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Conjectures on the distribution of prime numbers 33Problems on Chapter 2 36Chapter 3.. The Proof of the Prime Number Theorem 63§5.1.. The proof of the Prime Number Theorem 66§5.4.. A rev
Analytic Number Theory Exploring the Anatomy of Integers Jean-Marie De Koninck Florian Luca Graduate Studies in Mathematics Volume 134 American Mathematical Society Analytic Number Theory Exploring the Anatomy of Integers Jean-Marie De Koninck Florian Luca Graduate Studies in Mathematics Volume 134 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Daniel S Freed Rafe Mazzeo Gigliola Staffilani 2010 Mathematics Subject Classification Primary 11A05, 11A41, 11B05, 11K65, 11N05, 11N13, 11N35, 11N37, 11N60, 11B39 For additional information and updates on this book, visit www.ams.org/bookpages/gsm-134 Library of Congress Cataloging-in-Publication Data Koninck, J.-M de, 1948– Analytic number theory: exploring the anatomy of integers / Jean-Marie De Koninck, Florian Luca p cm – (Graduate studies in mathematics ; v 134) Includes bibliographical references and index ISBN 978-0-8218-7577-3 (alk paper) 1 Number theory 2 Euclidean algorithm 3 Integrals I Luca, Florian II Title QA241.K6855 2012 2011051431 512.7 4–dc23 Copying and reprinting Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA Requests can also be made by e-mail to reprint-permission@ams.org c 2012 by the American Mathematical Society All rights reserved The American Mathematical Society retains all rights except those granted to the United States Government Printed in the United States of America ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 17 16 15 14 13 12 Contents Preface ix Notation xiii Frequently Used Functions xvii Chapter 1 Preliminary Notions 1 §1.1 Approximating a sum by an integral 1 §1.2 The Euler-MacLaurin formula 2 §1.3 The Abel summation formula 5 §1.4 Stieltjes integrals 7 §1.5 Slowly oscillating functions 8 §1.6 Combinatorial results 9 §1.7 The Chinese Remainder Theorem 10 §1.8 The density of a set of integers 11 §1.9 The Stirling formula 11 §1.10 Basic inequalities 13 Problems on Chapter 1 15 Chapter 2 Prime Numbers and Their Properties 19 §2.1 Prime numbers and their polynomial representations 19 §2.2 There exist infinitely many primes 21 §2.3 A first glimpse at the size of π(x) 21 §2.4 Fermat numbers 22 §2.5 A better lower bound for π(x) 24 iii iv Contents §2.6 The Chebyshev estimates 24 §2.7 The Bertrand Postulate 29 §2.8 The distance between consecutive primes 31 §2.9 Mersenne primes 32 §2.10 Conjectures on the distribution of prime numbers 33 Problems on Chapter 2 36 Chapter 3 The Riemann Zeta Function 39 §3.1 The definition of the Riemann Zeta Function 39 §3.2 Extending the Zeta Function to the half-plane σ > 0 40 §3.3 The derivative of the Riemann Zeta Function 41 §3.4 The zeros of the Zeta Function 43 §3.5 Euler’s estimate ζ(2) = π2/6 45 Problems on Chapter 3 48 Chapter 4 Setting the Stage for the Proof of the Prime Number Theorem 51 §4.1 Key functions related to the Prime Number Theorem 51 §4.2 A closer analysis of the functions θ(x) and ψ(x) 52 §4.3 Useful estimates 53 §4.4 The Mertens estimate 55 §4.5 The M¨obius function 56 §4.6 The divisor function 58 Problems on Chapter 4 60 Chapter 5 The Proof of the Prime Number Theorem 63 §5.1 A theorem of D J Newman 63 §5.2 An application of Newman’s theorem 65 §5.3 The proof of the Prime Number Theorem 66 §5.4 A review of the proof of the Prime Number Theorem 69 §5.5 The Riemann Hypothesis and the Prime Number Theorem 70 §5.6 Useful estimates involving primes 71 §5.7 Elementary proofs of the Prime Number Theorem 72 Problems on Chapter 5 72 Chapter 6 The Global Behavior of Arithmetic Functions 75 §6.1 Dirichlet series and arithmetic functions 75 §6.2 The uniqueness of representation of a Dirichlet series 77 Contents v §6.3 Multiplicative functions 79 §6.4 Generating functions and Dirichlet products 81 §6.5 Wintner’s theorem 82 §6.6 Additive functions 85 §6.7 The average orders of ω(n) and Ω(n) 86 §6.8 The average order of an additive function 87 §6.9 The Erd˝os-Wintner theorem 88 Problems on Chapter 6 89 Chapter 7 The Local Behavior of Arithmetic Functions 93 §7.1 The normal order of an arithmetic function 93 §7.2 The Tur´an-Kubilius inequality 94 §7.3 Maximal order of the divisor function 99 §7.4 An upper bound for d(n) 101 §7.5 Asymptotic densities 103 §7.6 Perfect numbers 106 §7.7 Sierpin´ski, Riesel, and Romanov 106 §7.8 Some open problems of an elementary nature 108 Problems on Chapter 7 109 Chapter 8 The Fascinating Euler Function 115 §8.1 The Euler function 115 §8.2 Elementary properties of the Euler function 117 §8.3 The average order of the Euler function 118 §8.4 When is φ(n)σ(n) a square? 