) of Divisors applies when /3 >_ 0 but (y) is bounded above, and yields L
( l og
y )-G(fl) << dM (d( Y, z) )
(log Y)
« 1 + (-) +
( 7 . 12 )
and, for a suitable positive c,
where
L = exp{c
log logy. log log log logy}.
(7.13)
The corresponding formula for H(x, y, z) is again valid for x > z2. It will emerge later that (7.9) is now false - that is, a necessary and sufficient
condition for (7.9) is (y) - oo. Notice that for /3 < log 4 - 1 we have
1 g4(7.14)
G(f3)=/3+2Q()
Thus G(fl) > /3, indeed on the interval [0, log 4 - 1], G(/3) - /3 decreases from 6 to 0, where 6 = .086071 ... is the number defined in Lemma 0.3. We have Q(1) = Q'(1) = 0, so that G(/3) is continuously differentiable and convex for /3 > 0. For < 0, Taylor's theorem implies that
G(/3)=/3+
C
log 4
(log log
Y)-1
+
log y)-312)
in which the last term on the right is positive. If
(7.15)
is negative and
= o ((log log y)116) we deduce that z
(log y)-o(fl)
exp
toga (-_
4) log # y,
(7.16)
and we may insert this into (7.12), bearing in mind that
log -# y - E{a ' : a E d(y, z)}.
(7.17)
7 H(x, y, z)
242
We shall explain later how the factor L-1 on the left-hand side of (7.12) arises. If we assume that the upper bound in (7.12) is near the truth
then we are led by (7.16) and (7.17) to conjecture that there exists a probability distribution function such that if E R is fixed, and y, z -- co together, with z = z(y) determined by (7.5) and (7.7), then
>{a 1 : a E d(y, z)}.
d.# (d(y, z)) -
(7.18)
If in addition x -> oo sufficiently fast, we will have H(x, y, z)
(7.19) n<x
The conjecture contained in Divisors §2.2 is equivalent to (7.19) except
that we specified merely that x -> oo together with y and z which is plainly wrong (since we could put x = z, when H(x, y, z) = [z] - [y], or x < y, when H(x, y, z) = 0). We shall prove that (7.18) and (7.19) hold, with /logo
F(;)
2
eW dw.
(7.20)
This is the content of Theorem 7.9 below. Rather than set out a formal proof we lead up to our result in stages. We hope that in this way the reader will be able to see more easily which parts of the argument would apply to the case /3 E [0, log 4 - 1), /3 fixed, where it is an open problem to improve on (7.12). Hall and Tenenbaum (1989) proved that provided for any fixed co, we have, for x >- z2,
H(x,Y,z)'=
x(log y)-o(B)
1 + (-)+ ,
> -co(log log y)1/6
(7.21)
that is the upper bound in (7.12) is the correct order of magnitude. Our approach now is similar and is based on the principle that for suitable , the dominant contribution to H(x, y, z) comes from integers n for which t(n; y, z) = 1.
As usual, let S2(n, z) denote the number of prime factors of n which do not exceed z, counted according to multiplicity. A very crude first estimate of the probable order of magnitude of i(n; y, z) is i(n; y, z)
2f'(n,2)
to g(z
Zy)
(7.22)
g
This is of course not supposed to be more than a guide, but it suggests
7.2 Short intervals
243
the following strategy: let
k 0 = tog [--# log log z
(7.23)
.
J
Then if )(n, z) > ko, i(n; y, z) may well be large and n be counted by H(x, y, z) but these `safe bets' will be rare. More frequently S2(n, z) < ko
and then we expect i(n; y, z) = 0 or 1 with probabilities which can be estimated.
Notice that if $ is approximately log 4 - 1 then ko is approximately 2 log log z which is a threshold for the counting function Nk(x, z) = card{n : n < x, fl(n, z) = k}.
(7.24)
This (irrelevant) difficulty is caused by the prime 2 and we avoid it by working with S2,(n,z) which we define to be the number of odd prime factors of n which do not exceed z. We define Nk(x, z) analogously, and we have
ENk(x,z)<<x(logz)-Q((i+#)/log2)(log log
z)-1/2;
(7.25)
k-ko
see Divisors §2.6 for a detailed calculation. We replace z by y on
the right and we notice that for all 0 the exponent of logy is < -G(fl) with equality if and only if fi < log4 - 1 by (7.11). Provided of og og y) the sum in (7.25) is negligible compared with the right-hand side of (7.21). For k < ko we consider the sums
Rk = E i(n; y, z)
(7.26)
n<x ?i(n,z)=k Sk
=
(r(n;Yz))
7.27)
n<x 5(n,z)=k
Let Hl (x, y, z) denote the contribution to H(x, y, z) from integers n such that ?2(n, z) < ko. We have
H1(x,y,z) < 1: Rk
(7.28)
k
and, for any h < ko,
H, (x, y, z) > E Rk k
Sk.
(7.29)
k
For technical reasons to do with our upper bound for Sk we have to
7 H(x, y, z)
244
take h somewhat less than ko. If the sum over Sk is small relative to that over Rk then we obtain an asymptotic formula for Hl (x, y, z) : this clearly requires that usually z(n; y, z) = 0 or 1. The method works for all positive and for negative 1; suitably bounded below. As we try to reduce further, we find that we cannot cope satisfactorily with the sum over Sk in (7.29). The problem can be traced back to the crude model (7.22) for determining whether or not we expect i(n; y, z) < 1. For this to hold, we require not only that S2(n, z) should be suitably small but that the prime factors < z of n should have the right kind of distribution. This is rather vague, because this question (which
is likely to be the key to the problem) has not yet been resolved. A sufficient requirement would be uniformity of distribution, for example ?2,(n, t) <
to # log log t, (3 < t < z) g
(7.30)
but because of Erdos' law of the iterated logarithm for prime factors (see Divisors, Chapter 1; the generalization of this directly germane to (7.30) has been carried out by L. Bastick (1992)), the numbers n satisfying (7.30) are too rare. A practical condition of this sort is employed in Divisors §2.7: we add a small function of y on the right of (7.30); log L, with L as in (7.13) is suitable. This leads to the lower bound (7.12).
