Analytic Number Theory: Exploring the Anatomy of Integers

434 Pages·2012·3.14 MB·English

by  Jean-marie De Koninck

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 informationand updates on this book, visit
www.ams.org/bookpages/gsm-134
Library of Congress Cataloging-in-Publication Data
Koninck,J.-M.de,1948–
Analyticnumbertheory: exploringtheanatomyofintegers/Jean-MarieDeKoninck,Florian
Luca.
p.cm. –(Graduatestudiesinmathematics;v. 134)
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-7577-3(alk.paper)
1.Numbertheory. 2.Euclideanalgorithm. 3.Integrals. I.Luca,Florian. II.Title.
QA241.K6855 2012
512.7(cid:2)4–dc23 2011051431
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
actingforthem,arepermittedtomakefairuseofthematerial,suchastocopyachapterforuse
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
is permitted only under license from the American Mathematical Society. Requests for such
permissionshouldbeaddressedtotheAcquisitionsDepartment,AmericanMathematicalSociety,
201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by
[email protected].
(cid:2)c 2012bytheAmericanMathematicalSociety. Allrightsreserved.
TheAmericanMathematicalSocietyretainsallrights
exceptthosegrantedtotheUnitedStatesGovernment.
PrintedintheUnitedStatesofAmerica.
(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttp://www.ams.org/
10987654321 171615141312
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:
• Severalnumbertheory problemscan be formulatedin simpleterms
withverylittleornobackgroundrequiredtounderstandtheirstate-
ments.
• Ithasarichhistorythatgoesbackthousandsofyearswhenmankind
was learning to count (even before learning to write!).
• Some of the most famous minds of mathematics (including Pascal,
Euler,Gauss,andRiemann,tonameonlyafew)havebroughttheir
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 madeovertime, whilesome problemsremainas 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)tostudynumbertheoryproblemshasbroughtlightandeleganceto
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
whentwomathematicians,JacquesHadamardandCharlesJeandelaVall´ee
Poussin,oneFrench,theotherBelgian,independentlyusedcomplexanalysis
ix