目錄
Preface
Part III. Analytic Tools
9. Bernoulli Polynomials and the Gamma Function
9.1 Bernoulli Numbers and Polynomials
9.1.1 Generating Functions for Bernoulli Polynomials
9.1.2 Further Recurrences for Bernoulli Polynomials
9.1.3 Computing a Single Bernoulli Number
9.1.4 Bernoulli Polynomials and Fourier Series
9.2 Analytic Applications of Bernoulli Polynomials
9.2.1 Asymptotic Expansions
9.2.2 The Euler-MacLaurin Summation Formula
9.2.3 The Remainder Term and the Constant Term
9.2.4 Euler-MacLaurin and the Laplace Transform
9.2.5 Basic Applications of the Euler-MacLaurin Formula
9.3 Applications to Numerical Integration
9.3.1 Standard Euler-MacLaurin Numerical Integration
9.3.2 The Basic Tanh-Sinh Numerical Integration Method
9.3.3 General Doubly Exponential Numerical Integration
9.4 x-Bernoulli Numbers, Polynomials, and Functions
9.4.1 x-Bernoulli Numbers and Polynomials
9.4.2 x-Bernoulli Functions
9.4.3 The x-Euler-MacLaurin Summation Formula
9.5 Arithmetic Properties of Bernoulli Numbers
9.5.1 x-Power Sums
9.5.2 The Generalized Clausen-von Staudt Congruence
9.5.3 The Voronoi Congruence
9.5.4 The Kummer Congruences
9.5.5 The Almkvist-Meurman Theorem
9.6 The Real and Complex Gamma Functions
9.6.1 The Hurwitz Zeta Function
9.6.2 Definition of the Gamma Function
9.6.3 Preliminary Results for the Study of r(s)
9.6.4 Properties of the Gamma Function
9.6.5 Specific Properties of the Function w(s)
9.6.6 Fourier Expansions of S(s,x) and log(F(x))
9.7 Integral Transforms
9.7.1 Generalities on Integral Transforms
9.7.2 The Fourier Transform
9.7.3 The Mellin Transform
9.7.4 The Laplace Transform
9.8 Bessel Functions
9.8.1 Definitions
9.8.2 Integral Representations and Applications
9.9 Exercises for Chapter 9
10. Dirichlet Series and L-Functions
10.1 Arithmetic Functions and Dirichlet Series
10.1.1 Operations on Arithmetic Functions
10.1.2 Multiplicative Functions
10.2 The Analytic Theory of L-Series
10.2.1 Simple Approaches to Analytic Continuation
10.2.2 The Use of the Hurwitz Zeta Function S(s, x)
10.2.3 The Functional Equation for the Theta Function
10.2.4 The Functional Equation for Dirichlet L-Functions
10.2.5 Generalized Poisson Summation Formulas
10.2.6 Voronoi's Error Term in the Circle Problem
10.3 Special Values of Dirichlet L-Functions
10.3.1 Basic Results on Special Values
10.3.2 Special Values of L-Functions and Modular Forms
10.3.3 The P61ya-Vinogradov Inequality
10.3.4 Bounds and Averages for L(x, 1)
10.3.5 Expansions of ((s) Around s = k C Z < 1
10.3.6 Numerical Computation of Euler Products and Sums
10.4 Epstein Zeta Functions
10.4.1 The Nonholomorphic Eisenstein Series G(r, s)
10.4.2 The Kronecker Limit Formula
10.5 Dirichlet Series Linked to Number Fields
10.5.1 The Dedekind Zeta Function Sk(s)
10.5.2 The Dedekind Zeta Function of Quadratic Fields
10.5.3 Applications of the Kronecker Limit Formula
10.5.4 The Dedekind Zeta Function of Cyclotomic Fields