Our method requires asymptotic formulae for the sums of Rk and upper bounds for those of Sk. We tackle the former problem in the next section.
7.3 The asymptotic formula for E Rk
Theorem 7.1 Let Rk be as in (7.26), /3 = log 4 - 1 + / og log y, and ko be as in (7.23). Let x > exp(log z. log log z). Then for each fixed c, and c(loglogy)1/6, we have
ERk = (ci k
where
1+
+ oc
EO )) xlog #y
(7.31)
og og y ///
is the probability distribution function defined by (7.20) and if > 0, (1 + 2) exp(-c2/log2 4) if < 0.
The formula is reminiscent of Theorems 4.10 and 4.13 which is to be expected since we are concerned essentially with Poisson variables. All the formulae for H(x, y, z) in Divisors are valid for x > z2 and so
7.3 The asymptotic formula for E Rk
245
it would certainly be desirable to be able to cope with smaller values of x than those given above, although this is not our primary concern. We require several lemmas. The restriction on x occurs in the first lemma, and we make some further remarks about it. Lemma 7.2 Let A > 0 and 7 > 0 be fixed, and I w I < 3 -1i, z > zo(A, ri). Let x > exp(z log z. log log z). Then wn(n,z) = M(w)x + OA,n (x exp (-A log z ) )
(7.32)
n<x
where
M(w)
i
l( l p/ \1 - P /
11 \1
3
(7.33)
This is an application of Theorem 02 of Divisors, which is an elementary
result depending on Rankin's method. Notice that when w = 0 the sum on the left of (7.32) is closely related to Buchstab's function J(x, z) and is not asymptotic to M(0)x when log x < log z. These small values of x would require a reappraisal of this section and it is not clear that (7.31) has the correct main term. An examination of our method shows that in fact only values of w close to 1 in (7.32) are critical, when we might expect a wider range of validity. For IwI < 3, we define 1 \ '" /1 w\\ 1 / J(w)
= fl
3
- pl //
C1 -
pI
\
(7.34)
.
//
We put
Z=-
log 11- 1 I. 3
\\\
(7.35)
p/
Lemma 7.3 Let , > 0 be fixed. Then for z > zi (ri), j < (3 - rl) log log z and x > exp(2 log z. log log z) we have
Nj(x,z)=x( (j)+(f)) Zi fi j!
3
1
C1--"). p/
(7.36)
We shall apply Lemma 7.2 with 7 replaced by q/2. Let M(x, w) denote the sum on the left of (7.32) so that Proof of lemma
M(x, W) = E Nj(x, z)wi. 1=0
(7.37)
246
7 H(x,y,z)
When j = 0 we put w = 0 in (7.37) and fix A = 2. Let j >- 1 and C be the circle {w : lwl = r} where we assume r < 3 Cauchy's formula zj.
gives
N3(x'
z)
_
1
2ni
dw Jc M(x, w) wj+i /
(7.38)
.
We employ the formula (7.32) for M(x, w). The error term contributes
(7.39)
g
z
We write
M(w) = TT (1 3
- 1P ) J(w)ewz
(7.40)
so that the main term in (7.38) becomes
x 3SpSz 11
(1
1 \ 21ri Jc
J(w)exz
(7.41)
Let S(w, r) be defined implicitly by the equation
J(w) = J(r) + (w - r)J'(r) + 60(w, r),
(7.42)
i.e. as the remainder in Taylor's series. We have
f
wz d
J(w)e
J(r) -
W
J
+ J (r) ( (Z1 /
r2
1) !
f e(w'r)e C
+27ri j
Wz
dw wi+I
(7.43)
and we put r = j/Z so that the term involving J'(r) in (7.43) vanishes. By (7.35), Z = loglogz+0(1) and we assume that zi(A, q) is both > zo(A, z) and sufficiently large to imply r < 3 J(w) and its derivatives are bounded on compact subsets of the disc {w : lwl < 3} and so for such r we have 2ij.
.ff (w, r) <, lw - r12 .
(7.44)
We put w = re`O and we see that the final term in (7.43) is I
r2--
jf
n
I e`9 - 1 I2 erz cos Bde
0
(z)f
Kn
CZ J
2
Jon e2e 2jBZhn2de,
(7.45)
7.3 The asymptotic formula for E Rk
247
using the inequality cos0 < 1 - 202/7r2, 0 < 0 < it. The integral is j-3/2 whence by Stirling's formula this is
«n
j ZJ Z2.i
«ni1 Z'-.
(7.46)
Hence the expression in (7.41) is equal to
x( ( )+Onz))
(7.47) !
We substitute r = j/Z in (7.39) and employ Stirling's formula to compare this term with the error term in (7.47) bearing in mind that the product on the right of (7.47) is >> 1 /log z. If we fix A = 2, the error term arising from (7.39) is absorbed for the values of x stated. This completes the proof.
Lemma 7.4 Let Q(d) denote the number of odd prime factors of d counted according to multiplicity. Let ,j > 0 be fixed. Then uniformly for Jwi < 3-ri we have .
wi2(d)
= K(w)(z - y)(log
z)w-i
+ On
(z(log Z)aew-2)
(7.48)
y
where
K(w)=T(w)fJ(1-p) (1
1)
(7.49)
p
This result is of a standard type and may be derived from Perron's formula: the factor 1/1"(w) arises from Hankel's integral. We omit the proof.
Lemma 7.5 Let rl > 0 be fixed. Then uniformly for i < (3 - rl) log log z we have 1
z-y (loglogz)' + = (K()+o)). i!
y
Q(d)=i
+O n
(z(loglogz)') i!log2z
(7.50)
where l = log log z and K is given by (7.49).
