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哈代数论（英文版·第6版）

哈代数论（英文版·第6版）
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作者: （G.H.Hardy） 著 , （G.H.Hardy） 著
出版社: 人民邮电出版社
2009-11
版次: 1
ISBN: 9787115214270
定价: 59.00
装帧: 平装
开本: 32开
纸张: 胶版纸
页数: 621页
字数: 576千字
正文语种: 英语
丛书: 图灵原版数字·统计学系列
分类: 自然科学
1 张插图图片
24人买过
  • 《哈代数论(英文版·第6版)》是数论领域的一部传世名著，成书于作者在牛津大学、剑桥大学等学校授课的讲义。书中从各个不同角度对数论进行了阐述，内容包括素数、无理数、同余、费马定理、连分数、不定式、二次域、算术函数、分化等。新版修订了每章末的注解，简要介绍了数论最新的发展；增加了一章讲述椭圆曲线，这是数论中最重要的突破之一。还列出进一步阅读的文献。
    《哈代数论(英文版·第6版)》适合数学专业本科生、研究生和教师用作教材或参考书，也适合对数论感兴趣的专业人士阅读参考。 G.H.Hardy（1877-1947），20世纪上半叶享有世界声誉的数学大师，是英国数学界和英国分析学派的领袖，对数论和分析学的发展有巨大的贡献和重大的影响，除了自己的研究工作之外，他还培养和指导了众多数学大家，包括印度数学奇才拉马努金和我国数学家华罗庚。
    E.M.Wright（1906-2005），英国著名数学家，毕业于牛津大学，是G．H．Hardy的学生。生前担任英国名校阿伯丁大学校长多年。爱丁堡皇家学会会士、伦敦数学会会士。曾任JournalofGraphTheory和ZentralbtattfurMathematik的名誉主编。 Ⅰ.THESERIESOFPRIMES(1)
    1.1.Divisibilityofintegers
    1.2.Primenumbers
    1.3.Statementofthefundamentaltheoremofarithmetic
    1.4.Thesequenceofprimes
    1.5.Somequestionsconcerningprimes
    1.6.Somenotations
    1.7.Thelogarithmicfunction
    1.8.Statementoftheprimenumbertheorem

    Ⅱ.THESERIESOFPRIMES(2)
    2.1.FirstproofofEuclidssecondtheorem
    2.2.FurtherdeductionsfromEuclidsargument
    2.3.Primesincertainarithmeticalprogressions
    2.4.SecondproofofEuclidstheorem
    2.5.FermatsandMersennesnumbers
    2.6.ThirdproofofEuclidstheorem
    2.7.Furtherresultsonformulaeforprimes
    2.8.Unsolvedproblemsconcerningprimes
    2.9.Moduliofintegers
    2.10.Proofofthefundamentaltheoremorarithmetic
    2.11.Anotherproofofthefundamentaltheorem

    Ⅲ.FAREYSERIESANDATHEOREMOFMINKOWSKI
    3.1.ThedefinitionandsimplestpropertiesofaFareyseries
    3.2.Theequivalenceofthetwocharacteristicproperties
    3.3.FirstproofofTheorems28and
    3.4.Secondproofofthetheorems
    3.5.Theintegrallattice
    3.6.Somesimplepeopertiesofthefundamentallattice
    3.7.ThirdproofofTheorems28and
    3.8.TheFarevdissectionofthecontinuum
    3.9.AtheoremofMinkowski
    3.10ProofofMinkowskistheorem
    3.11.DevelopmentsofTheorem

    Ⅳ.IRRATIONALNUMBERS
    4.1.Somegeneralities
    4.2.Numbersknowntobeirrational
    4.3.ThetheoremofPythagorasanditsgeneralizations
    4.4.TheuseofthefundamentaltheoremintheproofsofTheorems43-
    4.5.Ahistoricaldigression3o
    4.6.Geometricalproofoftheirrationalityof√
    4.7.Somemoreirrationalnumbers
    Ⅴ.CONGRUENCESANDRESIDUES
    5.1.Highestcommondivisorandleastcommonmultiple
    5.2.Congruencesandclassesofresidues
    5.3.Elementaryorooertiesofcongruences
    5.4.Linearcongruences
    5.5.Eulersfunctionφ(m)
    5.6.AoolicationsofTheorems59and61totrigonometricalsums
    5.7.Ageneralprinciple
    5.8.Constructionoftheregularpolygonof17sides