10.5.5 The Nonvanishing of L(x, 1)
10.5.6 Application to Primes in Arithmetic Progression
10.5.7 Conjectures on Dirichlet L-Functions
10.6 Science Fiction on L-Functions
10.6.1 Local L-Functions
10.6.2 Global L-Functions
10.7 The Prime Number Theorem
10.7.1 Estimates for S(s)
10.7.2 Newman's Proof
10.7.3 Iwaniec's Proof
10.8 Exercises for Chapter 10
11. p-adic Gamma and L-Functions
11.1 Generalities on p-adic Functions
11.1.1 Methods for Constructing p-adic Functions
11.1.2 A Brief Study of Volkenborn Integrals
11.2 The p-adic Hurwitz Zeta Functions
11.2.1 Teichmfiller Extensions and Characters on Zv
11.2.2 The p-adic Hurwitz Zeta Function for x E CZp
11.2.3 The Function Sp(s, x) Around s = 1
11.2.4 The p-adic Hurwitz Zeta Function for x E Zp
11.3 p-adic L-Functions
11.3.1 Dirichlet Characters in the p-adic Context
11.3.2 Definition and Basic Properties of p-adic L-Fun
11.4.2 Bernoulli Numbers and Regular Primes
11.4.3 Strengthening of the Almkvist-Meurman Theorem
11.5 p-adic Log Gamma Functions
11.5.1 Diamond's p-adic Log Gamma Function
11.5.2 Morita's p-adic Log Gamma Function
11.5.3 Computation of some p-adic Logarithms
11.5.4 Computation of Limits of some Logarithmic Sums
11.5.5 Explicit Formulas for Cp(r/m) and Cv(x, r/m)
11.5.6 Application to the Value of Lp(x, 1)
11.6 Morita's p-adic Gamma Function
11.6.1 Introduction
11.6.2 Definitions and Basic Results
11.6.3 Main Properties of the p-adic Gamma Function
11.6.4 Mahler-Dwork Expansions Linked to Fp(x)
11.6.5 Power Series Expansions Linked to Fp(x)
11.6.6 The Jacobstahl-Kazandzidis Congruence
11.7 The Gross-Koblitz Formula and Applications
11.7.1 Statement and Proof of the Gross-Koblitz Formula
11.7.2 Application to Lp(x,O)
11.7.3 Application to the Stickelberger Congruence
11.7.4 Application to the Hasse-Davenport Product Relation
11.8 Exercises for Chapter 11
Part IV. Modern Tools
12. Applications of Linear Forms in Logarithms
12.1 Introduction
12.1.1 Lower Bounds
12.1.2 Applications to Diophantine Equations and Problems
12.1.3 A List of Applications
12.2 A Lower Bound for 12m - 3hi
12.3 Lower Bounds for the Trace of cn
12.4 Pure Powers in Binary Recurrent Sequences
12.5 Greatest Prime Factors of Terms of Some Recurrent Se quences
12.6 Greatest Prime Factors of Values of Integer Polynomials
12.7 The Diophantine Equation axn - byn = c
12.8 Simultaneous Pell Equations
12.8.1 General Strategy
12.8.2 An Example in Detail
12.8.3 A General Algorithm
12.9 Catalan's Equation
12.10 Thue Equations
12.10.1 The Main Theorem
12.10.2 Algorithmic Aspects
12.11 Other Classical Diophantine Equations
12.12 A Few Words on the Non-Archimedean Case
13. Rational Points on Higher-Genus Curves
13.1 Introduction
13.2 The Jacobian
13.2.1 Functions on Curves
13.2.2 Divisors
13.2.4 The Group Law: Cantor's Algorithm
13.2.5 The Group Law: The Geometric Point of View
13.3 Rational Points on Hyperelliptic Curves
13.3.1 The Method of Demtyanenko-Manin