This follows from the previous lemma in the same way as Lemma 7.3 follows from Lemma 7.2. Because K(i/l) - 2i/l for small i we leave
7 H(x, y, z)
248
the first error term on the right of (7.50) in this form, analogous to the left-hand term in (7.46). We proceed to an asymptotic formula for Rk. We assume henceforth that x > exp(log z. log log z). We have
Nj d
Rk
(7.51)
i+j=k y
52(d)=i
and we employ Lemma 7.3. We have
d = Y (1 + O(log y))
(7.52)
and we notice that the error term here is negligible compared with the term On(11Z) in (7.34), since We put
W = rI (1- 1)
= e Z,
(7.53)
3
and we have
/
(J Z)
Rk = xy-1 W i+j=k
\
I
/
\\\
+ On
(k)) Zjj! y
1.
(7.54)
5(d)=i
We apply Lemma 7.5 to the inner sum: since K(f)Y! (for the i in question) we may write
(K) +(h))
log z
=K
\/\ () (1+O))Y(logY)-1/
(7.55)
and we obtain
Rk = xW(logy)-'-0 i+j=k
+O n
J (z) K (I) iz (1+O (i))
(x(21og log z)k l
(7.56)
J The functions J(w) and K(w) are analytic for IwI < 3 and so their k! log3 z
derivatives are bounded in the disc I w I < 3 - n. Hence, when i + j = k,
J(Z ) K (I ) -J(2Z)K (2l) «n
l
2
(7.57)
We have
(k) I i
2
« 2k Ik_
(7.58)
7.3 The asymptotic formula for E Rk and we employ (7.56)-(7.58) to obtain
249
"1))
Rk=xW(logY)-'-fl (d(2Z)K(2l)+On
(
kZ)k
(7.59) (1
absorbing the final error term from (7.56). We require the following result which is a deduction from Theorem 1.8 of Norton (1978).
Lemma 7.6 Let v > 0 and Vk
e-°ki.
(7.60)
k
Then for I al < cv 6 , for any fixed c, Sv(a) 72,=,
f
oo
-U2/2 e
2
du + Oc
(1
e-'z/2) J
v
.
(7.61)
We shall also need
Lemma 7.7 Let k
Sy(a) _
"k
(v
(7.62)
k
Then for a as in the previous lemma, we have S' (a) <
Proof of Lemma 7.7
(-2 ((-a)+)2)
(7.63)
Put kl = v + oc Ji. Then vk
Sy(a) = kSv(a) - v E e°
Vki
k = (k1 - v)S,,(a) + ve° k i.
(7.64)
k
We consider two cases. If a > 0 we apply the bounds
1, k « 7,
S, (a)
(7.65)
which lead to (7.63) in this case. If a < 0 we deduce from (7.61) that S (a) <<
1 + a2
e "2/2 + loci
+
2/2
ea /2 <<
e
x212
1 + jai
(7.66)
and from Stirling's formula that V
z
vev k1! <
(7.67)
7 H(x, y, z)
250
Let
k1 = l + Z + a l + Z.
(7.68)
For k < k1 we have, uniformly for k1 < 3 log log z,
J(2Z)K(2l)=j(2Z)K(11)+0 21
(kil kI
(7.69)
and we insert this approximation into (7.59) and apply Lemmas 7.6, 7.7
with v = I + Z. We may fix n = 1 at this point because we shall have a << (log log z) 6 and k1 < (2 + o(1)) log log z. We note that, by (7.53), W(logz)-1
= e-('+z).
(7.70)
We obtain that, for any fixed c and Ian < c(loglogz)1/6, we have kl
Rk =x 109 -fl y J
2Z
k
k
kl 2l
fe
'2/2du+0c (1-1/2E1(a))
2n _00
(7.71)
where Ejoc)
_
l+ a,
a>0,
(1 + a2) exp(-a2/2)
a<0
We approximate J(k1/2Z)K(kj/2l) by J(1)K(1) = 1 with error << 1+ this is absorbed by the error term in (7.71). Finally, we put k1 = ko - 1 where ko is given by (7.23). We compute a from (7.68), noting that Z = log log z + 0(1). Thus a
21/2 log 2 +
0
(
(7.72)
7,1).
We clear our formula of /'s by writing F(l;) in the form (7.20). The error introduced by the approximation in (7.72) is absorbed. This proves Theorem 7.1.
7.4 The asymptotic formula for H(x, y, z)
We now have an asymptotic formula for the sum on the right of (7.28), and (by implication, provided h is suitably close to ko) for the first sum on the right of (7.29). Provided h is sufficiently close to ko, these sums will be asymptotically equal. We need an upper bound for Sk.
7.4 The asymptotic formula for H(x, y, z)
251
Lemma 7.8 For 0 < v < 1 let (T(n; Y, z)) vn(n^,). 2 n<x
T (v) _
(7.73)
Then provided x > z2,
T(v) « x(log y)-20+2v-2+(2v-1)' (log log y)2,
(7.74)
(where as usual 2+ denotes max(A, 0)).
Let us compare the right-hand side of (7.74) with what we should expect if the heuristic approximation (7.22) were exact. We should strike out the unimportant factor (log log y)2, and the exponent of logy would become
-2/3+4v-3,
(7.75)
that is as in (7.74) without the +. It will be clear from the proof that (7.74) is realistic, that is for small k the sum Sk is larger than the over-simplistic model would suggest. We shall see below that this effect begins to bite
as soon as k is substantially less than 2log log z, and so the method presented here fails if /3 is fixed < log 4 - 1, (cf. the remarks following (7.29)).
Proof of lemma
We put log -# y = 0 for convenience. Let din, d' I n where
y < d < d' < z. We put m = (d, d') so that m < Oy and write d = mt, d' = mt'. The Halberstam-Richert inequality (Divisors, Theorem 00) gives
E vn(n,z) . vii(mtt') nSx
X)°_i
(7.76)
mtt' (log
n-O(mod [d,d'])
where X = min(z, x/ [d, d'] ). We have x/ [d, d'] = xm/dd' >- m since dd < z2 < x, whence X > m. Hence
T(v) « x
v t(t)
)
m
2
v m) (log 2mv-i( E
(7.77)
y/m
Since v < 1 we have, trivially
At) y/m
t
<
m
z
v
In
In
<0+
m v
< 20.
