数论中的模函数和狄利克莱级数

Chapter1 Ellipticfunctions
1.1 Introduction
1.2 Doublyperiodicfunctions
1.3 Fundamentalpairsofperiods
1.4 Ellipticfunctions
1.5 Constructionofellipticfunctions
1.6 TheWeierstrassfunction
1.7 TheLaurentexpansionofganeartheorigin
1.8 Differentialequationsatisfiedbyξ
1.9 TheEisensteinseriesandtheinvariantsg2andg3
1.10 Thenumberse1，e2，e3
1.11 ThediscriminantA
1.12 KleinsmodularfunctionJ（τ）
1.13 InvarianceofJunderunimodulartransformations
1.14 TheFourierexpansionsofg2（τ）andg3（τ）
1.15 TheFourierexpansionsof△（τ）andJ（τ）
ExercisesforChapter1
Chapter2 TheModulargroupandmodularfunctions
2.1 M6biustransformations
2.2 Themodulargroup
2.3 Fundamentalregions
2.4 Modularfunctions
2.5 Specialvaluesof
2.6 Modularfunctionsasrationalfunctionsof
2.7 Mappingpropertiesof
2.8 ApplicationtotheinversionproblemforEisensteinseries
2.9 ApplicationtoPicardstheorem
ExercisesforChapter2
Chapter3 TheDedekindetafunction
3.1 Introduction
3.2 SiegeisproofofTheorem3.1
3.3 Infiniteproductrepresentationfor△（τ）
3.4 Thegeneralfunctionalequationforη（τ）
3.5 Isekistransformationformula
3.6 DeductionofDedekindsfunctionalequationfromIsekisformula
3.7 PropertiesofDedekindsums
3.8 ThereciprocitylawforDedekindsums
3.9 CongruencepropertiesofDedekindsums
3.1 0TheEisensteinseriesG2（τ）
ExercisesforChapter3
Chapter4 Congruencesforthecoefficientsofthemodularfunctionj
4.1 Introduction
4.2 ThesubgroupFo（q）
4.3 FundamentalregionofFo（p）
4.4 FunctionsautomorphicunderthesubgroupFo（p）
4.5 ConstructionoffunctionsbelongingtoFo（p）
4.6 Thebehavioroffpunderthegeneratorsofг
4.7 Thefunction（τ）=△（qτ）/△（τ）
4.8 Theunivalentfunctionφ（τ）
4.9 Invarianceofφ（τ）undertransformationsofг0（q）
4.1 0Thefunctionjpexpressedasapolynomialinφ
ExercisesforChapter4
Chapter5 Rademachersseriesforthepartitionfunction
5.1 Introduction
5.2 Theplanoftheproof
5.3 DedekindsfunctionalequationexpressedintermsofF
5.4 Fareyfractions
5.5 Fordcircles
5.6 Rademacherspathofintegration
5.7 Rademachersconvergentseriesforp（n）
ExercisesforChapter5
Chapter6 Modularformswithmultiplicativecoefficients
6.1 Introduction
6.2 Modularformsofweightk
6.3 Theweightformulaforzerosofanentiremodularform
6.4 RepresentationofentireformsintermsofG4andG6
6.5 ThelinearspaceMkandthesubspaceMk.o
6.6 Classificationofentireformsintermsoftheirzeros
6.7 TheHeckeoperatorsTn
6.8 Transformationsofordern
6.9 BehaviorofTnfunderthemodulargroup
6.10 MultiplicativepropertyofHeckeoperators
6.11 EigenfunctionsofHeckeoperators
6.12 Propertiesofsimultaneouseigenforms
6.13 Examplesofnormalizedsimultaneouseigenforms
6.14 RemarksonexistenceofsimultaneouseigenformsinM2k.0
6.15 EstimatesfortheFouriercoefficientsofentireforms
6.16 ModularformsandDirichletseries
Exerci