Arithmetical Functions

243 Pages·1970·6.56 MB·English

by  Prof. Dr. K. Chandrasekharan (auth.)

Die Grundlehren der
mathematischen Wissenschaften
in Einzeldarstellungen
mit besonderer Berucksichtigung
der Anwendungsgebiete
Band 167
Herausgegeben von
J. L. Doob . A. Grothendieck . E. Heinz· F. Hirzebruch
E. Hopf . H. Hopf . W. Maak . S. MacLane . W. Magnus
M. M. Postnikov . F. K. Schmidt . D. S. Scott . K. Stein
Geschiiftsfuhrende H erausgeber
B. Eckmann und B. L. van der Waerden
K. Chandrasekharan
Arithmetical Functions
Springer-Verlag Berlin Heidelberg New York 1970
Prof. Dr. K. Chandrasekharan
Eidgenossische Technische Hochschule Zurich
Geschiiftsfiihrende Herausgeber:
Prof. Dr. B. Eckmann
Eidgenossische Technische Hochschule Zurich
Prof. Dr. B. L. van der Waerden
Mathematisches Institut dec Universitiit Zurich
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ISBN 978-3-642-50028-2 ISBN 978-3-642-50026-8 (eBook)
DOI 10.1007/978-3-642-50026-8
© by Springer·Verlag Berlin' Heidelberg 1970. Library of Congress Catalog Card Number 72-102384.
Title No. 5150.
For Sarada
Preface
The plan of this book had its inception in a course of lectures
on arithmetical functions given by me in the summer of 1964
at the Forschungsinstitut fUr Mathematik of the Swiss Federal
Institute of Technology, Zurich, at the invitation of Professor
Beno Eckmann. My Introduction to Analytic Number Theory
has appeared in the meanwhile, and this book may be looked
upon as a sequel. It presupposes only a modicum of acquaintance
with analysis and number theory.
The arithmetical functions considered here are those associated
with the distribution of prime numbers, as well as the partition
function and the divisor function. Some of the problems posed
by their asymptotic behaviour form the theme. They afford a
glimpse of the variety of analytical methods used in the theory,
and of the variety of problems that await solution.
I owe a debt of gratitude to Professor Carl Ludwig Siegel,
who has read the book in manuscript and given me the benefit
of his criticism. I have improved the text in several places in
response to his comments. I must thank Professor Raghavan
Narasimhan for many stimulating discussions, and Mr. Henri
Joris for the valuable assistance he has given me in checking
the manuscript and correcting the proofs.
July 1970 K. Chandrasekharan
Contents
Chapter I
The prime number theorem and Selberg's method
§ 1. Selberg's fonnula . . . . . . 1
§ 2. A variant of Selberg's formula 6
§ 3. Wirsing's inequality . . . . . 12
§ 4. The prime number theorem. . 17
§ 5. The order of magnitude of the divisor function 19
Notes on Chapter I ........... . 21
Chapter II
The zeta-Junction of Riemann
§ 1. The functional equation . . . . . . 28
§ 2. The Riemann-von Mangoldt formula. 33
§ 3. The entire function ~ . 40
§ 4. Hardy's theorem . . 45
§ 5. Hamburger's theorem 51
Notes on Chapter II ... 54
Chapter III
Littlewood's theorem and Weyl's method
§ 1. Zero-free region of (. . . . . . . . . . . . . . 58
§ 2. Weyl's inequality . . . . . . . . . . . . . . . 60
§ 3. Some results of Hardy and Littlewood and of Weyl 69
§ 4. Littlewood's theorem. . . . . . . . 73
§ 5. Applications of Littlewood's theorem 78
Notes on Chapter III ..... 84
Chapter IV
Vinogradov's method
§ 1. A refinement of Littlewood's theorem 88
§ 2. An outline of the method. . . . . 88
§ 3. Vinogradov's mean-value theorem. . 90
§ 4. Vinogradov's inequality . . . . . . 99
§ 5. Estimation of sections of ((s) in the critical strip. 106
1 Chandrasekharan, Arithmetical Functions
x Contents
§ 6. Chudakov's theorem. . 108
§ 7. Approximation of n(x) . 110
Notes on Chapter IV .... 110
Chapter V
Theorems of Hoheisel and of Ingham