13.3.2 The Method of Chabauty-Coleman
13.3.3 Explicit Chabauty According to Flynn
13.3.4 When Chabauty Fails
13.3.5 Elliptic Curve Chabauty
13.3.6 A Complete Example
14. The Super-Fermat Equation
14.1 Preliminary Reductions
14.2 The Dihedral Cases (2, 2, r)
14.2.1 The Equation x2 - y2 = zr
14.2.2 The Equation x2 + y2 = zr
14.2.3 The Equations x2 + 3y2 = z3 and X2 + 3y2 = 4Z3
14.3 The Tetrahedral Case (2, 3, 3)
14.3.1 The Equation x3 + y3 = z2
14.3.2 The Equation x3 + y3 = 2z2
14.3.3 The Equation x3 - 2y3 = z2
14.4 The Octa.hedral Case (2, 3, 4)
14.4.1 The Equation x2 - y4 = z3
14.4.2 The Equation x2 + y4 = z3
14.5 Invariants, Covariants, and Dessins d'Enfants
14.5.1 Dessins d'Enfants, Klein Forms, and Covariants
14.5.2 The Icosahedral Case (2, 3, 5)
14.6 The Parabolic and Hyperbolic Cases
14.6.1 The Parabolic Case
14.6.2 General Results in the Hyperbolic Case
14.6.3 The Equations x4 + y4 = z3
14.6.4 The Equation x4 + y4 = z5
14.6.5 The Equation x6 - y4 = z2
14.6.6 The Equation x4 - y6 = z2
14.6.7 The Equation x6 + y4 = z2
14.6.8 Further Results
14.7 Applications of Mason's Theorem
14.7.1 Mason's Theorem
14.7.2 Applications
14.8 Exercises for Chapter 14
15. The Modular Approach to Diophantine Equations
15.1 Newforms
15.1.1 Introduction and Necessary Software Tools
15.1.2 Newforms
15.1.3 Rational Newforms and Elliptic Curves
15.2 Ribet's Level-Lowering Theorem
15.2.1 Definition of "Arises From"
15.2.2 Ribet's Level-Lowering Theorem
15.2.3 Absence of Isogenies
15.2.4 How to use Ribet's Theorem
15.3 Fermat's Last Theorem and Similar Equations
15.3.2 E Arises from a Curve with Complex Multiplication
15.3.3 End of the Proof of Theorem 15.3.1
15.3.4 The Equation x2 = yP + 2rZp for p > 7 and r > 2
15.3.5 The Equation x2 = yP + zp for p > 7
15.4 An Occasional Bound for the Exponent
15.5 An Example of Serre-Mazur-Kraus
15.6 The Method of Kraus
15.7 "Predicting Exponents of Constants"
15.7.1 The Diophantine Equation x2 - 2 = yP
15.7.2 Application to the SMK Equation
15.8 Recipes for Some Ternary Diophantine Equations
15.8.1 Recipes for Signature (p, p, p)
15.8.2 Recipes for Signature (p, p, 2)
15.8.3 Recipes for Signature (p, p, 3)
16. Catalan's Equation
16.1 Mihailescu's First Two Theorems
16.1.1 The First Theorem: Double Wieferich Pairs
16.1.2 The Equation (xp - 1)/(x - 1) = pyq
16.1.3 Mihailescu's Second Theorem: p | hq and q | hp
16.2 The + and - Subspaces and the Group S
16.2.1 The + and - Subspaces
16.2.2 The Group S
16.3 Mihailescu's Third Theorem: p < 4q2 and q < 4p2
16.4 Mihailescu's Fourth Theorem: p = 1 (mod q) or q = 1 (mod p)
16.4.1 Preliminaries on Commutative Algebra
16.4.2 Preliminaries on the Plus Part
16.4.3 Cyclotomic Units and Thaine's Theorem
16.4.4 Preliminaries on Power Series
16.4.5 Proof of Mihailescu's Fourth Theorem
16.4.6 Conclusion: Proof of Catalan's Conjecture
Bibliography
Index of Notation
Index of Names
General Index
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