Introduction to Analytic and Probabilistic Number Theory

658 Pages·2015·29.77 MB·English

by  Gerald Tenenbaum

THIRD EDITION
Gerald Tenenbaum
Graduate Studies
in Mathematics
Volume 163
American Mathematical Society

Introduction to Analytic
and Probabilistic
Number Theory

Introduction to Analytic
and Probabilistic
Number Theory
Third Edition
Gerald Tenenbaum
Translated by Patrick D. F. Ion
Graduate Studies
in Mathematics
Volume 163
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2010 Mathematics Subject Classification. Primary 11-02; Secondary llAxx, llJxx,
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Library of Congress Cataloging-in-Publication Data
Tenenbaum, Gerald.
a
(Introduction la theorie analytique et probabiliste des nombres. English) Introduction to
analytic and probabilistic number theory / Gerald Tenenbaum ; translated by Patrick Ion. -
Third edition.
pages cm. - (Graduate studies in mathematics; volume 163)
Includes bibliographical references and index.
ISBN 978-0-8218-9854-3 (alk. paper)
1. Number theory. 2. Probabilistic number theory. I. Title.
QA241.T42313 2015
512'.73-dc23 2014040135
This work was originally published in French by Editions Belin under the title Introduction d la
theorie analytique et probabiliste des nombres, Third edition © 2008. The present translation
was created under license for the American Mathematical Society and is published by permission.
Originally published in French as Introduction d la theorie analytique et probabiliste des nombres
Copyright @1990 G. Tenenbaum
English edition Copyright @1995 Cambridge University Press
Translated by C. B. Thomas, University of Cambridge
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Contents
Foreword xv
Preface to the third edition xix
Preface to the English translation xxi
Notation xxiii
Part I. Elementary Methods
Chapter 1.0. Some tools from real analysis 3
§0.1. Abel summation 3
§0.2. The Euler-Maclaurin summation formula 5
Exercises 8
Chapter 1.1. Prime numbers 11
§1 .1. Introduction 11
§1.2. Chebyshev's estimates 13
§1.3. p-adic valuation of n! 15
§1.4. Mertens' first theorem 16
§1.5. Two new asymptotic formulae 17
§1.6. Merten's formula 19
§1.7. Another theorem of Chebyshev 20
Notes 22
Exercises 23
-
vii
viii Contents
Chapter 1.2. Arithmetic functions 29
§2.1. Definitions 29
§2.2. Examples 30
§2.3. Formal Dirichlet series 31
§2.4. The ring of arithmetic functions 32
§2.5. The Mobius inversion formulae 34
§2.6. Von Mangoldt's function 36
§2. 7. Euler's totient function 37
Notes 39
Exercises 40
Chapter 1.3. Average orders 43
§3.1. Introduction 43
§3.2. Dirichlet's problem and the hyperbola method 44
§3.3. The sum of divisors function 46
§3.4. Euler's totient function 46
§3.5. The functions w and n 48
§3.6. Mean value of the Mobius function and Chebyshev's
summatory functions 49
§3. 7. Squarefree integers 52
§3.8. Mean value of a multiplicative function with values in [O, 1] 54
Notes 57
Exercises 59
Chapter 1.4. Sieve methods 67
§4.1. The sieve of Eratosthenes 67
§4.2. Brun's combinatorial sieve 68
§4.3. Application to twin primes 71
§4.4. The large sieve-analytic form 73
§4.5. The large sieve-arithmetic form 79
§4.6. Applications of the large sieve 82
§4.7. Selberg's sieve 84
§4.8. Sums of two squares in an interval 96
Notes 100
Exercises 105