    Ⅵ.FFRMATsTHEOREMANDITSCONSEOUENCES
    6.1.Fermatstheorem
    6.2.Somepropertiesofbinomialcoefficients
    6.3.AsecondproofofTheorem
    6.4.ProofofTheorem
    6.5.Quadraticresidues
    6.6.SoecialcasesofTheorem79：Wilsonstheorem
    6.7.Elementarypropertiesofquadraticresiduesandnon-residues
    6.8.Theorderofa(modm)IslS
    6.9.TheconverseofFermatstheorem
    6.10.Divisibilityof2P-1_1byp
    6.11.Gaussslemmaandthequadraticcharacterof
    6.12.Thelawofreciprocity
    6.13.Proofofthelawofreciprocity
    6.14.Testsfororimalitv
    6.15.FactorsofMersennenumbers;atheoremofEuler

    Ⅶ.GENERALPROPERTIESOFCONGRUENCES
    7.1.Rootsofcongruences
    7.2.Integralpolynomialsandidenticalcongruences
    7.3.Divisibilityofpolynomials(modm)
    7.4.Rootsofc~nmuencestoaorimemodulus
    7.5.Someapplicationsofthegeneraltheorems
    7.6.LagrangesproofofFermatsandWilsonstheorems
    7.7.Theresidueof{1/2(p-1)}!
    7.8.AtheoremofWolstenholme
    7.9.ThetheoremofvonStaudt
    7.10.ProofofvonStaudtstheorem

    Ⅷ.CONGRUENCESTOCOMPOSITEMODULI
    8.1.Linearcongruences
    8.2.Congruencesofhigherdegree
    8.3.Congruencestoaorime-oowermodulus
    8.4.Examoles
    8.5.Bauersidenticalcongruence
    8.6.Bauerscongruence：thecasep=
    8.7.AtheoremofLeudesdorf
    8.8.FurtherconsequencesofBauerstheorem
    8.9.Theresiduesof2P-1and(p-1)!tomodulusp

    Ⅸ.THEREPRESENTATIONOFNUMBERSBYDECIMALS
    9.1.Thedecimalassociatedwithagivennumber
    9.2.Terminatingandrecurringdecimals
    9.3.Representationofnumbersinotherscales
    9.4.Irrationalsdefinedbydecimals
    9.5.Testsfordivisibility
    9.6.Decimalswiththemaximumperiod
    9.7.Bachetsproblemoftheweights
    9.8.ThegameofNim
    9.9.Integerswithmissingdigits
    9.10.Setsofmeasurezero
    9.11.Decimalswithmissingdigits
    9.12.Normalnumbers
    9.13.Proofthatalmostallnumbersarenormal

    Ⅹ.CONTINUEDFRACTIONS
    10.1.Finitecontinuedfractions
    10.2.Convementstoacontinuedfraction
    10.3.Continuedfractionswithpositivequotients
    10.4.Simplecontinuedfractions
    10.5.Therepresentationofanirreduciblerationalfractionbyasimplecontinuedfraction
    10.6.ThecontinuedfractionalgorithmandEuclidsalgorithm
    10.7.Thedifferencebetweenthefractionanditsconvergents
    10.8.Infinitesimplecontinuedfractions
    10.9.Therepresentationofanirrationalnumberbyaninfinitecontinuedfraction
    10.10.Alemma
    10.11.Equivalentnumbers
    10.12.Periodiccontinuedfractions
    10.13.SomesoecialQuadraticsurds
    10.14.TheseriesofFibonacciandLucas
    10.15.Approximationbyconvergents

    Ⅺ.APPROXIMATIONOFIRRATIONALSBYRATIONALS
    11.1.Statementoftheoroblem
    11.2.Generalitiesconcerningtheproblem
    11.3.AnargumentofDirichlet
    11.4.Ordersofaporoximation
    11.5.Aloohrnienncltranscendentalnumbers
    11.6.Theexistenceoftranscendentalnumbers..
    11.7.Liouvillestheoremandtheconstructionoftranscendentalnumbers
    11.8.Themeasureoftheclosestapproximationstoanarbitraryirrational
    11.9.Anothertheoremconcerningtheconvergentstoacontinuedfraction
    11.10.Continuedfractionswithboundedquotients
    11.11.Furthertheoremsconcerningapproximation
    11.12.Simultaneousapproximation
    11.13.Thetranscendenceofe
    11.14.Thetranscendenceofπ

