Introduction to Analytic Number Theory

330 Pages·1988·16.762 MB·English

by  Aleksei Georgievich Postnikov

Introduction to
Analytic Number Theory
A. G. POSTNIKOV
Volume 68

TRANSLATIONS OF MATHEMATICAL MONOGRAPHS
68
VOLUME
Introduction to
Analytic Number Theory
A. G. POSTNIKOV
American Mathematical Society · Providence · Rhode Island
AJIEKCEf:I rEOPrHEBHl..f TIOCTHHKOB
BBE,Lt:EHHE B AHAJIHTHl..fECKYIO
TEOPHIO l..fHCEJI
«HAYKA>>, MOCKBA, 1971
Translated from the Russian by G. A. Kandall
Translation edited by Ben Silver
1980 Mathematics Subject Classification (1985 Revision). Primary 11-02,
11M45, 11N60, 11Q05.
ABSTRACT. This book is intended as a mid-level view of analytic num
ber theory, somewhere between a textbook and the latest monographs.
The author has attempted to give as broad a picture as possible of the
problems of analytic number theory, while avoiding specialization and
those topics already sufficiently well treated in the literature. This is
the meaning of the title, Introduction to analytic number theory. The
deep results in analytic number theory involve, of course, the use of
well-developed machinery. However, it is advisable for a young scholar
to have, in addition to a mastery of powerful tools, a supply of problems
to which he can apply these strong techniques. In this the author has
tried to be of assistance to his young colleagues. This book is directed
to scholars, teachers, and graduate students interested in number theory
and its connections with other branches of science.
Library of Congress Cataloging-in-Publication Data
Postnikov, A. G. <Aleksel Georgievichl
[Vvedenie v analiticheskufU teoriiU chisel. English]
Introduction to analytic number theory I A.G. Postnikov
translated from the Russian by G.A. Kandall ; translation edited by
Ben Si lverl.
p. cm. -- <Translations of mathematical monographs ; v. 68>
Translation of: Vvedenie v analiticheskufU teorifU chisel.
ISBN 0-8218-4521-7
1. Numbers, Theory of. I. Title. II. Series.
QA241.P6713 1987
512'. 73--dc19 87-33428
CIP
Copyright @1988 by the American Mathematical Society. All rights reserved.
Printed in the United States of America
The American Mathematical Society retains all rights
except those granted to the United States Government.
Copying and reprinting information can be found at the back of this volume.
The paper used in this book is acid-free and falls within the ~idelines
established to ensure permanence and durability. 1::9
Contents
Preface vn
Standard notation viii
Introduction 1
CHAPTER 1. Facts from analysis
§1.1. Tauberian theorems for power series and Dirichlet series 23
§1.2. The Hardy-Littlewood Tauberian theorem with Freud's
remainder term 30
§1.3. Ingham's Tauberian theorem 51
§1.4. The generalized Esseen inequality 62
CHAPTER 2. Additive problem~ with an increasing and an infinite
number of summands
§2.1. An additive sequence duplication problem 69
§2.2. Schneider's lemma 71
§2.3. The local limit theorem of probability theory and its
applications to number theory 77
§2.4. Additive problems with an increasing number of summands 81
§2.5. Bredikhin's theorem 92
§2.6. The asymptotic law for the distribution of basis elements for
free semigroups 99
§2.7. The Hardy-Ramanujan problem 116
§2.8. Ingham's additive theorem 129
CHAPTER 3. Theory of functions of a natural argument
§3.1. Metric theory of functions of a natural argument 135
§3.2. Wintner's theorem 138
§3.3. Almost periodic functions of a natural argument 144
§3.4. Independent functions of a natural argument 158
§3.5. Polyadic analysis and its applications 166
§3.6. Axer's theorem 178
iii
iv CONTENTS
CHAPTER 4. Theory of multiplicative functions
§4.1. Upper bounds of multiplicative functions 183
§4.2. Summation of values of Euler's function 188
§4.3. Wirsing's theorem 195
§4.4. The Turan-Kubilius inequality 215
§4.5. Delange's theorems 223
§4.6. Remarks on summation of multiplicative functions 240
§4. 7. The Erdos-Wintner theorem 246
§4.8. On the distribution of values of Euler's function 254
§4.9. Generalization of the characteristic function method
and the theory of multiplicative functions 269
§4.10. Distribution of values of unstable additive functions 273
§4.11. Asymptotic expansions for sums of multiplicative
functions 277
§4.12. A problem on summation of multiplicative functions in
which power reduction can be obtained 289
Bibliography 301
Appendix 311
This book is dedicated
to the memory of
Vyacheslav Vasil' evich Stepanov
Preface
The literature on analytic number theory is very rich and naturally reflects
the historical development of this science. As an introduction to analytic
number theory one can look to the supplementary material (problems and
solutions) in Vinogradov's textbook, Elements of number theory [148]. The
monograph Number theory [13] by Borevich and Shafarevich discusses the
foundations of the method of Dirichlet generating series (applied to algebraic
number theory). The book Elementary methods in analytic number theory [60]
by Gel'fond and Linnik can be regarded as an elementary reader on the sub
ject. One can find books in Russian dealing with Diophantine approximation
(Cassels [17]), the approximation of algebraic and transcendental numbers
(Gel'fond [59]), the method of trigonometric sums (Vinogradov [149]), the
distribution of primes and the theory of Dirichlet £-functions (Ingham [74],
Chudakov [20], Titchmarsh [139], and Prachar [118]), probabilistic number
theory (Kubilius [87]), and so on. Nevertheless, there are still some funda
