Number Theory

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