    Ⅻ.THEFUNDAMENIALTHEOREMOFARITHMETICINk(1)，k(i)，ANDk(O)
    12.1.Algebraicnumbersandintegers
    12.2.Therationalintegers，theGaussianintegers，andtheintegersofk(p)
    12.3.Euclidsalgorithm
    12.4.AoolicationofEuclidsalgorithmtothefundamentaltheoremink(1)
    12.5.HistoricalremarksonEuclidsalgorithmandthefundamentaltheorem
    12.6.ProoertiesoftheGaussianintegers
    12.7.Primesink(i)
    12.8.Thefundnmentaltheoremofarithmeticink(i)
    12.9.Theintegersofk(p)

    ⅩⅢ.SOMEDIOPHANTINEEQUATIONS
    13.1.Fermatslasttheorem
    13.2.Theeauationxz4-vz=zz
    13.3.Theequationx4-t-y4=z
    13.4.Theequationx3+y3=z
    13.5.Theequationx3+y3=3z
    13.6.Theexoressionofarationalasasumofrationalcubes
    13.7.Theequationx3+y3+z3=t

    ⅩⅣ.OUADRATICFIELDS(1)
    14.1.Algebraicfields
    14.2.Algebraicnumbersandintegers：orimitivepolynomials
    14.3.Thegeneralquadraticfieldk(√m)
    14.4.Unitiesandorimes
    14.5.Theunitiesofk(√2)
    14.6.Fieldsinwhichthefundamentaltheoremisfalse
    14.7.ComnlexEuclideanfields
    14.8.RealEuclideanfields
    14.9.RealEuclideanfields(continued)

    ⅩⅤ.OUADRATICFIELDS(2)
    15.1.Theorimesofk(i)
    15.2.Fermatstheoremink(i)
    15.3.Theprimesofk(p)
    15.4.Theprimesofk(√2)andk(√5)
    15.5.LucasstestfortheprimalityoftheMersennenumberM4n+
    15.6.Generalremarksonthearithmeticofquadraticfields
    15.7.Idealsinaquadraticfield
    15.8.Otherfields

    ⅩⅥ.THEARITHMETICALFUNCTIONSФ(n)，μ(n)，d(n)，σ(n)，r(n)
    16.1.ThefunctionФ(n)
    16.2.AfurtherproofofTheorem
    16.3.TheMrbiusfunction
    16.4.TheMrbiusinversionformula
    16.5.Furtherinversionformulae
    16.6.EvaluationofRamanuianssum
    16.7.Thefunctionsd(n)andcrk(n)
    16.8.Perfectnumbers
    16.9.Thefunctionr(n)
    16.10.Proofoftheformulaforr(n)

    ⅩⅦ.GENERATINGFUNCTIONSOFARITHMETICALFUNCTIONS
    17.1.ThegenerationofarithmeticalfunctionsbymeansofDirichletseries
    17.2.Thezetafunction
    17.3.Thebehaviourofξ(s)whens→
    17.4.MultiplicationofDirichletseries
    17.5.Thegeneratingfunctionsofsomespecialarithmeticalfunctions32~
    17.6.TheanalyticalinterpretationoftheM6biusformula
    17.7.ThefunctionA(n)
    17.8.Furtherexamplesofgeneratingfunctions
    17.9.Thegeneratingfunctionofr(n)
    17.10.Generatingfunctionsofothertypes

    ⅩⅧ.THEORDEROFMAGNITUDEOFARITHMETICALFUNCTIONS
    18.1.Theorderofd(n)
    18.2.Theaverageorderofd(n)
    18.3.Theorderofσ(n)
    18.4.TheorderofФ(n)
    18.5.TheaverageorderofФ(n)
    18.6.Thenumberofsquarefreenumbers
    18.7.Theorderofσ(n)

    ⅩⅨ.PARTITIONS
    19.1.Thegeneralproblemofadditivearithmetic
    19.2.Partitionsofnumbers
    19.3.Thegeneratingfunctionofp(n)
    19.4.Othergeneratingfunctions
    19.5.TwotheoremsofEuler
    19.6.Furtheralgebraicalidentities
    19.7.AnotherformulaforF(x)
    19.8.AtheoremofJacobi
    19.9.SpecialcasesofJacobisidentity
    19.10.ApplicationsofTheorem
    19.11.ElementaryproofofTheorem
    19.12.Congruencepropertiesofp(n)
    19.13.TheRogers-Ramanujanidentities
    19.14.ProofofTheorems362and
    19.15.Ramanujanscontinuedfraction

