The Lerch Zeta-Function

ISBN-13: 9789048161683 / Angielski / Miękka / 2010 / 189 str.

Antanas Laurincikas; Ramunas Garunkstis

Opis

The Lerch zeta-function is the first monograph on this topic, which is a generalization of the classic Riemann, and Hurwitz zeta-functions. Although analytic results have been presented previously in various monographs on zeta-functions, this is the first book containing both analytic and probability theory of Lerch zeta-functions. The book starts with classical analytical theory (Euler gamma-functions, functional equation, mean square). The majority of the presented results are new: on approximate functional equations and its applications and on zero distribution (zero-free regions, number of nontrivial zeros etc). Special attention is given to limit theorems in the sense of the weak convergence of probability measures for the Lerch zeta-function. From limit theorems in the space of analytic functions the universitality and functional independence is derived. In this respect the book continues the research of the first author presented in the monograph Limit Theorems for the Riemann zeta-function. This book will be useful to researchers and graduate students working in analytic and probabilistic number theory, and can also be used as a textbook for postgraduate students.

Spis treści

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.