![](https://csdnimg.cn/release/download_crawler_static/11143277/bg14.jpg)
Table of Contents xix
9.7.4 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
9.8 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
9.8.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
9.8.2 Integral Representations and Applications . . . . . . . . . . . 113
9.9 ExercisesforChapter9 ................................. 118
10. Dirichlet Series and L-Functions .......................... 151
10.1 Arithmetic Functions and Dirichlet Series. . . . . . . . . . . . . . . . . . 151
10.1.1 Operations on Arithmetic Functions . . . . . . . . . . . . . . . . 152
10.1.2 Multiplicative Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
10.1.3 Some Classical Arithmetical Functions . . . . . . . . . . . . . . 155
10.1.4 Numerical Dirichlet Series . . . . . . . . . . . . . . . . . . . . . . . . . 160
10.2 The Analytic Theory of L-Series . . . . . . . . . . . . . . . . . . . . . . . . . . 162
10.2.1 Simple Approaches to Analytic Continuation. . . . . . . . . 163
10.2.2 The Use of the Hurwitz Zeta Function ζ(s, x) ........ 168
10.2.3 The Functional Equation for the Theta Function . . . . . 169
10.2.4 The Functional Equation for Dirichlet L-Functions . . . 172
10.2.5 Generalized Poisson Summation Formulas . . . . . . . . . . . 177
10.2.6 Voronoi’s Error Term in the Circle Problem. . . . . . . . . . 182
10.3 Special Values of Dirichlet L-Functions . . . . . . . . . . . . . . . . . . . . 186
10.3.1 Basic Results on Special Values . . . . . . . . . . . . . . . . . . . . 186
10.3.2 Special Values of L-Functions and Modular Forms . . . . 193
10.3.3 The P´olya–Vinogradov Inequality . . . . . . . . . . . . . . . . . . 198
10.3.4 Bounds and Averages for L(χ, 1) ................... 200
10.3.5 Expansions of ζ(s) Around s = k ∈ Z
1
............. 205
10.3.6 Numerical Computation of Euler Products and Sums . 208
10.4 Epstein Zeta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
10.4.1 The Nonholomorphic Eisenstein Series G(τ,s)........ 211
10.4.2 The Kronecker Limit Formula . . . . . . . . . . . . . . . . . . . . . . 213
10.5 Dirichlet Series Linked to Number Fields . . . . . . . . . . . . . . . . . . 216
10.5.1 The Dedekind Zeta Function ζ
K
(s) ................. 216
10.5.2 The Dedekind Zeta Function of Quadratic Fields . . . . . 219
10.5.3 Applications of the Kronecker Limit Formula . . . . . . . . 223
10.5.4 The Dedekind Zeta Function of Cyclotomic Fields . . . . 230
10.5.5 The Nonvanishing of L(χ, 1) ....................... 235
10.5.6 Application to Primes in Arithmetic Progression . . . . . 237
10.5.7 Conjectures on Dirichlet L-Functions . . . . . . . . . . . . . . . 238
10.6 Science Fiction on L-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
10.6.1 Local L-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
10.6.2 Global L-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
10.7 The Prime Number Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
10.7.1 Estimates for ζ(s) ................................ 246
10.7.2 Newman’s Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
10.7.3 Iwaniec’s Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
10.8 Exercises for Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258