L-functions: Zeros and Values

American Institute Of Mathematics, Pasadena CA

Investigators

Abstract

The investigator is studying the statistical properties of zeros and values of L-functions in the context of random matrix theory. Random matrix models are being used to conjecture the full asymptotic expansion and analytic continuation for the moments of L-functions, and to make predictions for the statistical behavior of ranks of elliptic curves. A C++ L-function class library along with a front end application is also being developed for computing zeros and values of L-functions. This software is being used by the investigator to numerically confirm the predictions being made, and will be released freely to the public as a much needed tool for studying L-functions. Many problems in number theory can be described in terms of the properties of so-called L-functions. These functions, which encode profound information about various number theoretic problems, have remained largely unyielding to mathematical analysis. Many deep problems in number theory would be solved if one could understand these functions in detail. Surprisingly, a seemingly unrelated field known as random matrix theory, a subject that originally arose in connection to experimental physics, has recently been found by number theorists and physicists alike to provide a framework in which to model the behavior of L-functions. This mysterious connection has been used successfully to make hitherto unimaginable predictions for the behavior of L-functions. The work in this proposal is concerned with exploring the connections between these two fields, number theory and random matrix theory. To assist in this project, the investigator is also preparing a software package, to be made freely available to the public, for numerically studying L-functions. This award is being cofunded by the Algebra, Number Theory, and Combinatorics Program, the Numeric, Symbolic, and Geometric Computation Program, and the Computational Mathematics Program.