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
10.1.3 Some Classical Arithmetical Functions
10.1.4 Numerical Dirichlet Series
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 G

《数论》分为2卷，是Springer“数学研究生教材”丛书之239和240卷，是一套面向研究生的数论教程，主旨是全面介绍丢番图方程的解，包括多项式方程、有理数和代数数论等，其中特别强调了算术代数几何的现代理论。全书各章共有530例习题，部分习题有提示。
本书是其中的第2卷，由H.科恩著。共分2部分8章，内容包括伯努利多项式与伽玛函数、Dirichlet级数和L-函数、p-adicγ和l-函数、线性形式在对数中的应用、高亏格曲线上的有理点等。