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xviTable of Contents9.2 Analytic Applications of Bernoulli Polynomials . . . . . . . . . . . . . 6339.2.1 Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6349.2.2 The Euler–MacLaurin Summation Formula . . . . . . . . . . 6359.2.3 The Remainder Term and the Constant Term . . . . . . . . 6399.2.4 Euler–MacLaurin and the Laplace Transform . . . . . . . . 6429.2.5 Basic Applications of the Euler–MacLaurin Formula . . 6459.3 Applications to Numerical Integration . . . . . . . . . . . . . . . . . . . . . 6509.3.1 Standard Euler–MacLaurin Numerical Integration . . . . 6509.3.2 The Basic Tanh-Sinh Numerical Integration Method . . 6529.3.3 General Double Exponential Numerical Integration . . . 6549.4 χ-Bernoulli Numbers, Polynomials, and Functions . . . . . . . . . . 6579.4.1 χ-Bernoulli Numbers and Polynomials . . . . . . . . . . . . . . 6589.4.2 χ-Bernoulli Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.4.3 The χ-Euler–MacLaurin Summation Formula . . . . . . . . 6649.5 Arithmetic Properties of Bernoulli Numbers . . . . . . . . . . . . . . . 6679.5.1 χ-Power Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6679.5.2 The Generalized Clausen–von Staudt Congruence . . . . 6759.5.3 The Voronoi Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . 6789.5.4 The Kummer Congruences . . . . . . . . . . . . . . . . . . . . . . . . 6819.5.5 The Almkvist–Meurman Theorem . . . . . . . . . . . . . . . . . . 6839.6 The Real and Complex Gamma Function . . . . . . . . . . . . . . . . . . 6859.6.1 The Hurwitz Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . 6859.6.2 Definition of the Gamma Function. . . . . . . . . . . . . . . . . . 6919.6.3 Preliminary Results for the Study of Γ(s). . . . . . . . . . . . 6959.6.4 Properties of the Gamma Function . . . . . . . . . . . . . . . . . 6989.6.5 Specific Properties of the function ψ(s). . . . . . . . . . . . . . 7089.6.6 Fourier Expansions of ζ(s, x) and log(Γ(x)) . . . . . . . . . . 7139.7 Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7169.7.1 Generalities on Integral Transforms . . . . . . . . . . . . . . . . . 7179.7.2 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7189.7.3 The Mellin Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7209.7.4 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 7229.8 Bessel Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7239.8.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7239.8.2 Integral Representations and Applications . . . . . . . . . . . 7269.9 Exercises for Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73110. Dirichlet Series and L-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 76310.1 Arithmetic Functions and Dirichlet Series. . . . . . . . . . . . . . . . . . 76310.1.1 Operations on Arithmetic Functions . . . . . . . . . . . . . . . . 76410.1.2 Multiplicative Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 76610.1.3 Some Classical Arithmetical Functions . . . . . . . . . . . . . . 76710.1.4 Numerical Dirichlet Series . . . . . . . . . . . . . . . . . . . . . . . . . 77210.2 The Analytic Theory of L-Series. . . . . . . . . . . . . . . . . . . . . . . . . . 77410.2.1 Simple Approaches to Analytic Continuation. . . . . . . . . 775
Table of Contentsxvii10.2.2 The Use of the Hurwitz Zeta Function ζ(s, x) . . . . . . . . 77910.2.3 The Functional Equation for the Theta Function . . . . . 78110.2.4 The Functional Equation for Dirichlet L-Functions . . . 78410.2.5 Generalized Poisson Summation Formulas . . . . . . . . . . . 78910.2.6 Voronoi’s Error Term in the Circle Problem. . . . . . . . . . 79410.3 Special Values of Dirichlet L-Functions . . . . . . . . . . . . . . . . . . . . 79810.3.1 Basic Results on Special Values . . . . . . . . . . . . . . . . . . . . 79810.3.2 Special Values of L-Functions and Modular Forms . . . . 80510.3.3 The Polya–Vinogradov Inequality . . . . . . . . . . . . . . . . . . 81010.3.4 Bounds and Averages for L(χ, 1) . . . . . . . . . . . . . . . . . . . 81210.3.5 Expansions of ζ(s) Around s = k ∈ Z 1 . . . . . . . . . . . . . 81710.3.6 Numerical Computation of Euler Products and Sums . 82010.4 Epstein Zeta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82310.4.1 The Nonholomorphic Eisenstein Series G(τ, s). . . . . . . . 82310.4.2 The Kronecker Limit Formula. . . . . . . . . . . . . . . . . . . . . . 82610.5 Dirichlet Series Linked to Number Fields . . . . . . . . . . . . . . . . . . 82810.5.1 The Dedekind Zeta Function ζ K (s) . . . . . . . . . . . . . . . . . 82810.5.2 The Dedekind Zeta Function of Quadratic Fields . . . . . 83110.5.3 Applications of the Kronecker Limit Formula . . . . . . . . 83510.5.4 The Dedekind Zeta Function of Cyclotomic Fields . . . . 84310.5.5 The Nonvanishing of L(χ, 1) . . . . . . . . . . . . . . . . . . . . . . . 84810.5.6 Application to Primes in Arithmetic Progression . . . . . 85010.5.7 Conjectures on Dirichlet L-Functions . . . . . . . . . . . . . . . 85110.6 Science-Fiction on L-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 85210.6.1 Local L-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85210.6.2 Global L-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85310.7 The Prime Number Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85810.7.1 Estimates for ζ(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85810.7.2 Newman’s Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86310.7.3 Iwaniec’s Proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86710.8 Exercises for Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87111. p-adic Gamma and L-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 88711.1 Generalities on p-adic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 88711.1.1 Methods for Constructing p-adic Functions . . . . . . . . . . 88711.1.2 A Brief Study of Volkenborn Integrals. . . . . . . . . . . . . . . 88811.2 The p-adic Hurwitz Zeta functions . . . . . . . . . . . . . . . . . . . . . . . . 89211.2.1 Teichmüller Extensions and Characters on Z p . . . . . . . . 89211.2.2 The p-adic Hurwitz Zeta Function for x ∈ CZ p . . . . . . . 89311.2.3 The Function ζ p (s, x) Around s = 1. . . . . . . . . . . . . . . . . 90011.2.4 The p-adic Hurwitz Zeta Function for x ∈ Z p . . . . . . . . 90211.3 p-adic L-Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91211.3.1 Dirichlet Characters in the p-adic Context . . . . . . . . . . . 91211.3.2 Definition and Basic Properties of p-adic L-Functions . 91311.3.3 p-adic L-Functions at Positive Integers . . . . . . . . . . . . . . 917
Page 5 and 6: vKoblitz formula for Morita’s p-a
Page 8: viiiinstance) A(X) = A d X d +A d
Page 11 and 12: Table of Contentsxi3.6.1 Introducti
Page 13 and 14: Table of Contentsxiii6.3.1 The Gene
Page 15: Table of Contentsxv8.2.3 The Fundam
Page 19 and 20: Table of Contentsxix12.8.3 A Genera
Page 21: Table of Contentsxxi16.4 Mihăilesc

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