Non-vanishing of L-Functions and Applications

This volume develops methods for proving the non-vanishing of certain L-functions at points in the critical strip. It begins at a very basic level and continues to develop, providing readers with a theoretical foundation that allows them to understand the latest discoveries in the field.

This book systematically develops some methods for proving the non-vanishing of certain L-functions at points in the critical strip. Researchers in number theory, graduate students who wish to enter into the area and non-specialists who wish to acquire an introduction to the subject will benefit by a study of this book. One of the most attractive features of the monograph is that it begins at a very basic level and quickly develops enough aspects of the theory to bring the reader to a point where the latest discoveries as are presented in the final chapters can be fully appreciated.?

Well-written monograph on a difficult but attractive subject in number theory Provides an excellent easy-to-read introduction to the modern analytic theory of L-functions Each chapter is accompanied by exercises ? Includes supplementary material: sn.pub/extras

Autorentext

M. Ram Murty is a Professor of Mathematics at the Queen's University in Kingston, ON, Canada.

V. Kumar Murty is a Professor of Mathematics at the University of Toronto.

Klappentext

This monograph brings together a collection of results on the non-vanishing of- functions.Thepresentation,thoughbasedlargelyontheoriginalpapers,issuitable forindependentstudy.Anumberofexerciseshavealsobeenprovidedtoaidinthis endeavour. The exercises are of varying di?culty and those which require more e?ort have been marked with an asterisk. The authors would like to thank the Institut d Estudis Catalans for their encouragementof thiswork throughtheFerranSunyeriBalaguerPrize.Wewould also like to thank the Institute for Advanced Study, Princeton for the excellent conditions which made this work possible, as well as NSERC, NSF and FCAR for funding. Princeton M. Ram Murty August, 1996 V. Kumar Murty xi Introduction Since the time of Dirichlet and Riemann, the analytic properties of L-functions have been used to establish theorems of a purely arithmetic nature. The dist- bution of prime numbers in arithmetic progressions is intimately connected with non-vanishing properties of various L-functions. With the subsequent advent of the Tauberian theory as developed by Wiener and Ikehara, these arithmetical t- orems have been shown to be equivalent to the non-vanishing of these L-functions on the line Re(s)=1. In the 1950 s, a new theme was introduced by Birch and Swinnerton-Dyer. Given an elliptic curve E over a number ?eld K of ?nite degree over Q,they associated an L-function to E and conjectured that this L-function extends to an entire function and has a zero at s = 1 of order equal to the Z-rank of the group of K-rational points of E. In particular, the L-function vanishes at s=1ifand only if E has in?nitely many K-rational points.

Zusammenfassung

From the book reviews: "This is the softcover reprint of a monograph that was awarded the Ferran Sunyer i Balaguer prize in 1996. It is devoted to a recurring theme in number theory, namely that the non-vanishing of L-functions implies important arithmetical results. ... Giving a well-informed overview of related results it will continue to be an important source of information for graduate students and researchers ... ." (Ch. Baxa, Monatshefte für Mathematik, 2014)

Inhalt
1 The Prime Number Theorem and Generalizations.- 2 Artin L-Functions.- 3 Equidistribution and L-Functions.- 4 Modular Forms and Dirichlet Series.- 5 Dirichlet L-Functions.- 6 Non-Vanishing of Quadratic Twists of Modular L-Functions.- 7 Selberg's Conjectures.- 8 Suggestions for Further Reading.