Non-vanishing of L-Functions and Applications

1. Front Matter

   Pages i-xi

   PDF

2. Introduction

   • M. Ram Murty, V. Kumar Murty

   Pages 1-3

3. The Prime Number Theorem and Generalizations

   • M. Ram Murty, V. Kumar Murty

   Pages 5-23

4. Artin L-Functions

   • M. Ram Murty, V. Kumar Murty

   Pages 25-64

5. Equidistribution and L-Functions

   • M. Ram Murty, V. Kumar Murty

   Pages 65-73

6. Modular Forms and Dirichlet Series

   • M. Ram Murty, V. Kumar Murty

   Pages 75-92

7. Dirichlet L-Functions

   • M. Ram Murty, V. Kumar Murty

   Pages 93-132

8. Non-Vanishing of Quadratic Twists of Modular L-Functions

   • M. Ram Murty, V. Kumar Murty

   Pages 133-176

9. Selberg’s Conjectures

   • M. Ram Murty, V. Kumar Murty

   Pages 177-185

10. Suggestions for Further Reading

    • M. Ram Murty, V. Kumar Murty

    Pages 187-191

11. Back Matter

    Pages 192-196

    PDF

This monograph brings together a collection of results on the non-vanishing of L­ functions. The presentation, though based largely on the original papers, is suitable for independent study. A number of exercises have also been provided to aid in this endeavour. The exercises are of varying difficulty and those which require more effort have been marked with an asterisk. The authors would like to thank the Institut d'Estudis Catalans for their encouragement of this work through the Ferran Sunyer i Balaguer Prize. We would 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 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 distri­ 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 the­ 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.

• Number theory
• alegbraic geometry
• arithmetic
• number theory
• prime number

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, Vol. 173, 2014)

• Department of Mathematics, Queen’s University, Kingston, Canada

  M. Ram Murty

• Department of Mathematics, University of Toronto, Toronto, Canada

  V. Kumar Murty

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