    ⅩⅩ.THEREPRESENTATIONOFANUMBERBYTWOORFOURSQUARES
    20.1.Waringsproblem：thenumbersg(k)andG(k)
    20.2.Squares
    20.3.SecondproofofTheorem
    20，4.ThirdandfourthproofsofTheorem
    20.5.Thefour-squaretheorem
    20.6.Quaternions
    20.7.Preliminarytheoremsaboutintegralquatemions
    20.8.Thehighestcommonfight-handdivisoroftwoquatemions
    20.9.PrimequatemionsandtheproofofTheorem
    20.10.Thevaluesofg(2)andG(2)
    20.11.LemmasforthethirdproofofTheorem
    20.12.ThirdproofofTheorem369：thenumberofrepresentations
    20.13.Representationsbyalargernumberofsquares

    ⅩⅩⅠ.REPRESENTATIONBYCUBESANDHIGHERPOWERS
    21.1.Biquadrates
    21.2.Cubes：theexistenceofG(3)andg(3)
    21.3.Aboundforg(3)
    21.4.Higherpowers
    21.5.Alowerboundforg(k)
    21.6.LowerboundsforG(k)
    21.7.Sumsaffectedwithsigns：thenumberv(k)
    21.8.Upperboundsforv(k)
    21.9.TheproblemofProuhetandTarry：thenumberP(k，j)
    21.10.EvaluationofP(k，j)forparticularkandj
    21.11.FurtherproblemsofDiophantineanalysis

    ⅩⅩⅡ.THESERIESOFPRIMES(3)
    22.1.Thefunctions0(x)and$(x)
    22.2.Proofthat0(x)and~(x)areoforderx
    22.3.Bertrandspostulateandaformulaforprimes
    22.4.ProofofTheorems7and
    22.5.Twoformaltransformations
    22.6.Animportantsum
    22.7.Thesum12p~1andtheproductFI(1-p-1)
    22.8.Mertensstheorem
    22.9.ProofofTheorems323and
    22.10.Thenumberofprimefactorsofn
    22.11.Thenormalorderofto(n)andf2(n)
    22.12.Anoteonroundnumbers
    22.13.Thenormalorderofd(n)
    22.14.Selbergstheorem
    22.15.ThefunctionsR(x)andV(ξ)
    22.16.CompletionoftheproofofTheorems434，6，and
    22.17.ProofofTheorem
    22.18.Productsofkprimefactors
    22.19.Primesinaninterval
    22.20.Aconjectureaboutthedistributionofprimepairsp，p+

    ⅩⅩⅢ.KRONECKERSTHEOREM
    23.1.Kroneckerstheoreminonedimension
    23.2.Proofsoftheone-dimensionaltheorem
    23.3.Theproblemofthereflectedray
    23.4.Statementofthegeneraltheorem
    23.5.Thetwoformsofthetheorem
    23.6.Anillustration
    23.7.Lettenmeyersproofofthetheorem
    23.8.Estermannsproofofthetheorem
    23.9.Bohrsproofofthetheorem
    23.10.Uniformdistribution

    ⅩⅩⅣ.GEOMETRYOFNUMBERS
    24.1.Introductionandrestatementofthefundamentaltheorem
    24.2.Simpleapplications
    24.3.ArithmeticalproofofTheorem
    24.4.Bestpossibleinequalities
    24.5.Thebestpossibleinequalityforξ2+n
    24.6.Thebestpossibleinequalityfor|ξn|
    24.7.Atheoremconcerningnon-homogeneousforms
    24.8.ArithmeticalproofofTheorem
    24.9.Tchebotarefstheorem
    24.10.AconverseofMinkowskisTheorem

    ⅩⅩⅤ.ELLIPTICCURVES
    25.1.Thecongruentnumberproblem
    25.2.Theadditionlawonanellipticcurve
    25.3.Otherequationsthatdefineellipticcurves
    25.4.Pointsoffiniteorder
    25.5.Thegroupofrationalpoints
    25.6.Thegroupofpointsmodulop.
    25.7.Integerpointsonellipticcurves
    25.8.TheL-seriesofanellipticcurve
    25.9.Pointsoffiniteorderandmodularcurves
    25.10.EllipticcurvesandFermatslasttheorem

