Analytic Theory of L-functions

American Institute Of Mathematics, Pasadena CA

Investigators

Abstract

L-functions are fundamental objects in analytic number theory which encode arithmetic information. The simplest examples are the Riemann zeta-function, Dirichlet L-functions, and the L-functions associated with modular forms. Understanding the statistical behavior of the values and zeros of these L-functions is the primary theme of Dr. Conrey.s research. He and his collaborators propose a variety of projects each involved with some aspect of L-functions. One involves a new formula with the divisor function d(n) which generalizes the classical Voronoi formula and has an application in the Nyman-Beurling approach to the Riemann Hypothesis; Conrey will develop further applications to moment formulae and to the question of understanding low degree L-functions. Conrey will use the Asymptotic Large Sieve (invented with Iwaniec and Soundararajan) to prove a conjecture about the mean square of all Dirichlet L-functions multiplied by an arbitrary Dirichlet polynomial. A third project involves Mazur.s conjecture that the symmetric power L-functions associated with an elliptic curve seldom vanish at their central point. Professor Conrey's research is in the area of number theory. Modern Number Theory has surprisingly diverse applications, from enabling secure internet transactions, to the construction of optimal networks, and even to the question of cataloguing the various types of bodies in the study of low dimensional topology. One of the most successful tools invented by number theorists is the zeta-function. Its original purpose was to help with the study of prime numbers. Now it, and its analogues, are ubiquitous in number theory. However, there are still some very basic properties of zeta-functions which we do not understand, and which if we did would lead to much progress. The main question is Why do all of the zeros of zeta-functions occur on just one line? Professor Conrey's research is centered on the study of the zeros of zeta-functions. As part of this project, Professor Conrey will also continue his work with Math Teachers' Circles, which are a collection of 39 problem solving groups all across the country that involve professional mathematicians and Middle School math teachers working together to build communities of problem solvers.