§ 1. The difference between consecutive primes . . 112
§ 2. Landau's formula for the Chebyshev function '" 113
§ 3. Hoheisel's theorem . 124
§ 4. Two auxiliary lemmas . . . . . . . . 126
§ 5. Ingham's theorem. . . . . . . . . . 130
§ 6. An application of Chudakov's theorem. 138
Notes on Chapter V .......... . 139
Chapter VI
Dirichlet's L-functions and Siegefs theorem
§ 1. Characters and L-functions . 143
§ 2. Zeros of L-functions. . . . . . . 145
§ 3. Proper characters . . . . . . . . 146
§ 4. The functional equation of L(s, X) . 149
§ 5. Siegel's theorem. 155
Notes on Chapter VI. ....... . 164
Chapter VII
Theorems of Hardy-Ramanujan and of Rademacher on the partition function
§ 1. The partition function 166
§ 2. A simple case. . . . . . . . . . . . . . 166
§ 3. A bound for p(n) . . . . . . . . . . . . 169
§ 4. A property of the generating function of p(n) 170
§ 5. The Dedekind 1/-function .... 174
§ 6. The Hardy-Ramanujan formula . 178
§ 7. Rademacher's identity 185
Notes on Chapter VII . . . . . . . 191
Chapter VIII
Dirichlet's divisor problem
§ 1. The average order of the divisor function. 194
§ 2. An application of Perron's formula . . . 195
§ 3. An auxiliary function . . . . . . . . . 198
§ 4. An identity involving the divisor function. 200
§ 5. Voronoi's theorem .......... . 202
Contents XI
§ 6. A theorem of A. S. Besicovitch . . . . 204
§ 7. Theorems of Hardy and of Ingham .. 205
§ 8. Equiconvergence theorems of A. Zygmund 209
§ 9. The Voronoi identity. 223
Notes on Chapter VIII. 226
A list of books 229
Subject index . 230
I'
Chapter I
The prime number theorem and Selberg's method
§ 1. Selberg's formula. Let n(x) denote, for any real x, the number of
primes not exceeding x. The prime number theorem is the assertion that
(~)=
lim 1. (1)
x~oo x/logx
A fundamental formula discovered by Atle Selberg has made a proof
of (1) possible without the use of the properties of the zeta-function of
Riemann, and without the use of the theory of functions of a complex
variable. We shall prove Selberg's formula in this chapter, and indicate
some of its consequences. We shall also prove an inequality due to
E. Wirsing, which, when combined with a variant of Selberg's formula,
gives a proof of the prime number theorem.
We recall the definitions and simple properties of a number of
well-known arithmetical functions. An arithmetical function is any
complex-valued function defined on the set of all positive integers. We
shall be concerned, almost always, with integer-valued arithmetical
functions.
The Mobius function fJ. is defined, for any positive integer n, by the
following three properties:
(i) fJ.(1)= 1;
(ii) fJ.(n) = ( -It, if n is a product of k different primes;
(iii) fJ.(n)=O, if n is divisible by a square different from l.
As a simple consequence of the definition, we have
l, if n= 1,
I fJ.(d) = { . (2)
0, If n> 1 ,
dill
where the summation is over all the positive divisors d of n.
There are two elementary inversion formulas governing the Mobius
function:
2 I The prime number theorem and Selberg's method
(The first Mobius inversion formula). If f is an arithmetical function, and
g(n) = IJ(d) ,
din
then
(3)
and conversely.
(The second Mobius inversion formula). If f is a function defined for
x~ 1, and
then
L (~), ~
f(x) = J1(n)g for x 1, (4)
n~x n
and conversely.
If we set f(x) = 1, f(x)=x, f(x)=xlogx, f(x)=xlog2 x, in formula
(4), and note that
L ~ +O(~),
= logx+'Y (5)
n x
n~x
where 'Y is Euler's constant, while
n x)
L -log- = _1l og2X+C +0 (l-Og- ,
1
n~x n 2 x
(6)
x
I
log - = O(x) ,
n
n~x
where C and C2 are constants, we successively obtain the following
1
estimates:
L J1(n) = 0(1) , L -J1I(n)o g-x =0(1), I -J1l(no) g2x- = 2Iogx+0(1),
n~x n n::;:;x n n n~x n n
(7)
and
(8)