    APPENDIX
    1.Anotherformulaforpn
    2.AgeneralizationofTheorem
    3.Unsolvedproblemsconcerningprimes
    ALISTOFBOOKS
    INDEXOFSPECIALSYMBOLSANDWORDS
    INDEXOFNAMES
    GENERALINDEX
  • 内容简介:
    《哈代数论(英文版·第6版)》是数论领域的一部传世名著，成书于作者在牛津大学、剑桥大学等学校授课的讲义。书中从各个不同角度对数论进行了阐述，内容包括素数、无理数、同余、费马定理、连分数、不定式、二次域、算术函数、分化等。新版修订了每章末的注解，简要介绍了数论最新的发展；增加了一章讲述椭圆曲线，这是数论中最重要的突破之一。还列出进一步阅读的文献。
    《哈代数论(英文版·第6版)》适合数学专业本科生、研究生和教师用作教材或参考书，也适合对数论感兴趣的专业人士阅读参考。
  • 作者简介:
    G.H.Hardy（1877-1947），20世纪上半叶享有世界声誉的数学大师，是英国数学界和英国分析学派的领袖，对数论和分析学的发展有巨大的贡献和重大的影响，除了自己的研究工作之外，他还培养和指导了众多数学大家，包括印度数学奇才拉马努金和我国数学家华罗庚。
    E.M.Wright（1906-2005），英国著名数学家，毕业于牛津大学，是G．H．Hardy的学生。生前担任英国名校阿伯丁大学校长多年。爱丁堡皇家学会会士、伦敦数学会会士。曾任JournalofGraphTheory和ZentralbtattfurMathematik的名誉主编。
  • 目录:
    Ⅰ.THESERIESOFPRIMES(1)
    1.1.Divisibilityofintegers
    1.2.Primenumbers
    1.3.Statementofthefundamentaltheoremofarithmetic
    1.4.Thesequenceofprimes
    1.5.Somequestionsconcerningprimes
    1.6.Somenotations
    1.7.Thelogarithmicfunction
    1.8.Statementoftheprimenumbertheorem

    Ⅱ.THESERIESOFPRIMES(2)
    2.1.FirstproofofEuclidssecondtheorem
    2.2.FurtherdeductionsfromEuclidsargument
    2.3.Primesincertainarithmeticalprogressions
    2.4.SecondproofofEuclidstheorem
    2.5.FermatsandMersennesnumbers
    2.6.ThirdproofofEuclidstheorem
    2.7.Furtherresultsonformulaeforprimes
    2.8.Unsolvedproblemsconcerningprimes
    2.9.Moduliofintegers
    2.10.Proofofthefundamentaltheoremorarithmetic
    2.11.Anotherproofofthefundamentaltheorem

    Ⅲ.FAREYSERIESANDATHEOREMOFMINKOWSKI
    3.1.ThedefinitionandsimplestpropertiesofaFareyseries
    3.2.Theequivalenceofthetwocharacteristicproperties
    3.3.FirstproofofTheorems28and
    3.4.Secondproofofthetheorems
    3.5.Theintegrallattice
    3.6.Somesimplepeopertiesofthefundamentallattice
    3.7.ThirdproofofTheorems28and
    3.8.TheFarevdissectionofthecontinuum
    3.9.AtheoremofMinkowski
    3.10ProofofMinkowskistheorem
    3.11.DevelopmentsofTheorem

    Ⅳ.IRRATIONALNUMBERS
    4.1.Somegeneralities
    4.2.Numbersknowntobeirrational
    4.3.ThetheoremofPythagorasanditsgeneralizations
    4.4.TheuseofthefundamentaltheoremintheproofsofTheorems43-
    4.5.Ahistoricaldigression3o
    4.6.Geometricalproofoftheirrationalityof√
    4.7.Somemoreirrationalnumbers
    Ⅴ.CONGRUENCESANDRESIDUES
    5.1.Highestcommondivisorandleastcommonmultiple
    5.2.Congruencesandclassesofresidues
    5.3.Elementaryorooertiesofcongruences
    5.4.Linearcongruences
    5.5.Eulersfunctionφ(m)
    5.6.AoolicationsofTheorems59and61totrigonometricalsums
    5.7.Ageneralprinciple
    5.8.Constructionoftheregularpolygonof17sides