(7.78)
7 H(x, y, z)
252
This is insufficient for the application and we apply a result of Shiu (1980), (Divisors, Theorem 03) (see Lemma 1.9) which yields
E
(
ve(t)
m \1og In
y/m
l v-1
(7.79)
provided 8y/m >_ (z/m)" for some fixed K > 0. We put x = Z, and the constraint reduces to y/m >- 9-2(1 + 0). We roll (7.78) and (7.79) into a universal upper bound, viz. VD(t)
y/m
t
/
« 0 I log 8
\
y v-1
\ 1-v
(7.80)
(log m)
I JJJ
whence
T (v) << x log 2/i y(log log
y)2-2v j
vi(m)
msOy
m
(log
)
2m)v-1 (log
2v-2
m
The inner sum on the right is <<
I
1: vu(m)(log 2m)v-1 E(log m
2h)2v-2
h<m
Y
<<
Y E(log 2h)2v-2 E vn(m)(log 2m)oh
«
my//h
/
E (log 2h)2v-2 I log _y
<<
2y ) 2v-2
h
(log y)v-1+(2v-1)+(loglogy)B
(7.82)
where B = 0 unless v = 21 when B = 1. We insert this into (7.81) to obtain the result stated. Theorem 7.9 Let
z=y+ylog-fly, fl>0
(7.83)
fi=log4-1+
(7.84)
and
log logy ,
Then for each fixed c we have, uniformly for c > -c(log log y)1/6
d_# (d(Y, z)) = (F(c) +
y)) log -# y
(7.85)
7.4 The asymptotic formula for H(x, y, z)
253
where 1
FO _
7
z
J
e
" du
(7.86)
00
and
y) «,
2 + log log log y
oglogy
e
2/ 1og2 4
(7.87)
if < 0; otherwise
y) «
+ log log logy
(7.88)
og log y
Moreover, provided x > exp(log z. log log z), (7.85) remains valid if we substitute x-'H(x,y,z) on the left-hand side. Thus (7.18) and (7.19) hold as conjectured. The central problem which
remains is to extend the range of negative
for which we have an
asymptotic formula for dA' (d(y, z)). There are interesting questions still to be answered for positive . For > 0, we have 1 + 0 ((1 + )-ie_ 2/1og24)
(7.89)
and so when exceeds about Jlog log log y there is no point in writing instead of 1 on the right-hand side of (7.85). As increases further the error term (7.88) becomes unsatisfactory, requiring = of og logy) for (7.85) to be non-trivial. Of course we know from Divisors, Theorem 21 that provided -* oo as y -* oo, we have
ddf (d(y, z)) ^-
log-fl
y
(7.90)
and it is reasonable to ask for a uniform error term in this formula, valid for large positive . An easy argument (Divisors, p.38) gives
d,# (d(y, z)) = (1 + O(log" y))
log-It
y
(7.91)
and we bridge the gap between this result and Theorem 7.9 in Theorem 7.10 below.
Proof of Theorem 7.9 We prove the formula for H(x, y, z), from which (7.85) follows. We assume x > exp(log z. log log z).
We begin by considering the second sum on the right of (7.29). We have, for v < 1, v'' > Sk < T(v) k
(7.92)
7 H(x, y, z)
254
with T (v) as in (7.73). We put v = z and h = ko - g. By (7.23), 2k° << (log y)1+p and so (7.92) and Lemma 7.8 give
J:Sk << 2-gx(logy)-fl(loglogy)2.
(7.93)
k
We put g = 4 log log logy if >_ 0, g = 41og log logy + 2 if < 0: in each case the sum (7.93) is absorbed by the error term implicit in (7.85). (There is no point at this stage in ducking under the error term in (7.31)
by making g bigger.) Next we consider the first sum on the right of (7.29), and we write
- E R. k
k
(7.94)
h
We require an upper bound for the second sum here and we employ (7.59). It is familiar that for all t, k > 0 we have, uniformly, tk/k! < e`/,[t whence (7.59) yields, (we may assume ko < z log log z), Rk <<
x log y log log y
(7.95)
0 to obtain
We apply this when
Rk << hsk
log log log y x og og Y
log-p
y
(7.96)
which is absorbed by the error term in the theorem. When
< 0 we
observe that ko < 2 log log z, so that (7.59) gives
E Rk
<<
gx(log y)-2-'
(21og log z)k0
h5k
«
(log gx og logy
ko! Y)-Q((1+#)/log2)
(7.97)
by Stirling's formula and the definition of Q, that is Q(t) = t log t - t + 1, (t > 0). The exponent of logy above is -G(/3), as in (7.15), and we see that the sum (7.97) is again absorbed by the error term in the theorem. We deduce from (7.28), (7.29), (7.96), (7.97) and Theorem 7.1 that
#y,
(7.98)
with y) as in the statement of the theorem. We notice that Theorem 7.1 requires 1 < c(log log y) 6 whereas our present hypothesis is 1
> -c(loglogy)1/6. The reason for this is that Theorem 7.10 (below) already gives a stronger result than Theorem 7.9 when > 3 log log logy.
7.4 The asymptotic formula for H(x, y, z)
255
Thus when dealing with positive in this proof we may assume that < (log log y)1/6 throughout; hence the error term in (7.88) does not depend on c. Let H2(x, y, z) be the contribution to H(x, y, z) from the integers n for which S2(n, z) z ko. By (7.25) we have H2(x, Y, Z) < E Nk (X, z) <
(log Y)-Q((i+f)/log 2).
(7.99)
k>ko
When < 0 this is negligible in comparison with the right-hand side of (7.97) and is therefore absorbed. When > 0 we have only to observe that the exponent of logy in (7.99) is We assemble (7.98) and (7.99) to obtain our result. The reader will observe that we could obtain a better result for positive
(we should need g larger) if the error term in Theorem 7.1 were improved.
