數論中的模函式和狄利克萊級數

《數論中的模函式和狄利克萊級數》是由世界圖書出版公司出版編著的實體書。主要講述了模函式和狄利克萊級數在數論中的體現。

基本介紹

• 書名：數論中的模函式和狄利克萊級數
• 頁數：204頁
• 出版時間：第2版 (2009年4月1日)
• 裝幀：平裝

圖書信息

出版社: 世界圖書出版公司;

外文書名: Modular Functions and Dirichlet Series in Number Theory (2nd Edition)

:

正文語種: 英語

開本: 24

ISBN: 7510004403, 9787510004407

條形碼: 9787510004407

尺寸: 22 x 14.8 x 1.2 cm

重量: 299 g

作者簡介

作者：(美國) 阿波斯托爾 (Apostol.T.M.)

內容簡介

《數論中的模函式和狄利克萊級數(第2版)》講述了：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

目錄

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