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.
Readership
Graduate students and research mathematicians interested in
analytic number theory.