We conclude with a formula for H(x, y, z) which is more precise than Theorem 7.9 for large positive . Theorem 7.10 Let x >- z2 and from t >- 0,
> 0. Let b be fixed and t be determined
2Q(1 + t) + t log 4 =
and put A = 2Q(1 + t). Then for
,/log log y
(7.100)
og log y,
H(x, y, z) = (1 + Oe ((log y)-A(log log y)2)) x log -# Y.
(7.101)
Put X = 1; / og log y and A = A(X). Then A(X) is an increasing function and A(X) - X as X oo. We show later that X2
X2 2X3 < A(X) < log 4 log 4
,
3
X>0
(7.102)
so that (7.101) is more precise than the result given by Theorem 7.9 when > 3 og og log y. (Of course for fixed it is worse than trivial because of the factor (log log y)2.)
Proof
We have
H(x, y, z) < E i(n; y, z)
(7.103)
n<x
and this is sufficient as an upper bound. For kl at our disposal,
H(x, y, z) > E(Rk - Sk) k
(7.104)
7 H(x, y, z)
256
where Rk and Sk are as in (7.26) and (7.27). Hence
i(n;Y, z) - > Rk - > Sk.
H(x,Y, z) >n5x
k>kl
(7.105)
k
We have, uniformly for w < wo for each fixed w0, Y)2w-2-/3
E i(n; y, z)w°(na) << x(log
(7.106)
n<x
(we leave this as an exercise) whence, provided w > 1,
1: Rk « w-k' x(log
Y)2w-2-/1.
(7.107)
k>kl
We put k1 = K log logy and minimize the exponent 2w - 2 - /3 - K log w of logy on the right by choosing w = K/2: we require K > 2. This gives
E Rk << x(log
Y)-1-2Q(K/2).
(7.108)
k>kl
We apply Lemma 7.8 with v = 2 to obtain «x(logy)-2/f-1+K1°g2(loglogy)2
Sk
(7.109)
k
and we put K = 2 + 2t, where t >- 0. The exponents of logy in (7.108) and (7.109) are equal when (7.100) holds, to -/3 - A, and this gives the result stated. The constant implied by Vinogradov's notation is uniform provided w < w0 in (7.106) and this is equivalent to < b og log y, that is /3 < log 4 - 1 + b, where w0 = wo(b). Notice that if < c(log log y)1/6 then (7.102) implies (log
K, e ,2/ loge 4.
(7.110)
> e 2/log'4 >> 1 - FO
(7.111)
Y)-A
Moreover in any case (log
y)-A
by (7.89). We could therefore write instead of 1 in (7.101) if we wished. It would be desirable, and this is a more delicate question, to obtain a result in which appeared as the main term with error o(1 - F(c)). It remains to prove (7.102). Let A(X) be defined as above, with X instead of / og log y on the right of (7.100). Then X and t increase from 0 to 0o together, and X = dX = 2 log(1 + t) + log 4 > log 4.
(7.112)
7.4 The asymptotic formula for H(x, y, z)
257
The function A(X) E C°°(R+), and we have A(X) + t log 4 = X.
(7.113)
From (7.100), t - X (2 log X) as X -* oo whence A(X) - X as X -> oo from (7.113). Notice that for X sufficiently large, the error term in (7.101)
exceeds that in (7.91). This occurs when A(X) < X + log4 - 2, that is when t > (log 2)-1 - I or X > .78584..., which is therefore a suitable value of b in the statement of the theorem. By Taylor's theorem, we have
x2 A(X) = A(0) + XA'(0) +
A"(0) +
i
3
A"'(OX)
(7.114)
for some 0 E (0,1). We compute the derivatives from (7.113) using the relations dt t'
dX
t" t"'
+ 3X2X-5 =
(1 + t)2
{X-4 + 6X-s}.
(7.115)
Since X increases with t and X, t"' plainly decreases whence 0 < t"' < t"'(0), and -t" (0) log 4 < A"'(0X) < 0. We have t(0) log 4 = 3.9997... < 4, and (7.102) follows.
Bibliography
L. Bastick (1992) D.Phil Thesis, York University, U.K. F.A. Behrend (1932-3) Uber numeri abundantes I, II. Sitzungsberichte der PreuJi. Akad. phys-math. Masse. 322-328, 280-293.
F.A. Behrend (1935) On sequences of integers not divisible one by another. J. London Math. Soc. 10, 42-4. F.A. Behrend (1948) Generalization of an inequality of Heilbronn and Rohrbach. B. Amer. Math. Soc. 54, 681-4. A.S. Besicovitch (1934) On the density of certain sequences. Math. Annalen. 110, 336-41.
K. Bognar (1970) On a problem of statistical group theory. Studia Sci. Math. Hung. 5, 29-36. J.D. Bovey (1977) On the size of prime factors of integers. Acta Arith. 33, 65-80. S. Chowla (1934) On abundant numbers. J. Indian Math. Soc. (2) 1, 41-4.
H. Davenport (1933) Uber numeri abundantes. Sitzungsberichte Akad. Wiss. Berlin. 27, 830-7.
H. Davenport & P. Erdos (1937) On sequences of positive integers. Acta Arith. 2, 147-51.
H. Davenport & P. Erdos (1951) On sequences of positive integers. J. Indian Math. Soc. (2) 15, 19-24. L.E. Dickson (1931) Even abundant numbers. Amer. J. Math. 35, 413-426. Y. Dupain, R.R. Hall and G. Tenenbaum (1982) Sur 1'equirepartition modulo 1 de certaines fonctions de diviseurs. J. London Math. Soc. (2) 26, 397-411. P.D.T.A. Elliott (1979-80) Probabilistic number theory. 1,11. Springer-Verlag. P. Erdos (1934) On the density of the abundant numbers. J. London Math. Soc. 9, 278-82. P. Erdos (1935a) On primitive abundant numbers. J. London Math. Soc. 10, 49-58. P. Erdos (1935b) Note on sequences of integers no one of which is divisible by any other. J. London Math. Soc. 10, 126-8. P. Erdos (1936) A generalization of a theorem of Besicovitch. J. London Math. Soc. 11, 92-8. P. Erdos (1946) On the coefficients of the cyclotomic polynomial. B. Amer. Math. Soc. 52, 179-84. 258
Bibliography
259
P. Erdos (1948a) On the density of some sequences of integers. B. Amer. Math. Soc. 54, 685-92. P. Erdos (1948b) Integers with exactly k prime factors. Ann. Math. 11 49, 53-66.
