Topics in Arithmetical Functions

281 Pages·1980·8.56 MB·English

by  J.-M. De Koninck and A. Ivić (Eds.)

TOPICS IN ARITHMETICAL FUNCTIONS
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NORTH-HOLLAND
MATHEMATICS STUDIES 43
Notas de Maternatica (72)
Editor: Leopoldo Nachbin
Universidade Federal do Rio de Janeiro
and University of Rochester
Topics in
Arithmetical Functions
Asymptotic formulae for sums of reciprocals of
arithmetical functions and related results
J.-M. DE KONINCK
University of Lava1
Quebec, Canada
and
A. lVlc
University ofB elgrade
Belgrade, Yugoslavia
1980
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD
North-Holland Publishing Company, 1980
All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem,
or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording
or otherwise, without the prior permission of the copyright o wner.
ISBN: 0444860495
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PRINTED IN THE NETHERLANDS
I NT RO DU CT IO N
The object of this monograph is to examine several topics in the
theory of arithmetical functions. The principal topic to be studied is that
of asymptotic formulae for the sum
where f(n) is a non-negative arithmetical function, and the sununation is
over those n not exceeding P for which f(n)*O . These sums possess
an intrinsic interest apart from their number-theoretic applications.
Two of the most interesting classes of arithmetical functions, na-
mely multiplicative and additive functions, require different techniques
for the estimation of (1). Since the reciprocal of a multiplicative func-
tion is itself multiplicative, we see that for the case of multiplicative
functions f , the estimation of (1) may be regarded as a special case of
the estimation of
where g(n) is multiplicative. Since an extensive literature already
exists for this problem, we will deal with reciprocals of multiplicative
functions in Chapter 1 only. There, we use important analytic tools such
vi J.M. DE KONINCK AND A. IVIC
as the convolution method and the method of complex integration in order
to prove the basic Lemma 1.1, which is repeatedly used in later chapters.
When f is additive, the sum (1) is much more difficult to esti-
mate, and no significant results were hown for a long time. The first
horn result is due to R.L. Duncan C11, who proved the inequality
(3)
A general method for estimating reciprocals of additive functions
was introduced by the first named author in his 1972 doctoral dissertation
at Temple University, Pa., parts of which appeared in De Koninck C11. This
method, which rests on the fact that zf(n) is multiplicative whenever f
is additive, is explained in Chapter 2 and yields improvements of (3) to
(4) ZI l/w(n) = A1x/loglog3c t.. .tA~/(loglogz)'t O(x/(loglogz)'tl) ,
n a
where IV is arbitrary but fixed, A1=l , and the remaining Ails are
computable.
Subsequent papers, a number of which are due to one or both of the
authors, contained further results which give more than enough material
for a systematic account of sums of reciprocals of arithmetical functions.
This topic seems particularly well suited for a monograph such as this,
since it allows treatment in considerable depth without being too wide in
scope. Furthermore, this branch of analytic number theory is quite new,
and no general self-contained publication has yet appeared on this subject.
We have included a number of our hitherto unpublished results, as
well as some sharpened asymptotic formulae. In particular, we improve (4)
INTRODUCTION vi i
in Chapter 5 to
where M is arbitrary but fixed, L .(z) (j= 1,.. . ,M) is a slowly oscil-
3
lating function admitting an asymptotic expansion of the form
(6) Lj (z) a . (log log z) - '+. . .t a .( log log x)4 t O( (log logxrN-l) ,
1J3 N, 3
where all the constants are computable, and N is an arbitrary but fixed
integer. Although no claim is made that (5) is the best possible asympto-
tic formula, it certainly will be difficult to improve.
A large number of additive functions arise from the logarithms of
positive multiplicative functions. In Chapter 3 we give asymptotic formu-
lae for reciprocals of logarithms of some of the most important multiplica-
tive functions. In addition we establish asymptotic formulae for recipro-
cals of many interesting additive functions. The methods we develop are
also used to estimate
(7)
where both g and f are additive; this is the subject considered in
Chapter 4.
We have devoted Chapter 7 to the study of reciprocals in "short"
intervals. These are sums of the type
viii J.M. DE KONINCK AND A. IVIC
.
where the interval is "short" in the sense that h o(z) as x+- The
method used allows us to obtain estimates for other sums as well, such as
(9) A (z,h) = 1 1 ,
4 zwath,f (n) =q
where q is a fixed integer, and f is a suitable arithmetical function.
This estimate is established by using both general results and methods of
analytic number theory. We hope that digressions of this sort will make
the book more interesting for the general reader. Even in these digres-
sions, "reciprocals of arithmetical functions" remains the thread which
holds the whole together.
Analytic methods are also employed in Chapter 8 to estimate
where f belongs to a certain class of non-negative additive functions,
and 9 is a suitable subset of the set of natural numbers. It is only
in Chapter 6 that the analytic approach is abandoned, and special elemen-
tary methods are used to deal with large additive functions such as
, B(n) 1 ap . Apart from the asymptotic formula (Theo-
Pal In
rem 6.2)
the results are not as sharp as those obtained in other chapters, although
we are certain that sharper results (such as those stated as open problems
in Chapter 9) can be obtained.
INTRODUCTION ix
This monograph does not resolve all the major problems connected
with asymptotic formulae for reciprocals of arithmetical functions. In
fact we give a list of open problems in the last chapter. It is our mo-
dest hope that this monograph will induce further research in this interest-
ing field.
Although this book is intended primarily for specialists, we think
that it will be of interest to the more general reader as well. Apart
from a general knowledge of analytic number theory and the theory of arith-
metical functions, which may be found in a number of standard texts such
as Hardy and Wright C11, Ayoub 111, Grosswald C11, etc., the reader needs
only to possess a basic knowledge of calculus and of complex analysis. The
text is, with some minor exceptions, completely self-contained; we have
tried to make the exposition as clear as possible, without omitting impor-
tant details, by deleting routine calculations and repetition of similar
arguments.
Each chapter is followed by a section of Notes, where all necessa-
ry clarifications and discussions are given, together with appropriate
references. We have tried to include in these references all the papers
which deal with asymptotic formulae for reciprocals of arithmetical func-
tions and related topics.
The notation is kept standard throughout and is fully explained in
the next section.
A number of mathematicians have kindly read the manuscript and
have made many useful critical remarks and suggestions. We take this op-
portunity to thank Dr. E. Brinitzer-Scriba, Professor P. Erdos, Professor