The proposer will continue his investigations in multiplicative number theory and the analytic theory of L-functions. The long term goal of this research is to shed light on the sizes and distribution of zeros of L-functions, and to explore the consequences of such results for arithmetical problems. Recently the proposer obtained a ``weak subconvexity" result for a large class of L-functions, and he will investigate generalizations and extensions of that result. Together with Conrey and Iwaniec, the proposer is developing an asymptotic large sieve which promises to give new results on zeros and moments of L-functions in certain families. He will also investigate problems relating to the equidistribution of modular forms. Jointly with Andrew Granville, the proposer has been studying mean values of multiplicative functions over the last ten years, and this activity has found a number of striking applications. The proposer will continue these investigations, and together with Granville is working on a unified treatment of many topics in analytic number theory from this viewpoint.
The proposer works broadly in the area of analytic number theory. L-functions are of central importance in this area, and encode a great deal of arithmetic information. A key example is the Riemann zeta function which contains much information about the distribution of prime numbers, and an understanding of its zeros is one of the fundamental problems in mathematics. The proposer's work is motivated by a desire to understand the behavior of such L-functions. The problems in number theory are of intrinsic interest to mathematicians, and pose formidable challenges. Moreover, progress in number theory has had applications in cryptography and theoretical computer science.
