数论中的模函数和狄利克莱级数(第2版)(Modular Functions and Dirichlet Series in Number Theory 2nd ed)

基本信息·出版社：世界图书出版公司

·页码：204 页

·出版日期：2009年

·ISBN：7510004403/9787510004407

·条形码：9787510004407

·包装版本：2版

·装帧：平装

·开本：24

·正文语种：英语

·外文书名：Modular Functions and Dirichlet Series in Number Theory 2nd ed

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内容简介《数论中的模函数和狄利克莱级数(第2版)》讲述了：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

编辑推荐《数论中的模函数和狄利克莱级数(第2版)》是由世界图书出版公司出版的。

目录

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 Klein'smodularfunctionJ(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 ApplicationtoPicard'stheorem

ExercisesforChapter2

Chapter3 TheDedekindetafunction

3.1 Introduction

3.2 Siegei'sproofofTheorem3.1

3.3 Infiniteproductrepresentationfor△(τ)

3.4 Thegeneralfunctionalequationforη(τ)

3.5 Iseki'stransformationformula

3.6 DeductionofDedekind'sfunctionalequationfromIseki'sformula

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 Rademacher'sseriesforthepartitionfunction

5.1 Introduction

5.2 Theplanoftheproof

5.3 Dedekind'sfunctionalequationexpressedintermsofF

5.4 Fareyfractions

5.5 Fordcircles

5.6 Rademacher'spathofintegration

5.7 Rademacher'sconvergentseriesforp(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

……[看更多目录]

序言This iS the second volume of a 2.volume textbook*which evolved from a course（Mathematics 160）offered at the Ca“fornia 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.

Most of the present volume iS devoted to elliptic functions and modular functions with some of their number.theoretic applications.Among the major topics treated are Rademacher'S convergent series for the partition function.Lehner’S congruences for the Fourier coefficients of the modular functionJ，and Hecke’S theory of entire forms with multiplicative Fourier coeflicients.The last chapter gives an account of Bohr’s theory ofequivalence of general Dirichlet series.

Both volumes of this work emphasize classical aspects of a subject which in recent years has undergone a great deaI of modern development.It iS hoped that these volumes wilI help the nonspecialist become acquainted with an important and fascinating part of mathematics and，at the same time.will provide some of the background that belongs to the repertory of every specialist in the field.

This volume.Iike the first，iS dedicated to the students who have taken this course and have gone on to make notable contributions tO number theory and other parts of mathematics.

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