    Ⅵ.FFRMATsTHEOREMANDITSCONSEOUENCES
    6.1.Fermatstheorem
    6.2.Somepropertiesofbinomialcoefficients
    6.3.AsecondproofofTheorem
    6.4.ProofofTheorem
    6.5.Quadraticresidues
    6.6.SoecialcasesofTheorem79：Wilsonstheorem
    6.7.Elementarypropertiesofquadraticresiduesandnon-residues
    6.8.Theorderofa(modm)IslS
    6.9.TheconverseofFermatstheorem
    6.10.Divisibilityof2P-1_1byp
    6.11.Gaussslemmaandthequadraticcharacterof
    6.12.Thelawofreciprocity
    6.13.Proofofthelawofreciprocity
    6.14.Testsfororimalitv
    6.15.FactorsofMersennenumbers;atheoremofEuler

    Ⅶ.GENERALPROPERTIESOFCONGRUENCES
    7.1.Rootsofcongruences
    7.2.Integralpolynomialsandidenticalcongruences
    7.3.Divisibilityofpolynomials(modm)
    7.4.Rootsofc~nmuencestoaorimemodulus
    7.5.Someapplicationsofthegeneraltheorems
    7.6.LagrangesproofofFermatsandWilsonstheorems
    7.7.Theresidueof{1/2(p-1)}!
    7.8.AtheoremofWolstenholme
    7.9.ThetheoremofvonStaudt
    7.10.ProofofvonStaudtstheorem

    Ⅷ.CONGRUENCESTOCOMPOSITEMODULI
    8.1.Linearcongruences
    8.2.Congruencesofhigherdegree
    8.3.Congruencestoaorime-oowermodulus
    8.4.Examoles
    8.5.Bauersidenticalcongruence
    8.6.Bauerscongruence：thecasep=
    8.7.AtheoremofLeudesdorf
    8.8.FurtherconsequencesofBauerstheorem
    8.9.Theresiduesof2P-1and(p-1)!tomodulusp

    Ⅸ.THEREPRESENTATIONOFNUMBERSBYDECIMALS
    9.1.Thedecimalassociatedwithagivennumber
    9.2.Terminatingandrecurringdecimals
    9.3.Representationofnumbersinotherscales
    9.4.Irrationalsdefinedbydecimals
    9.5.Testsfordivisibility
    9.6.Decimalswiththemaximumperiod
    9.7.Bachetsproblemoftheweights
    9.8.ThegameofNim
    9.9.Integerswithmissingdigits
    9.10.Setsofmeasurezero
    9.11.Decimalswithmissingdigits
    9.12.Normalnumbers
    9.13.Proofthatalmostallnumbersarenormal

    Ⅹ.CONTINUEDFRACTIONS
    10.1.Finitecontinuedfractions
    10.2.Convementstoacontinuedfraction
    10.3.Continuedfractionswithpositivequotients
    10.4.Simplecontinuedfractions
    10.5.Therepresentationofanirreduciblerationalfractionbyasimplecontinuedfraction
    10.6.ThecontinuedfractionalgorithmandEuclidsalgorithm
    10.7.Thedifferencebetweenthefractionanditsconvergents
    10.8.Infinitesimplecontinuedfractions
    10.9.Therepresentationofanirrationalnumberbyaninfinitecontinuedfraction
    10.10.Alemma
    10.11.Equivalentnumbers
    10.12.Periodiccontinuedfractions
    10.13.SomesoecialQuadraticsurds
    10.14.TheseriesofFibonacciandLucas
    10.15.Approximationbyconvergents

    Ⅺ.APPROXIMATIONOFIRRATIONALSBYRATIONALS
    11.1.Statementoftheoroblem
    11.2.Generalitiesconcerningtheproblem
    11.3.AnargumentofDirichlet
    11.4.Ordersofaporoximation
    11.5.Aloohrnienncltranscendentalnumbers
    11.6.Theexistenceoftranscendentalnumbers..
    11.7.Liouvillestheoremandtheconstructionoftranscendentalnumbers
    11.8.Themeasureoftheclosestapproximationstoanarbitraryirrational
    11.9.Anothertheoremconcerningtheconvergentstoacontinuedfraction
    11.10.Continuedfractionswithboundedquotients
    11.11.Furthertheoremsconcerningapproximation
    11.12.Simultaneousapproximation
    11.13.Thetranscendenceofe
    11.14.Thetranscendenceofπ

