An Invitation to Modern Number Theory

515 Pages·2006·5.38 MB·English

by  Steven J. Miller, Ramin Takloo-Bighash

An Invitation to
Modern Number Theory

An Invitation to
Modern Number Theory
Steven J. Miller and
Ramin Takloo-Bighash
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright(cid:2)c 2006byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,41WilliamStreet,Princeton,NewJersey08540
IntheUnitedKingdom: PrincetonUniversityPress,3MarketPlace,Woodstock,OxfordshireOX20
1SY
AllRightsReserved
LibraryofCongressCataloging-in-PublicationData
Miller,StevenJ.
Aninvitationtomodernnumbertheory/StevenJ.MillerandRaminTakloo-Bighash.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN-13:978-0-691-12060-7(cloth:alk.paper)
ISBN-10:0-691-12060-9(cloth:alk.paper)
1.Numbertheory.I.Takloo-Bighash,Ramin.II.Title.
QA241.M53442006
512.7—dc22 2005052165
BritishLibraryCataloging-in-PublicationDataisavailable
ThisbookhasbeencomposedinTimesRomaninLATEX
Thepublisherwouldliketoacknowledgetheauthorsofthisvolumeforprovidingthecamera-ready
copyfromwhichthisbookwasprinted.
Printedonacid-freepaper.∞
pup.princeton.edu
PrintedintheUnitedStatesofAmerica
109876543
To our wives, for their patience and constant encouragement, and to
ourstudentsandcolleagues,whoseinsightsandexuberancemadethis
projectbothpossibleandenjoyable.