mental questions in analytic number theory that have not been adequately
discussed in the literature in any systematic way-these are questions directly
or indirectly connected with the concept of a numerical semigroup, in other
words, with general additive number theory. It is to this aspect of number
theory that the present book is devoted.
It is assumed that the reader is familiar with the fundamentals of analy
sis, number theory, and probability theory. The necessary analysis is covered
by Fikhtengol'ts [48], [49], [50], the number theory by Vinogradov [148] or
Bukhshtab [16], and the probability theory by the first eight chapters of Gne
denko [61].
The author thanks G. A. Freiman for his simple proof of the Hardy
Ramanujan asymptotic formula. He is also grateful to K. Yu. Bulota for
taking on the difficult task of editing this book; his constructive criticism was
most helpful. Finally, the author expresses his sincere gratitude to all who
helped in the preparation of this volume.
v
Standard Notation
To avoid being repetitious let us agree on the meaning of the following
notation (unless stated otherwise):
Euler's constant,
I
X4(n) the nonprincipal character modulo 4,
cp(n) Euler's function,
r(n) the number of natural divisors of n,
a(n) the sum of the natural divisors of n,
v(n) the number of distinct prime divisors of n,
r(n) the number of representations of n as a sum of two squares,
ap(n) the exponent of the prime p in the canonical factorization of n,
µ(n) the Mobius function,
lnp n=pa.,
A(n) von Mangoldt's function, A(n) = { ' a.
0 4
' n..,...p'
(x) the distance from the real number x to the nearest integer,
[x] the integral part of the real number x,
{x} the fractional part of x ({ x } = x - [x]),
r(s) the gamma-function,
s-(s) the zeta-function,
7r(x) the number of primes not exceeding x.
vi
Introduction
We will give here an overview of the content of this book and reflect on
these thoughts in the course of our exposition.
Modern life and, in particular, our mathematical life are inconceivable with
out a continual "separation" of machinery from problems. Consequently, be
fore entering into a discussion of the problems of analytic number theory it is
convenient to present a chapter of an auxiliary nature.
The first chapter is devoted to an exposition of some facts from analysis.
Here we will be concerned mainly with Tauberian theorems.
In analytic number theory the method of generating functions has broad
applicability (power series are mainly used in additive number theory, Dirich
let series in the theory of multiplicative functions, and characteristic functions
in questions concerning the distribution of values of arithmetical functions).
An indispensable feature of the method of generating functions is the use of
inversion (which leads in various problems to explicit formulas). The gener
ating series of the classical problems of number theory admit of investigation
in the whole complex plane; hence the most convenient tools for inversion are
Cauchy's formula for the coefficients of power series and Perron's formula for
the summation function of a Dirichlet series. In general problems, information
about the Dirichlet generating series is much more meager. This is the reason
for using Tauberian theory in general number theory problems.
In the first chapter we also discuss Esseen 's theorem pertaining to the
method of characteristic functions.
Many problems of analytic number theory are of an additive nature. To
clarify this we formulate several problems.
We first consider direct additive problems.
Suppose we are given two finite or infinite sequences of real numbers, ar
ranged in increasing order:
>.i, >.2, >.3, ... ' (1)
µi, µ2, µ3, .. . (2)
2 INTRODUCTION
(we may assume that each of these two sequences has at most one limit point
at infinity). We form the set of pairwise sums
(3)
To a point r of this set we attach a multiplicity equal to the number of
representations of r in the form (3). One problem is to obtain properties of
the set of pairwise sums from properties of sequences (1) and (2).
This general formulation includes many problems, among which we distin
guish two types.
a) Problems on the distribution of multiplicities. These problems will be
called local. Here we want to study the quantity
2:
p(r) = i. (4)
r=>..;+µ;
A more general problem is the following. Suppose we are given two functions,
fi(x) and h(x), of a real variable. We want to study the quantity
2:
p(r) = 1i(>.i)h(µj)· (4')
r=>..;+µ;
We will adopt this terminology: a function f(x) defined on some sequence
will be called a law, and the quantity p( r) defined by ( 4') will be called the
convolution of Ii and h and will be denoted by p = Ii * h.
b) We denote by n1 ( u) and n2 (u ) the number of elements of the sequences
(1) and (2), respectively, that do not exceed u (u is an asymptotically increas
ing parameter); we denote by q(u) the number of solutions of the inequality
>.i + µj ~ u. The problem is to study q(u), given n1(u) and n2(u). Such
problems will be called integral. Obviously,
L L
= =
q(u) n2(u - >.i) n1(u - µj)·
Bachmann's book Die Analytische Zahlentheorie (6], which appeared at
the end of the last century, played a large role in the development of analytic
number theory. On pp. 447-449 of that book the following theorem is proved.
THEOREM. Suppose A, B, a, and /3 are given positive numbers. Let q(N)
denote the number of pairs ( x, y) of natural numbers such that Ax°'+ Byf3 ~ N.
Then
(N)l/fJ)
_ N1/a+l/f3 _1_r(I/a)f(l//3) _ ((N)l/a
q(N)- Al/aBl/f3a+/3f(l/a+l//3) O A + B '
0<0<1.