Advanced Analytic Number Theory: L-Functions
About this Title
Carlos Julio Moreno, The City University of New York (CUNY), New York, NY
Publication: Mathematical Surveys and Monographs
Publication Year 2005: Volume 115
ISBNs: 978-0-8218-4266-9 (print); 978-1-4704-1342-2 (online)
MathSciNet review: MR2135107
MSC: Primary 11Mxx; Secondary 11F66, 11M36, 11R42
Since the pioneering work of Euler, Dirichlet, and Riemann, the analytic properties of L-functions have been used to study the distribution of prime numbers. With the advent of the Langlands Program, L-functions have assumed a greater role in the study of the interplay between Diophantine questions about primes and representation theoretic properties of Galois representations. The present book provides a complete introduction to the most significant class of L-functions: the Artin-Hecke L-functions associated to finite-dimensional representations of Weil groups and to automorphic L-functions of principal type on the general linear group. In addition to establishing functional equations, growth estimates, and non-vanishing theorems, a thorough presentation of the explicit formulas of Riemann type in the context of Artin-Hecke and automorphic L-functions is also given.
The survey is aimed at mathematicians and graduate students who want to learn about the modern analytic theory of L-functions and their applications in number theory and in the theory of automorphic representations. The requirements for a profitable study of this monograph are a knowledge of basic number theory and the rudiments of abstract harmonic analysis on locally compact abelian groups.
Graduate students and research mathematicians interested in analytic number theory.
Table of Contents
- I. Hecke L-functions
- II. Artin-Hecke L-functions
- III. Analytic properties of L-functions
- IV. The explicit formulas
- V. Bounds on discriminants and conductors
- VI. Non-vanishing theorems