Contents
Foreword xi
Preface xiii
Notation xix
PART1. BASICNUMBERTHEORY 1
Chapter1. ModpArithmetic,GroupTheoryandCryptography 3
1.1 Cryptography 3
1.2 EfficientAlgorithms 5
1.3 ClockArithmetic:ArithmeticModulon 14
1.4 GroupTheory 15
1.5 RSARevisited 20
1.6 Eisenstein’sProofofQuadraticReciprocity 21
Chapter2. ArithmeticFunctions 29
2.1 ArithmeticFunctions 29
2.2 AverageOrder 32
2.3 CountingtheNumberofPrimes 38
Chapter3. ZetaandL-Functions 47
3.1 TheRiemannZetaFunction 47
3.2 ZerosoftheRiemannZetaFunction 54
3.3 DirichletCharactersandL-Functions 69
Chapter4. SolutionstoDiophantineEquations 81
4.1 DiophantineEquations 81
4.2 EllipticCurves 85
4.3 HeightFunctionsandDiophantineEquations 89
4.4 CountingSolutionsofCongruencesModulop 95
4.5 ResearchProjects 105
PART2. CONTINUEDFRACTIONSANDAPPROXIMATIONS 107
Chapter5. AlgebraicandTranscendentalNumbers 109
viii CONTENTS
5.1 Russell’sParadoxandtheBanach-TarskiParadox 109
5.2 Definitions 110
5.3 CountableandUncountableSets 112
5.4 Propertiesofe 118
5.5 Exponent(orOrder)ofApproximation 124
5.6 Liouville’sTheorem 128
5.7 Roth’sTheorem 132
Chapter6. TheProofofRoth’sTheorem 137
6.1 Liouville’sTheoremandRoth’sTheorem 137
6.2 EquivalentFormulationofRoth’sTheorem 138
6.3 Roth’sMainLemma 142
6.4 PreliminariestoProvingRoth’sLemma 147
6.5 ProofofRoth’sLemma 155
Chapter7. IntroductiontoContinuedFractions 158
7.1 DecimalExpansions 158
7.2 DefinitionofContinuedFractions 159
7.3 RepresentationofNumbersbyContinuedFractions 161
7.4 InfiniteContinuedFractions 167
7.5 PositiveSimpleConvergentsandConvergence 169
7.6 PeriodicContinuedFractionsand
QuadraticIrrationals 170
7.7 ComputingAlgebraicNumbers’ContinuedFractions 177
7.8 FamousContinuedFractionExpansions 179
7.9 ContinuedFractionsandApproximations 182
7.10 ResearchProjects 186
PART3. PROBABILISTICMETHODSANDEQUIDISTRIBUTION 189
Chapter8. IntroductiontoProbability 191
8.1 ProbabilitiesofDiscreteEvents 192
8.2 StandardDistributions 205
8.3 RandomSampling 211
8.4 TheCentralLimitTheorem 213
Chapter9. ApplicationsofProbability: Benford’sLawand
HypothesisTesting 216
9.1 Benford’sLaw 216
9.2 Benford’sLawandEquidistributedSequences 218
9.3 RecurrenceRelationsandBenford’sLaw 219
9.4 RandomWalksandBenford’sLaw 221
9.5 StatisticalInference 225
9.6 Summary 229
Chapter10. DistributionofDigitsofContinuedFractions 231
10.1 SimpleResultsonDistributionofDigits 231
10.2 MeasureofαwithSpecifiedDigits 235
CONTENTS ix
10.3 TheGauss-KuzminTheorem 237
10.4 DependenciesofDigits 244
10.5 Gauss-KuzminExperiments 248
10.6 ResearchProjects 252
Chapter11. IntroductiontoFourierAnalysis 255
11.1 InnerProductofFunctions 256
11.2 FourierSeries 258
11.3 ConvergenceofFourierSeries 262
11.4 ApplicationsoftheFourierTransform 268
11.5 CentralLimitTheorem 273
11.6 AdvancedTopics 276
Chapter12. {nkα}andPoissonianBehavior 278
12.1 DefinitionsandProblems 278
12.2 Densenessof{nkα} 280
12.3 Equidistributionof{nkα} 283
12.4 SpacingPreliminaries 288
12.5 PointMassesandInducedProbabilityMeasures 289
12.6 NeighborSpacings 290
12.7 PoissonianBehavior 291
12.8 NeighborSpacingsof{nkα} 296
12.9 ResearchProjects 299
PART4. THECIRCLEMETHOD 301
Chapter13. IntroductiontotheCircleMethod 303
13.1 Origins 303
13.2 TheCircleMethod 309
13.3 Goldbach’sConjectureRevisited 315
Chapter14. CircleMethod: HeuristicsforGermainPrimes 326
14.1 GermainPrimes 326
14.2 Preliminaries 328
14.3 TheFunctionsFN(x)andu(x) 331
14.4 ApproximatingFN(x)ontheMajorArcs 332
14.5 IntegralsovertheMajorArcs 338
14.6 MajorArcsandtheSingularSeries 342
14.7 NumberofGermainPrimesandWeightedSums 350
14.8 Exercises 353
14.9 ResearchProjects 354
PART5. RANDOMMATRIXTHEORYANDL-FUNCTIONS 357
Chapter15. FromNuclearPhysicstoL-Functions 359
15.1 HistoricalIntroduction 359
15.2 EigenvaluePreliminaries 364
x CONTENTS
15.3 Semi-CircleLaw 368
15.4 AdjacentNeighborSpacings 374
15.5 ThinSub-families 377
15.6 NumberTheory 383
15.7 SimilaritiesbetweenRandomMatrixTheoryandL-Functions 389
15.8 SuggestionsforFurtherReading 390
Chapter16. RandomMatrixTheory: EigenvalueDensities 391
16.1 Semi-CircleLaw 391
16.2 Non-Semi-CircleBehavior 398
16.3 SparseMatrices 402
16.4 ResearchProjects 403
Chapter17. RandomMatrixTheory: Spacingsbetween
AdjacentEigenvalues 405
17.1 Introductiontothe2×2GOEModel 405
17.2 DistributionofEigenvaluesof2×2GOEModel 409
17.3 GeneralizationtoN ×N GOE 414
17.4 ConjecturesandResearchProjects 418
Chapter18. TheExplicitFormulaandDensityConjectures 421
18.1 ExplicitFormula 422
18.2 DirichletCharactersfromaPrimeConductor 429
18.3 SummaryofCalculations 437
AppendixA. AnalysisReview 439
A.1 ProofsbyInduction 439
A.2 CalculusReview 442
A.3 ConvergenceandContinuity 447
A.4 Dirichlet’sPigeon-HolePrinciple 448
A.5 MeasuresandLength 450
A.6 Inequalities 452
AppendixB. LinearAlgebraReview 455
B.1 Definitions 455
B.2 ChangeofBasis 456
B.3 OrthogonalandUnitaryMatrices 457
B.4 Trace 458
B.5 SpectralTheoremforRealSymmetricMatrices 459
AppendixC. HintsandRemarksontheExercises 463
AppendixD. ConcludingRemarks 475
Bibliography 476
Index 497