P. Erdos (1959) Some remarks on prime factors of integers. Can. J. Math. 11, 161-7.
P. Erdos (1965) On the distribution of divisors of integers in the residue classes (mod d). B. Math. Soc. Grece. 6, 27-36
P. Erdos (1969) On the distribution of prime divisors. Aequationes Math. 2, 177-83.
P. Erdos (1979) Some unconventional problems in number theory. Asterisque. 61, 73-82. P. Erdos & R.R. Hall (1974) Some distribution problems concerning the divisors of integers. Acta Arith. 26, 175-188.
P. Erdos & R.R. Hall (1976a) Proof of a conjecture about the distribution of divisors of integers in residue classes. Math. Proc. Camb. Phil. Soc. 79, 281-9. P. Erdos & R.R. Hall (1976b) Probabilistic methods in group theory H. Houston J. Math. 2, 173-180.
P. Erdos & R.R. Hall (1978) Some new results in probabilistic group theory. Comment. Math. Helvetici. 53, 448-457. P. Erdos & R.R. Hall (1980) On the Mobius function. J. Reine Angew. Math. 315, 121-6.
P. Erdos, R.R. Hall. & G. Tenenbaum (1994) On the densities of sets of multiples. J. Reine Angew. Math. 454, 119-141. P. Erdos & M. Kac (1939) On the Gaussian law of errors in the theory of additive functions. Proc. Nat. Acad. Sci. U.S.A. 25, 206-7. P. Erdos & M. Kac (1940) The Gaussian law of errors in the theory of additive number-theoretic functions. Amer. J. Math. 62, 738-42. P. Erdos & A. Renyi (1965) Probabilistic methods in group theory. J. d'Analyse Math. 14, 127-138. P. Erdos, A. Sarkozy & E. Szemeredi (1967a) On a theorem of Behrend. J. Austr. Math. Soc. 7, 9-16. P. Erdos, A. Sarkozy & E. Szemeredi (1967b) On an extremal problem concerning primitive sequences. J. London Math. Soc. 42, 484-8. P. Erdos & P. Turin (1948) On a problem in the theory of uniform distribution I, II. Indag. Math. 10, 370-8, 406-13.
J. Franel (1924) Les suites de Farey et le probleme des nombres premiers. Gottinger Nachr. 198-201. G. Freud (1952-4) Restglied eines Tauberschen Satzes I, II, III. Acta Math. Acad. Sci. Hungar. 2, 299-308, 299-307, 275-89.
J. Friedlander & A. Granville (1989) Limitations to the equi-distribution of primes I. Ann. Math. 129, 363-382. J. Friedlander, A. Granville, A. Hildebrand & H. Maier (1991) Oscillation theorems for primes in arithmetic progressions and for sifting functions. J. Amer. Math. Soc. 4, 25-86. G.R.H. Greaves, R.R. Hall, M.N. Huxley & J.C. Wilson (1993) Multiple Franel Integrals. Mathematika. 40, 50-69. H. Halberstam & H-E. Richert (1974) Sieve methods. Academic Press, New York.
260
Bibliography
H. Halberstam & H-E. Richert (1979) On a result of R.R. Hall. J. Number Theory. 11, 76-89. H. Halberstam & K.F. Roth (1966) Sequences. Oxford University Press. R.R. Hall (1972) On a theorem of Erdos and Renyi concerning Abelian groups. J. London Math. Soc. (2) 5, 143-53.
R.R. Hall (1974a) Halving an estimate obtained from Selberg's upper bound method. Acta Arith. 25, 247-351. R.R. Hall (1974b) The divisors of integers I. Acta Arith. 26, 41-46. R.R. Hall (1975a) The divisors of integers II. Acta Arith. 28, 129-35. R.R. Hall (1975b) Sums of imaginary powers of the divisors of integers. J. London Math. Soc. 9, 571-80. R.R. Hall (1976) The distribution of f(d) (mod 1). Acta Arith. 31, 91-97. R.R. Hall (1977) Extensions of a theorem of Erdos-=Renyi in probabilistic group theory. Houston J. Math. 3, 225-234. R.R. Hall (1978) A new definition of the density of integer sequences. J. Austr. Math. Soc. A. 26, 487-500. R.R. Hall (1981) The divisor density of integer sequences. J. London Math. Soc. (2) 24, 41-53. R.R. Hall (1989) The distribution of square-free numbers. J. Reine Angew. Math. 394, 107-17. R.R. Hall (1990a) Large irregularities in sets of multiples and sieves. Mathematika. 37, 119-135. R.R. Hall (1990b) Sets of multiples and Behrend sequences, in A tribute to Paul Erdos. (ed. A. Baker, B. Bollobas & A. Hajnal.) Cambridge University Press. R.R. Hall (1992) On some conjectures of Erdos in Asterisque I. J. Number Theory. 42, 313-319. R.R. Hall (1994) S2 theorems for the complex divisor function. Math. Proc. Camb. Phil. Soc. 115, 145-57. R.R. Hall & A. Sudbery (1972) On a conjecture of Erdos and Renyi concerning Abelian groups. J. London Math. Soc. (2) 6, 177-89.
R.R. Hall & G. Tenenbaum (1986) Les ensembles de multiples et la densite divisorielle. J. Number Theory. 22, 308-33. R.R. Hall & G. Tenenbaum (1988) Divisors. Cambridge University Press.