119 §8.5 The distribution of the values of φ(n)/n 121 §8.6 The local behavior of the Euler function 122 Problems on Chapter 8 124 Chapter 9 Smooth Numbers 127 §9.1 Notation 127 §9.2 The smallest prime factor of an integer 127 §9.3 The largest prime factor of an integer 131 §9.4 The Rankin method 137 §9.5 An application to pseudoprimes 141 §9.6 The geometric method 145 §9.7 The best known estimates on Ψ(x, y) 146 vi Contents §9.8 The Dickman function 147 §9.9 Consecutive smooth numbers 149 Problems on Chapter 9 150 Chapter 10 The Hardy-Ramanujan and Landau Theorems 157 §10.1 The Hardy-Ramanujan inequality 157 §10.2 Landau’s theorem 159 Problems on Chapter 10 164 Chapter 11 The abc Conjecture and Some of Its Applications 167 §11.1 The abc conjecture 167 §11.2 The relevance of the condition ε > 0 168 §11.3 The Generalized Fermat Equation 171 §11.4 Consecutive powerful numbers 172 §11.5 Sums of k-powerful numbers 172 §11.6 The Erd˝os-Woods conjecture 173 §11.7 A problem of Gandhi 174 §11.8 The k-abc conjecture 175 Problems on Chapter 11 176 Chapter 12 Sieve Methods 179 §12.1 The sieve of Eratosthenes 179 §12.2 The Brun sieve 180 §12.3 Twin primes 184 §12.4 The Brun combinatorial sieve 187 §12.5 A Chebyshev type estimate 187 §12.6 The Brun-Titchmarsh theorem 188 §12.7 Twin primes revisited 190 §12.8 Smooth shifted primes 191 §12.9 The Goldbach conjecture 192 §12.10 The Schnirelman theorem 194 §12.11 The Selberg sieve 198 §12.12 The Brun-Titchmarsh theorem from the Selberg sieve 201 §12.13 The Large sieve 202 §12.14 Quasi-squares 203 §12.15 The smallest quadratic nonresidue modulo p 204 Problems on Chapter 12 206 Contents vii Chapter 13 Prime Numbers in Arithmetic Progression 217 §13.1 Quadratic residues 217 §13.2 The proof of the Quadratic Reciprocity Law 220 §13.3 Primes in arithmetic progressions with small moduli 222 §13.4 The Primitive Divisor theorem 224 §13.5 Comments on the Primitive Divisor theorem 227 Problems on Chapter 13 228 Chapter 14 Characters and the Dirichlet Theorem 233 §14.1 Primitive roots 233 §14.2 Characters 235 §14.3 Theorems about characters 236 §14.4 L-series 240 §14.5 L(1, χ) is finite if χ is a non-principal character 242 §14.6 The nonvanishing of L(1, χ): first step 243 §14.7 The completion of the proof of the Dirichlet theorem 244 Problems on Chapter 14 247 Chapter 15 Selected Applications of Primes in Arithmetic Progression 251 §15.1 Known results about primes in arithmetical progressions 251 §15.2 Some Diophantine applications 254 §15.3 Primes p with p − 1 squarefree 257 §15.4 More applications of primes in arithmetic progressions 259 §15.5 Probabilistic applications 261 Problems on Chapter 15 263 Chapter 16 The Index of Composition of an Integer 267 §16.1 Introduction 267 §16.2 Elementary results 268 §16.3 Mean values of λ and 1/λ 270 §16.4 Local behavior of λ(n) 273 §16.5 Distribution function of λ(n) 275 §16.6 Probabilistic results 276 Problems on Chapter 16 279 Appendix: Basic Complex Analysis Theory 281 §17.1 Basic definitions 281 viii Contents §17.2 Infinite products 283 §17.3 The derivative of a function of a complex variable 284 §17.4 The integral of a function along a path 285 §17.5 The Cauchy theorem 287 §17.6 The Cauchy integral formula 289 Solutions to Even-Numbered Problems 291 Solutions to problems from Chapter 1 291 Solutions to problems from Chapter 2 295 Solutions to problems from Chapter 3 303 Solutions to problems from Chapter 4 309 Solutions to problems from Chapter 5 312 Solutions to problems from Chapter 6 318 Solutions to problems from Chapter 7 321 Solutions to problems from Chapter 8 334 Solutions to problems from Chapter 9 338 Solutions to problems from Chapter 10 351 Solutions to problems from Chapter 11 353 Solutions to problems from Chapter 12 356 Solutions to problems from Chapter 13 377 Solutions to problems from Chapter 14 384 Solutions to problems from Chapter 15 392 Solutions to problems from Chapter 16 401 Bibliography 405 Index 413 Preface Number theory is one of the most fascinating topics in mathematics, and there are various reasons for this Here are a few: • Several number theory problems can be formulated in simple terms with very little or no background required to understand their state- ments • It has a rich history that goes back thousands of years when mankind was learning to count (even before learning to write!) • Some of the most famous minds of mathematics (including Pascal, Euler, Gauss, and Riemann, to name only a few) have brought their contributions to the development of number theory • Like many other areas of science, but perhaps more so with this one, its development suffers from an apparent paradox: giant leaps have been made over time, while some problems remain as of today completely impenetrable, with little or no progress being made All this explains in part why so many scientists and so many amateurs have worked on famous problems and conjectures in number theory The long quest for a proof of Fermat’s Last Theorem is only one example And what about “analytic number theory”? The use of analysis (real or complex) to study number theory problems has brought light and elegance to this field, in particular to the problem of the distribution of prime numbers Through the centuries, a large variety of tools has been developed to grasp a better understanding of this particular problem But the year 1896 saw a turning point in the history of number theory Indeed, that was the year when two mathematicians, Jacques Hadamard and Charles Jean de la Vall´ee Poussin, one French, the other Belgian, independently used complex analysis ix