Advanced Analytic Number Theory: L-Functions is a broad introduction and survey of the theory of 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 GLn . The approach is a generalization of the ideas of J. Tate (Tate's thesis) and A. Weil, who used abstract harmonic analysis to understand the analytic properties and functional equations of L-functions arising in number theory and algebraic geometry. Moreover, the book also emphasizes the use of a more modern technique: the local Langlands' correspondence for GLn.
The book is aimed at mathematicians and advanced graduate students with some background in a variety of topics: abstract harmonic analysis, topological groups, theory of distributions, representations of finite groups, algebraic number theory, complex analysis, etc. The author does a good job keeping the book as self-contained as possible by including short introductions to topics needed in the development of the text. For example, the book opens with an introduction to abstract harmonic analysis and locally compact topological groups in order to refresh the memory of the reader and set-up some notation used throughout the volume. However, the word advanced in the title of the book should not be underestimated. The topic at hand and the approach used is advanced indeed and requires much from the reader.
The following topics are covered in the book: Hecke L-functions, Zeta distributions, Artin-Hecke L-functions; relative local/global Weil groups (and representations of such); Archimedean and non-Archimedean local L-factors, functional equations, root numbers (the appendix is dedicated to the local theory of root numbers); analytic properties: Jensen's formula, classical convexity estimate, Riemann's product formula; Riemann's, Von Mangoldt's, Delsarte's and Weil's explicit formulas. The last two chapters are devoted to two interesting applications of the theory: bounds on discriminants/conductors (Stark's and Odlyzko's methods) and non-vanishing theorems (method of Hadamard and de La Vallée Poussin, the method of Eisenstein series, etc.).
It is the opinion of the reviewer that this is a very recommendable book to all those who intend to learn the modern techniques used in the theory of automorphic L-functions. Throughout the volume, the author writes many remarks about the history of the field and the development of the concepts, which give a nice perspective to the reader. The exposition is clear and appealing, and the choice of topics is quite interesting, making this book a very powerful resource.
Álvaro Lozano-Robledo is H. C. Wang Assistant Professor at Cornell University.