R.R. Hall & G. Tenenbaum (1989) The set of multiples of a short interval, in Number Theory New York Seminar. (ed. D.V. Chudnovsky, G.V. Chudnovsky, H. Cohn & M.B. Nathanson.) Springer-Verlag.
R.R. Hall & G. Tenenbaum (1992) On Behrend sequences. Math. Proc. Camb. Phil. Soc. 112, 467-82.
G.H. Hardy (1915) The mean value of the modulus of an analytic function. P. London Math. Soc. (2) 14, 269-77. G.H. Hardy (1949) Divergent series. Oxford University Press. G.H. Hardy, J.E. Littlewood & G. Polya (1934) Inequalities. Cambridge University Press.
G.H. Hardy & S. Ramanujan (1917) The normal number of prime factors of a number n. Quart. J. Math. 48, 76-92. H. Heilbronn (1937) On an inequality in the elementary theory of numbers. Proc. Camb. Phil. Soc. 33, 207-9.
Bibliography
261
A. Hildebrand & H. Maier (1989) Irregularities in the distribution of primes in short intervals. J. Reine Angew. Math. 397, 162-93. J. Karamata (1931) Neuer Beweis and Verallgemeinerung der Tauberschen Satze, welche die Laplaceshe and Stieltjesche Transformation betreffen. J. Reine Angew. Math. 164, 27-40. 1. Katai (1976a) The distribution of divisors (mod 1). Acta Math. Acad. Sci. Hung. 27, 149-52. I. Katai (1976b) Distribution mod 1 of additive functions on the set of divisors. Acta Arith. 30, 2, 9-12. J.C. Kluyver (1903) An analytical expression for the greatest common divisor of two integers. Proc. Roy. Acad. Amsterdam. 5, 658-62. J. Kubilius (1964) Probabilistic methods in the theory of numbers. Amer. Math. Soc. Monographs. No. 11 Providence R.I. L. Kuipers & H. Niederreiter (1974) Uniform distribution of sequences. Wiley, New York. E. Landau (1927) Vorlesungen fiber Zahlentheorie. S. Hirzel. Leipzig. H. Maier (1985) Primes in short intervals. Michigan Math. J. 32, 221-25.
H. Maier & G. Tenenbaum (1984) On the set of divisors of an integer. Invent. Math. 76, 121-8.
R.J. Miech (1967) On a conjecture of Erdos and Renyi. Illinois J. Math. 11, 114-27.
K.K. Norton (1978) Estimates for partial sums of the exponential series. J. Math. & Appl. 63, 265-96.
A. Perelli & U. Zannier (1989) An extremal property of the Mobius function. Arch. Math. 53, 20-29. S. Pillai (1939) On numbers which are not multiples of any other in the set. Proc. Indian Acad. Sci. A 10, 392-4. K. Prachar (1957) Primzahlverteilung. Springer-Verlag. H-E. Richert (1967) Zur Abschatzung der Riemannschen Zeta-funktion in der Nahe der Vertikalen a = 1. Math. Annalen. 169, 97-101
H. Rohrbach (1937) Beweis einer zahlentheoretischen Ungleichung. J. Reine Angew. Math. 177, 193-6.
I.Z. Ruzsa (1976) Probabilistic generalization of a number theoretic inequality. Amer. Math. Monthly. 83, 723-5. I.Z. Ruzsa (1988) On an additive property of squares and primes. Acta Arith. 49, 281-89. I.Z. Ruzsa & G. Tenenbaum (199x) A note on Behrend sequences. Acta Math. Hung. (to appear). 1. Schoenberg (1928) Uber die asymptotische Verteilung reeller Zahlen mod 1. Math. Zeitschrift. 28, 171-99. P. Shin (1980) A Brun-Titchmarsh theorem for multiplicative functions. J. Reine Angew. Math. 313, 161-70. G. Tenenbaum (1979) Lois de repartition des diviseurs 4. Ann. Inst. Fourier. 29, 1-15.
G. Tenenbaum (1980) Lois de repartition des diviseurs 2. Acta Arith. 38, 1-36. G. Tenenbaum (1982) Sur la densite divisorielle d'une suite d'entiers. J. Number Theory. 15, 331-46.
262
Bibliography
G. Tenenbaum (1984) Sur la probabilite qu'un entier possede un diviseur dans un intervalle donne. Composito Math. 51, 243-63. G. Tenenbaum (1988) Un probleme de probabilite conditionelle en Arithmetique, Acta Arith.49, 165-87. G. Tenenbaum (1990) Introduction a la theorie analytique et probabiliste des nombres, Institut Elie Cartan 13, Nancy. English translation: Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995. G. Tenenbaum (199x) On block Behrend sequences. Math. Proc. Cam. Phil. Soc. (to appear). G. Tenenbaum (199y) Uniform distribution on divisors and Behrend sequences. l'Enseignement Mathematique (to appear). E.C. Titchmarsh (1939) The theory of functions. Oxford University Press. E.C. Titchmarsh (1951) The theory of the Riemann zeta-function. Oxford University Press.
J.H. van Lint & H-E. Richert (1965) On primes in arithmetic progressions. Acta Arith. 11, 209-216. I.M. Vinogradov (1958) A new estimate for i'(1+it) (in Russian). Izv. Akad. Nauk. S.S.S.R. Ser. Math. 22, 161-164. J.C. Wilson (1994) On Franel-Kluyver integrals of order three. Acta Arith. 66, 71-87.
