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Number Theory: An Introduction to Mathematics

(ff) #1

  • I The Expanding Universe of Numbers............................ Part A

    • 0 Sets,RelationsandMappings

    • 1 NaturalNumbers............................................

    • 2 IntegersandRationalNumbers

    • 3 RealNumbers

    • 4 Metric Spaces

    • 5 ComplexNumbers

    • 6 QuaternionsandOctonions

    • 7 Groups

    • 8 RingsandFields

    • 9 Vector Spaces and Associative Algebras

    • 10 Inner Product Spaces........................................

    • 11 FurtherRemarks

    • 12 SelectedReferences

    • AdditionalReferences



  • II Divisibility..................................................

    • 1 GreatestCommonDivisors

    • 2TheB ́ezout Identity

    • 3 Polynomials

    • 4 EuclideanDomains..........................................

    • 5 Congruences

    • 6 SumsofSquares

    • 7 FurtherRemarks

    • 8 SelectedReferences

    • AdditionalReferences



  • III More on Divisibility.......................................... viii Contents

    • 1 TheLawofQuadraticReciprocity

    • 2 QuadraticFields

    • 3 Multiplicative Functions

    • 4 Linear Diophantine Equations

    • 5 FurtherRemarks

    • 6 SelectedReferences

    • AdditionalReferences



  • IV Continued Fractions and Their Uses............................

    • 1 TheContinuedFractionAlgorithm.............................

    • 2 Diophantine Approximation

    • 3 PeriodicContinuedFractions..................................

    • 4 Quadratic Diophantine Equations..............................

    • 5 The Modular Group

    • 6 Non-EuclideanGeometry.....................................

    • 7 Complements...............................................

    • 8 FurtherRemarks

    • 9 SelectedReferences

    • AdditionalReferences



  • V Hadamard’s Determinant Problem.............................

    • 1 WhatisaDeterminant?

    • 2 HadamardMatrices..........................................

    • 3 TheArtofWeighing.........................................

    • 4 SomeMatrixTheory.........................................

    • 5 ApplicationtoHadamard’sDeterminantProblem.................

    • 6 Designs....................................................

    • 7 Groups and Codes

    • 8 FurtherRemarks

    • 9 SelectedReferences



  • VI Hensel’sp-adic Numbers......................................

    • 1 ValuedFields...............................................

    • 2 Equivalence

    • 3 Completions................................................

    • 4 Non-ArchimedeanValuedFields...............................

    • 5 Hensel’sLemma

    • 6 LocallyCompactValuedFields................................

    • 7 FurtherRemarks

    • 8 SelectedReferences



  • VII The Arithmetic of Quadratic Forms............................. Part B

    • 1 Quadratic Spaces............................................

    • 2 TheHilbertSymbol

    • 3 TheHasse–MinkowskiTheorem...............................

    • 4 Supplements

    • 5 FurtherRemarks

    • 6 SelectedReferences



  • VIII The Geometry of Numbers....................................

    • 1 Minkowski’s Lattice Point Theorem

    • 2 Lattices....................................................

    • 3 Proof of the Lattice Point Theorem; Other Results................

    • 4 Voronoi Cells

    • 5 DensestPackings............................................

    • 6 Mahler’sCompactnessTheorem...............................

    • 7 FurtherRemarks

    • 8 SelectedReferences

    • AdditionalReferences



  • IX The Number of Prime Numbers................................

    • 1 FindingtheProblem

    • 2 Chebyshev’sFunctions.......................................

    • 3 ProofofthePrimeNumberTheorem

    • 4 The Riemann Hypothesis

    • 5 Generalizations and Analogues

    • 6 AlternativeFormulations

    • 7 SomeFurtherProblems

    • 8 FurtherRemarks

    • 9 SelectedReferences

    • AdditionalReferences



  • X A Character Study...........................................

    • 1 PrimesinArithmeticProgressions

    • 2 Characters of Finite Abelian Groups

    • 3 Proof of the Prime Number Theorem for Arithmetic Progressions

    • 4 Representations of Arbitrary Finite Groups

    • 5 Characters of Arbitrary Finite Groups..........................

    • 6 InducedRepresentationsandExamples

    • 7 Applications................................................

    • 8 Generalizations

    • 9 FurtherRemarks

    • 10 SelectedReferences



  • XI Uniform Distribution and Ergodic Theory....................... x Contents

    • 1 UniformDistribution

    • 2 Discrepancy................................................

    • 3 Birkhoff’s Ergodic Theorem

    • 4 Applications................................................

    • 5 Recurrence.................................................

    • 6 FurtherRemarks

    • 7 SelectedReferences

    • AdditionalReference



  • XII Elliptic Functions............................................

    • 1 Elliptic Integrals............................................

    • 2 TheArithmetic-GeometricMean

    • 3 Elliptic Functions

    • 4 ThetaFunctions.............................................

    • 5 Jacobian Elliptic Functions...................................

    • 6 The Modular Function

    • 7 FurtherRemarks

    • 8 SelectedReferences



  • XIII Connections with Number Theory..............................

    • 1 SumsofSquares

    • 2 Partitions..................................................

    • 3 CubicCurves...............................................

    • 4 Mordell’sTheorem

    • 5 FurtherResultsandConjectures

    • 6 SomeApplications

    • 7 FurtherRemarks

    • 8 SelectedReferences

    • AdditionalReferences



  • Notations........................................................

  • Axioms..........................................................

  • Index............................................................

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