数论中的模函数和狄利克莱级数

Chapter1

Ellipticfunctions

1.1Introduction

1.2Doublyperiodicfunctions

1.3Fundamentalpairsofperiods

1.4Ellipticfunctions

1.5Constructionofellipticfunctions

1.6TheWeierstrassfunction

1.7TheLaurentexpansionofneartheorigin

1.8Differentialequationsatisfiedby

1.9TheEisensteinseriesandtheinvariantsand

1.10Thenumbersel,e2,e3

1.11Thediscriminant

1.12KleinsmodularfunctionJ()

1.13InvarianceofJunderunimodulartransformations

1.14TheFourierexpansionsofg2()andg3()

1.15TheFourierexpansionsofA()andJ()ExercisesforChapter1

Chapter2TheModulargroupandmodularfunctions

2.1M6biustransformations

2.2ThemodulargroupF

2.3Fundamentalregions

2.4Modularfunctions

2.5SpecialvaluesofJ

2.6ModularfunctionsasrationalfunctionsofJ

2.7MappingpropertiesofJ

2.8ApplicationtotheinversionproblemforEisensteinseries

2.9ApplicationtoPicardstheoremExercisesforChapter2

Chapter3TheDedekindetafunction

3.1Introduction

3.2SiegersproofofTheorem3.1

3.3InfiniteproductrepresentationforA(r)

3.4Thegeneralfunctionalequationforq(r)

3.5Isekistransformationformula

3.6DeductionofDedekindsfunctionalequationfromIsekisformula

3.7PropertiesofDedekindsums

3.8ThereciprocitylawforDedekindsums

3.9CongruencepropertiesofDedekindsums

3.10TheEisensteinseriesG2(z)ExercisesforChapter

Chapter4Conyruencesforthecoefficientsofthemodularfunction

4.1Introduction

4.2ThesubgroupFo(q)

4.3FundamentalregionofFo(p)

4.4FunctionsautomorphicunderthesubgroupFo(p)

4.5ConstructionoffunctionsbelongingtoFo(p)

4.6Thebehaviorof∫punderthegeneratorsofF

4.7Thefunctionφ(τ)=A(qτ)/A(τ)

4.8Theunivalentfunctionφ(τ)

4.9Invarianceofφ(ι)undertransformationsofFo(q)

4.10Thefunctionjpexpressedasapolynomialin

ExercisesforChapter4

Chapter5

Rademachersseriesforthepartitionfunction

5.1Introduction

5.2Theplanoftheproof

5.3DedekindsfunctionalequationexpressedintermsofF

5.4Fareyfractions

5.5Fordcircles

5.6Rademacherspathofintegration

5.7Rademachersconvergentseriesforp(n)

ExercisesforChapter5

Chapter6

Modularformswithmultiplicativecoefficients

6.1Introduction

6.2Modularformsofweightk

6.3Theweightformulaforzerosofanentiremodularform

6.4RepresentationofentireformsintermsofG4andG6

6.5ThelinearspaceMkandthesubspaceMk.o

6.6Classificationofentireformsintermsoftheirzeros

6.7TheHeckeoperatorsTn

6.8Transformationsofordern

6.9BehaviorofTnfunderthemodulargroup

6.10MultiplicativepropertyofHeckeoperators

6.11EigenfunctionsofHeckeoperators

6.12Propertiesofsimultaneouseigenforms

6.13Examplesofnormalizedsimultaneouseigenforms

6.14RemarksonexistenceofsimultaneouseigenformsinM2k,o

6.15EstimatesfortheFouriercoefficientsofentireforms

6.16ModularformsandDirichletseries

ExercisesforChapter6

Chapter7

Kroneckerstheoremwithapplications

7.1Approximatingrealnumbersbyrationalnumbers

7.2Dirichletsapproximationtheorem

7.3Liouvillesapproximationtheorem

7.4Kroneckersapproximationtheorem:theone-dimensionalcase

7.5ExtensionofKroneckerstheoremtosimultaneousapproximation

7.6ApplicationstotheRiemannzetafunction

7.7Applicationstoperiodicfunctions

ExercisesforChapter7

Chapter8

GeneralDirichletseriesandBohrsequivalencetheorem

8.1Introduction

8.2Thehalf-planeofconvergenceofgeneralDirichletseries

8.3BasesforthesequenceofexponentsofaDirichletseries

8.4Bohrmatrices

8.5TheBohrfunctionassociatedwithaDirichletseries

8.6ThesetofvaluestakenbyaDirichletseriesf(s)onalineσ=σ0

8.7EquivalenceofgeneralDirichletseries

8.8EquivalenceofordinaryDirichletseries

8.9EqualityofthesetsUι(σo)andUg(σo)forequivalentDirichletseries

8.10ThesetofvaluestakenbyaDirichletseriesinaneighborhoodofthelineσ=σ0

8.11Bohrsequivalencetheorem

8.12ProofofTheorem8.15

8.13ExamplesofequivalentDirichletseries.ApplicationsofBohrstheoremtoL-series

8.14ApplicationsofBohrstheoremtotheRiemannzetafunctionSupplementtoChapter3

Bibliography

Indexofspecialsymbols

Index

　　Thisisthesecondvolumeofa2-volumetextbook*whichevolvedfromacourse(Mathematics160)offeredattheCaliforniaInstituteofTechnologyduringthelast25years.　　Thesecondvolumepresupposesabackgroundinnumbertheorycomparabletothatprovidedinthefirstvolume,togetherwithaknowledgeofthebasicconceptsofcomplexanalysis.