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书名 数论中的基本方法
分类 科学技术-自然科学-数学
作者 M.B.Nathanson
出版社 世界图书出版公司
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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.

此书为英文版。

目录

Preface

Notationandconventions

Ⅰ 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 Notes

Ⅱ 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 Notes

Ⅲ 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 Notes

References

Index

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