解析数论导论

图书标准信息

• 作者 [美]阿波斯托尔 著
• 出版社 世界图书出版公司
• 出版时间 2012-01
• 版次 1
• ISBN 9787510040627
• 定价 45.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 360页

【内容简介】

　　《解析数论导论》是一部为本科生提供学习数论的基本思想和技巧的教程，重点强调解析数论。前五章讲述可约性、收敛和算术函数等基本概念。紧下来的章节讲述序列中素数的狄利克莱定理、高斯和、二次剩余、狄利克莱级数和欧拉积及其在黎曼zeta函数和狄利克莱函数中的应用，并且引进了划分的概念。书中每章末都收集了大量练习。前十章，除去第一章，任何具备基本微积分知识的人都可以读懂；最后四章需要对复函数理论（包括复积分和留数积分）一定的了解。本书由（美）阿波斯托尔著。
　　

【目录】

HistoricalIntroduction

Chapter1

TheFundamentalTheoremofArithmetic

1．1Introduction

1．2Divisibility

1．3Greatestcommondivisor

1．4Primenumbers

1．5Thefundamentaltheoremofarithmetic

1．6Theseriesofreciprocalsoftheprimes

1．7TheEuclideanalgorithm

1．8Thegreatestcommondivisorofmorethantwo，numbers

ExercisesforChapter1

Chapter2

ArithmeticalFunctionsandDirichletMultiplication

2．1Introduction

2．2TheM6biusfunction(n)

2．3TheEulertotientfunction(n)

2．4Arelationconnectingandu

2．5Aproductformulafor(n)

2．6TheDirichletproductofarithmeticalfunctions

2．7DirichletinversesandtheM6biusinversionformula

2．8TheMangoldtfunctionA(n)

2．9Muitiplicativefunctions

2．10MultiplicativefunctionsandDirichletmultiplication

2．11Theinverseofacompletelymultiplicativefunction

2．12Liouville'sfunction)

2．13Thedivisorfunctionsa，(n)

2．14Generalizedconvolutions

2．15Formalpowerseries

2．16TheBellseriesofanarithmeticalfunction

2．17BellseriesandDirichletmultiplication

2．18Derivativesofarithmeticalfunctions

2．19TheSelbergidentity

ExercisesforChapter2

Chapter3

AveragesofArithmeticalFunctions

3．1Introduction

3．2Thebigohnotation．Asymptoticequalityoffunctions

3．3Euler'ssummationformula

3．4Someelementaryasymptoticformulas

3．5Theaverageorderofdin)

3．6Theaverageorderofthedivisorfunctionsa，(n)

3．7Theaverageorderof~0(n)

3．8Anapplicationtothedistributionoflatticepointsvisiblefromtheorigin

3．9Theaverageorderof/4n)andofA(n)

3．10ThepartialsumsofaDirichletproduct

3．11Applicationstopin)andA(n)

3．12AnotheridentityforthepartialgumsofaDirichletproduct

ExercisesforChapter3

Chapter4

SomeElementaryTheoremsontheDistributionofPrime

Numbers

4．1Introduction

4．2Chebyshev'sfunctions(x)and(x)

4．3Relationsconnecting/x)andn(x)

4．4Someequivalentformsoftheprimenumbertheorem

4．5Inequalitiesfor(n)andp，

4．6Shapiro'sTauberiantheorem

4．7ApplicationsofShapiro'stheorem

4．8Anasymptoticformulaforthepartialsums，(I/p)

4．9ThepartialsumsoftheM6biusfunction91

4．10Briefsketchofanelementaryproofoftheprimenumbertheorem

4．11Selbcrg'sasymptoticformula

ExercisesforChapter4

Chapter5

Congruences

5．1Definitionandbasicpropertiesofcongruences

5．2Residueclassesandcompleteresiduesystems

5．3Linearcongruences

Chapter6

FiniteAbelianGroupsandTheirCharacters

Chapter7

Dirichlet'sTheoremonPrimesinArithmeticProgressions

Chapter8

PeriodicArithmeticalFunctionsandGaussSums

Chapter9

QuadraticResiduesandtheQuadraticReciprocityLaw

Chapter10

PrimitiveRoots

Chapter11

DirichletSeriesandEulerProducts

Chapter12

TheFunctions(s)andL(s，x)

Chapter13

AnalyticProofofthePrimeNumberTheorem