数论中的模函数和狄利克莱级数

　　This is the second volume of a 2-volume textbook* which evolved from a course （Mathematics 160） offered at the California Institute of Technology during the last 25 years.The second volume presupposes a background in number theory com-parable to that provided in the first volume， together with a knowledge of the basic concepts of complex analysis

暂缺《数论中的模函数和狄利克莱级数》作者简介

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