内容

目次

Preface. 1: Euler Gamma-Function. 1.1. Definition and Analytic Continuation. 1.2. Representation by an Infinite Product. 1.3. Functional Equation. 1.4. Complementary Formula. 1.5. Asymptotic Formulas. 1.6. Hypergeometric Function. Notes. 2: Functional Equation. 2.1. Definition of the Lerch Zeta-Function. 2.2. Analytic Continuation. 2.3. Functional Equation. 2.4. Application of the Euler-Maclaurin Formula. Notes. 3: Moments. 3.1. Approximation of L(lambda, alpha, s) by a Finite Sum. 3.2. Montgomery Vaughan Theorem. 3.3. Mean Square of L(lambda, alpha, s). 3.4. Mean Square of L(lambda, alpha, s) with Respect to alpha. Notes. 4: Approximate Functional Equation. 4.1. Proof of the Approximate Functional Equation. 4.2. Application of the Approximate Functional Equation to the Mean Square of L(lambda, alpha, s). Notes. 5: Statistical Properties. 5.1. Limit Theorems on the Complex Plane. 5.2. Limit Theorems in the Space of Analytic Functions. 5.3. Joint Limit Theorems in the Space of Analytic Functions with Rational alpha. 6: Universality. 6.1. Case of Trancendental alpha. 6.2. Case of Rational alpha. 6.3. Joint Universality of Lerch Zeta-Functions. 6.4. Effectivization Problem of the Universality Theorem. Notes. 7: Functional Independence. 7.1 The One-Dimensional Case. 7.2. Joint Functional Independence. Notes. 8: Distribution of Zeros. 8.1. Zero-Free Regions on the Right. 8.2.< Zero-Free Region on the Left. 8.3. Number of Nontrivial Zeros. 8.4. Estimates of the Number of Nontrivial Zeros. 8.5. Sums over Nontrivial Zeros. Notes. References. Notation. Subject Index.