D. Wolke (1971) Multiplikative Funktionen auf schnell wachsenden Folgen. J. Reine Angew. Math. 251, 54-67.
Index
appalling notation, 89 Asterisque, 129 asymptotic density, 3-6, 9, 11-13, 170, 172,
divisor density, 3, 170-175, 177-179, 183, 187-189, 192, 193, 199, 201, 202, 204-206, 218 divisor uniform distribution, 203, 204 div isor uniforml y distributed , 203-207 , 210 ,
174, 179, 187, 203, 206, 210, 215
211, 213-216, 219
Bastick, 244 Behrend sequences, 12 Behrend, 2, 13, 15, 16, 18, 26 Behrend inequality, 15, 82-84 Behrend sequence, 12, 14-16, 18, 21, 26, 29,
Divisors, 7, 27, 32, 33, 37, 39, 42, 49, 50, 52, 53, 55, 56, 155, 214, 217, 227, 230, 232, 239-245, 251-253 Double Variance, 233 Dupain, 213
33-36, 37, 39, 40, 41, 44-47, 49, 51, 52, 54-58, 60-65, 67, 72, 82, 86, 122-124, 126, 128, 146, 153, 155, 156, 166, 167, 171, 178, 179, 193, 205, 225, 226
elementary multilinear function, 91, 93, 95, 101
Elliott, 127, 146
Behrend's inequality, 15, 21, 64, 67, 69, 71,
Erdds, 2, 5-7, 11, 12, 26, 27, 29, 30, 33, 36, 37, 46-48, 58, 66, 99, 126-131, 137-139, 146, 149, 153, 154, 156, 161, 167, 168, 170, 171, 179, 192-194, 196, 213, 216, 218, 226, 238, 244
85, 86
Behrend's lemma, 16 Bernoulli trials, 73, 77, 80, 81, 177 Bertrand's postulate, 80 Besicovitch, 5, 7, 8, 12, 179 B esi covit ch sequence, 11 , 12 , 26 , 27 , 29 ,
block sequence, 47 Bognar, 168 Borel-Caratheodory inequality, 197 Brun's sieve, 88 Buchstab, 100, 245
faithful family, 112-114 Fejer kernel, 212 Fejer mean, 211 Franel, 98 Freud, 218 Friedlander, 101
Central Limit Theorem, 177 checkpoints, 38, 39, 48 complete family, 92, 94, 101, 108, 113, 117,
Granville, 101 Greaves, 99 group character, 131-134
32-35, 58, 63, 96, 122, 126, 239
118
Halberstam, 2, 7, 11, 16, 28, 29, 42, 49, 78, Davenport, 5, 11, 12, 66, 179 deficiency, 92, 94, 95, 104-106 derived sequences, 3, 67, 104, 105, 111, 112, 117
Dickman's theorem, 206 discrepancy, 96, 203-206, 213, 214, 217,
160, 214, 232, 251 Hall, 7, 12, 23, 25, 27, 30, 33, 36, 47, 48, 50, 99, 111, 130, 131, 138, 139, 153, 155, 160, 168, 170, 173, 174, 183, 188, 189, 193, 196, 203, 204, 206, 208, 213, 217, 226, 229, 231, 233, 238, 240, 242
Hankel's integral, 247
218, 226-228
263
264
Index
Hardy, 8, 11, 41, 46, 116, 126, 156, 166, 167, 178, 179, 182, 186, 189, 205, 209, 217, 218, 223, 224, 227, 231, 232 Heilbronn, 13, 14, 70, 80 Hildebrand, 101 Huxley, 99 Jensen's inequality, 238 k-minimal, 106-108 k-primitive, 86, 87 Kac, 127, 146, 154, 156, 161
Karamata, 8, 11, 179, 182, 186, 189, 218 Kluyver, 98 Kubilius, 139 Kuipers, 206 Katai, 214-217 Landau, 98, 99 Littlewood, 8, 11, 116, 179, 182, 186, 189, 218
logarithmic density, 3-5, 12, 61, 63, 66, 67, 83, 175, 179, 187, 192, 199, 201, 210, 216,219,223-225 Maier, 47, 101, 155 Markoff's inequality, 76, 133, 135, 148, 151, 215, 218 maximally incomplete, 119, 120 Miech, 129, 168 minimally complete, 106, 117 multilinear function, 88-91, 103 multiplicatively non-increasing, 18, 20, 85, 86
Niederreiter, 206 non-trivial, 1, 12, 13 non-trivial sequence, 21, 61, 83, 86, 87, 97, 116-118, 120 Norton, 249 Oberwohlfach, 64 orthogonality relation, 131, 132 Perelli, 99 perfect sequence, 96, 122-125 Pillai, 2
Prachar, 143 primitive sequence, 1-3, 15, 18, 21, 33-35, 62, 86, 87, 96, 97, 102, 117, 118, 120-123,125,174 Polya, 116
Ramanujan, 41, 46, 126, 156, 166, 167, 178, 205, 209, 217, 223, 224, 227, 231 Rankin, 43, 245 Renyi, 126, 128-130, 146, 149, 167, 168 Richert, 7, 28, 42, 49, 160, 197, 214, 232, 251
Rohrbach, 13, 14, 70, 80 Roth, 2, 11, 16, 29, 78 Ruzsa, 15, 16, 18, 22 Sarkozy, 2 Second Mean Value Theorem, 195 Selberg, 42 Shin, 42, 43, 140, 252 short interval, 32, 240 Siegel zero, 138, 143, 144 slowly switching, 192, 198 span, 92-95, 104, 108, 112, 113, 120 Sperner chains, 114 split, 19-21 Stirling's formula, 112, 147, 152, 247, 249, 254 Sudbery, 168 symmetric decreasing rearrangement, 116 Szemeredi, 2
taut sequence, 14, 15, 21 Tenenbaum, 7, 12, 27, 30, 32, 33, 47, 48, 50, 56, 64, 139, 155, 173, 174, 183, 188, 196, 206, 210, 213, 226, 239, 240, 242 threshold, 138, 240, 243 tip of an iceberg, 89 Titchmarsh, 196, 197, 221 total decomposition set, 22, 24, 87-91, 93, 94, 101, 104, 110, 116, 118, 239 Turan, 59, 141, 213, 216, 218, 227, 238
van Lint, 160 Vinogradov, 197
weak block sequence, 47, 49 Weierstrass' Approximation Theorem, 186 Weyl, 210, 213, 214, 218, 219, 229 Wilson, 99 witness, 60, 63 Wolke, 43 Zannier, 99
