數(shù)論中的基本方法

內(nèi)容概要

This second edition of "Categories Work" adds two new chapters on topics of active interest. One is on symmetric monoidal categories and braided monoidal categories and the coherence theorems for them--items of interest in their own right and also in view of their use in string theory in quantum field theory. The second new chapter describes 2-categories and the higher-dimensional categories that have recently come into prominence. In addition, the bibliography has been expanded to cover some of the many other recent advances concerning categories.

書籍目錄

PrefaceNotationandconventionsⅠ AFirstCourseinNumberTheory  1 DivisibilityandPrimes    1.1 DivisionAlgorithm    1.2 GreatestCommonDivisors    1.3 TheEuclideanAlgorithmandContinuedFractions    1.4 TheFundamentalTheoremofArithmetic    1.5 Euclid'sTheoremandtheSieveofEratosthenes    1.6 ALinearDiophantineEquation    1.7 Notes  2 Congruences    2.1 TheRingofCongruenceClasses    2.2 LinearCongruences    2.3 TheEulerPhiFunction    2.4 ChineseRemainderTheorem    2.5 Euler'sTheoremandFermat'sTheorem    2.6 PseudoprimesandCarmichaelNumbers    2.7 PublicKeyCryptography    2.8 Notes  3 PrimitiveRootsandQuadraticReciprocity    3.1 PolynomialsandPrimitiveRoots    3.2 PrimitiveRootstoCompositeModuli    3.3 PowerResidues    3.4 QuadraticResidues    3.5 QuadraticReciprocityLaw    3.6 QuadraticResiduestoCompositeModuli    3.7 Notes  4 FourierAnalysisonFiniteAbelianGroups    4.1 TheStructureofFiniteAbelianGroups    4.2 CharactersofFiniteAbelianGroups    4.3 ElementaryFourierAnalysis    4.4 PoissonSummation    4.5 TraceFormulaeonFiniteAbelianGroups    4.6 GaussSumsandQuadraticReciprocity    4.7 TheSignoftheGaussSum    4.8 Notes  5 TheabcConjecture    5.1 IdealsandRadicals    5.2 Derivations    5.3 Mason'sTheorem    5.4 TheabcConjecture    5.5 TheCongruenceabcConjecture    5.6 NotesⅡ DivisorsandPrimesinMultiplicativeNumberTheory  6 ArithmeticFunctions    6.1 TheRingofArithmeticFunctions    6.2 MeanValuesofArithmeticFunctions    6.3 TheMSbiusFunction    6.4 MultiplicativeFunctions    6.5 ThemeanvalueoftheEulerPhiFunction    6.6 Notes  7 DivisorFunctions    7.1 DivisorsandFactorizations    7.2 ATheoremofRamanujan    7.3 SumsofDivisors    7.4 SumsandDifferencesofProducts    7.5 SetsofMultiples    7.6 AbundantNumbers    7.7 Notes  8 PrimeNumbers    8.1 Chebyshev'sTheorems    8.2 Mertens'sTheorems    8.3 TheNumberofPrimeDivisorsofanInteger    8.4 Notes  9 ThePrimeNumberTheorem    9.1 GeneralizedVonMangoldtFunctions    9.2 Selberg'sFormulae    9.3 TheElementaryProof    9.4 IntegerswithkPrimeFactors    9.5 Notes  10 PrimesinArithmeticProgressions    10.1 DirichletCharacters    10.2 DirichletL-Functions    10.3 PrimesModulo4    10.4 TheNonvanishingofL(1,X)    10.5 NotesⅢ ThreeProblemsinAdditiveNumberTheory  11 Waring'sProblem    11.1 SumsofPowers    11.2 StableBases    11.3 Shnirel'man'sTheorem    11.4 Waring'sProblemforPolynomials    11.5 Notes  12 SumsofSequencesofPolynomials    12.1 SumsandDifferencesofWeightedSets    12.2 LinearandQuadraticEquations    12.3 AnUpperBoundforRepresentations    12.4 Waring'sProblemforSequencesofPolynomials    12.5 Notes  13 Liouville'sIdentity    13.1 AMiraculousFormula    13.2 PrimeNumbersandQuadraticForms    13.3 ATernaryForm    13.4 ProofofLiouville'sIdentity    13.5 TwoCorollaries    13.6 Notes  14 SumsofanEvenNumberofSquares    14.1 SummaryofResults    14.2 ARecursionFormula    14.3 SumsofTwoSquares    14.4 SumsofFourSquares    14.5 SumsofSixSquares    14.6 SumsofEightSquares    14.7 SumsofTenSquares    14.8 Notes  15 PartitionAsymptotics    15.1 TheSizeofp(n)    15.2 PartitionFunctionsforFiniteSets    15.3 UpperandLowerBoundsforlogp(n)    15.4 Notes  16 AnInverseTheoremforPartitions    16.1 DensityDeterminesAsymptotics    16.2 AsymptoticsDetermineDensity    16.3 AbelianandTauberianTheorems    16.4 NotesReferencesIndex

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