    Ⅻ.THEFUNDAMENIALTHEOREMOFARITHMETICINk(1)，k(i)，ANDk(O)
    12.1.Algebraicnumbersandintegers
    12.2.Therationalintegers，theGaussianintegers，andtheintegersofk(p)
    12.3.Euclidsalgorithm
    12.4.AoolicationofEuclidsalgorithmtothefundamentaltheoremink(1)
    12.5.HistoricalremarksonEuclidsalgorithmandthefundamentaltheorem
    12.6.ProoertiesoftheGaussianintegers
    12.7.Primesink(i)
    12.8.Thefundnmentaltheoremofarithmeticink(i)
    12.9.Theintegersofk(p)

    ⅩⅢ.SOMEDIOPHANTINEEQUATIONS
    13.1.Fermatslasttheorem
    13.2.Theeauationxz4-vz=zz
    13.3.Theequationx4-t-y4=z
    13.4.Theequationx3+y3=z
    13.5.Theequationx3+y3=3z
    13.6.Theexoressionofarationalasasumofrationalcubes
    13.7.Theequationx3+y3+z3=t

    ⅩⅣ.OUADRATICFIELDS(1)
    14.1.Algebraicfields
    14.2.Algebraicnumbersandintegers：orimitivepolynomials
    14.3.Thegeneralquadraticfieldk(√m)
    14.4.Unitiesandorimes
    14.5.Theunitiesofk(√2)
    14.6.Fieldsinwhichthefundamentaltheoremisfalse
    14.7.ComnlexEuclideanfields
    14.8.RealEuclideanfields
    14.9.RealEuclideanfields(continued)

    ⅩⅤ.OUADRATICFIELDS(2)
    15.1.Theorimesofk(i)
    15.2.Fermatstheoremink(i)
    15.3.Theprimesofk(p)
    15.4.Theprimesofk(√2)andk(√5)
    15.5.LucasstestfortheprimalityoftheMersennenumberM4n+
    15.6.Generalremarksonthearithmeticofquadraticfields
    15.7.Idealsinaquadraticfield
    15.8.Otherfields

    ⅩⅥ.THEARITHMETICALFUNCTIONSФ(n)，μ(n)，d(n)，σ(n)，r(n)
    16.1.ThefunctionФ(n)
    16.2.AfurtherproofofTheorem
    16.3.TheMrbiusfunction
    16.4.TheMrbiusinversionformula
    16.5.Furtherinversionformulae
    16.6.EvaluationofRamanuianssum
    16.7.Thefunctionsd(n)andcrk(n)
    16.8.Perfectnumbers
    16.9.Thefunctionr(n)
    16.10.Proofoftheformulaforr(n)

    ⅩⅦ.GENERATINGFUNCTIONSOFARITHMETICALFUNCTIONS
    17.1.ThegenerationofarithmeticalfunctionsbymeansofDirichletseries
    17.2.Thezetafunction
    17.3.Thebehaviourofξ(s)whens→
    17.4.MultiplicationofDirichletseries
    17.5.Thegeneratingfunctionsofsomespecialarithmeticalfunctions32~
    17.6.TheanalyticalinterpretationoftheM6biusformula
    17.7.ThefunctionA(n)
    17.8.Furtherexamplesofgeneratingfunctions
    17.9.Thegeneratingfunctionofr(n)
    17.10.Generatingfunctionsofothertypes

    ⅩⅧ.THEORDEROFMAGNITUDEOFARITHMETICALFUNCTIONS
    18.1.Theorderofd(n)
    18.2.Theaverageorderofd(n)
    18.3.Theorderofσ(n)
    18.4.TheorderofФ(n)
    18.5.TheaverageorderofФ(n)
    18.6.Thenumberofsquarefreenumbers
    18.7.Theorderofσ(n)

    ⅩⅨ.PARTITIONS
    19.1.Thegeneralproblemofadditivearithmetic
    19.2.Partitionsofnumbers
    19.3.Thegeneratingfunctionofp(n)
    19.4.Othergeneratingfunctions
    19.5.TwotheoremsofEuler
    19.6.Furtheralgebraicalidentities
    19.7.AnotherformulaforF(x)
    19.8.AtheoremofJacobi
    19.9.SpecialcasesofJacobisidentity
    19.10.ApplicationsofTheorem
    19.11.ElementaryproofofTheorem
    19.12.Congruencepropertiesofp(n)
    19.13.TheRogers-Ramanujanidentities
    19.14.ProofofTheorems362and
    19.15.Ramanujanscontinuedfraction

