Advanced Analytic Number Theory: L-Functions (Mathematical Surveys & Monographs) (Mathematical Surveys and Monographs)

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. This 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 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. 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. -- Book Jacket. Hecke L-functions -- Artin-hecke L-functions -- Analytic Properties Of L-functions -- The Explicit Formulas -- Bounds On Discriminants And Conductors -- Non-vanishing Theorems -- The Local Theory Of Root Numbers. Carlos Julio Moreno. Includes Bibliographical References (p. 283-290) And Index.