    ⅩⅩ.THEREPRESENTATIONOFANUMBERBYTWOORFOURSQUARES
    20.1.Waringsproblem：thenumbersg(k)andG(k)
    20.2.Squares
    20.3.SecondproofofTheorem
    20，4.ThirdandfourthproofsofTheorem
    20.5.Thefour-squaretheorem
    20.6.Quaternions
    20.7.Preliminarytheoremsaboutintegralquatemions
    20.8.Thehighestcommonfight-handdivisoroftwoquatemions
    20.9.PrimequatemionsandtheproofofTheorem
    20.10.Thevaluesofg(2)andG(2)
    20.11.LemmasforthethirdproofofTheorem
    20.12.ThirdproofofTheorem369：thenumberofrepresentations
    20.13.Representationsbyalargernumberofsquares

    ⅩⅩⅠ.REPRESENTATIONBYCUBESANDHIGHERPOWERS
    21.1.Biquadrates
    21.2.Cubes：theexistenceofG(3)andg(3)
    21.3.Aboundforg(3)
    21.4.Higherpowers
    21.5.Alowerboundforg(k)
    21.6.LowerboundsforG(k)
    21.7.Sumsaffectedwithsigns：thenumberv(k)
    21.8.Upperboundsforv(k)
    21.9.TheproblemofProuhetandTarry：thenumberP(k，j)
    21.10.EvaluationofP(k，j)forparticularkandj
    21.11.FurtherproblemsofDiophantineanalysis

    ⅩⅩⅡ.THESERIESOFPRIMES(3)
    22.1.Thefunctions0(x)and$(x)
    22.2.Proofthat0(x)and~(x)areoforderx
    22.3.Bertrandspostulateandaformulaforprimes
    22.4.ProofofTheorems7and
    22.5.Twoformaltransformations
    22.6.Animportantsum
    22.7.Thesum12p~1andtheproductFI(1-p-1)
    22.8.Mertensstheorem
    22.9.ProofofTheorems323and
    22.10.Thenumberofprimefactorsofn
    22.11.Thenormalorderofto(n)andf2(n)
    22.12.Anoteonroundnumbers
    22.13.Thenormalorderofd(n)
    22.14.Selbergstheorem
    22.15.ThefunctionsR(x)andV(ξ)
    22.16.CompletionoftheproofofTheorems434，6，and
    22.17.ProofofTheorem
    22.18.Productsofkprimefactors
    22.19.Primesinaninterval
    22.20.Aconjectureaboutthedistributionofprimepairsp，p+

    ⅩⅩⅢ.KRONECKERSTHEOREM
    23.1.Kroneckerstheoreminonedimension
    23.2.Proofsoftheone-dimensionaltheorem
    23.3.Theproblemofthereflectedray
    23.4.Statementofthegeneraltheorem
    23.5.Thetwoformsofthetheorem
    23.6.Anillustration
    23.7.Lettenmeyersproofofthetheorem
    23.8.Estermannsproofofthetheorem
    23.9.Bohrsproofofthetheorem
    23.10.Uniformdistribution

    ⅩⅩⅣ.GEOMETRYOFNUMBERS
    24.1.Introductionandrestatementofthefundamentaltheorem
    24.2.Simpleapplications
    24.3.ArithmeticalproofofTheorem
    24.4.Bestpossibleinequalities
    24.5.Thebestpossibleinequalityforξ2+n
    24.6.Thebestpossibleinequalityfor|ξn|
    24.7.Atheoremconcerningnon-homogeneousforms
    24.8.ArithmeticalproofofTheorem
    24.9.Tchebotarefstheorem
    24.10.AconverseofMinkowskisTheorem

    ⅩⅩⅤ.ELLIPTICCURVES
    25.1.Thecongruentnumberproblem
    25.2.Theadditionlawonanellipticcurve
    25.3.Otherequationsthatdefineellipticcurves
    25.4.Pointsoffiniteorder
    25.5.Thegroupofrationalpoints
    25.6.Thegroupofpointsmodulop.
    25.7.Integerpointsonellipticcurves
    25.8.TheL-seriesofanellipticcurve
    25.9.Pointsoffiniteorderandmodularcurves
    25.10.EllipticcurvesandFermatslasttheorem

    APPENDIX
    1.Anotherformulaforpn
    2.AgeneralizationofTheorem
    3.Unsolvedproblemsconcerningprimes
    ALISTOFBOOKS
    INDEXOFSPECIALSYMBOLSANDWORDS
    INDEXOFNAMES
    GENERALINDEX
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