Mean values f L-functions

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

-functions play a unifying role in number theory, as they connect many otherwise different problems. Recently it has become increasingly apparent that L-functions most naturally fit into families, and that many properties of an individual L-function can be studied through the family. In particular, the study of mean values of L-functions can be used to understand the sizes (such as nonvanishing and subconvexity) of L-functions at special values, which have great arithmetical interest. Furthermore, there is evidence that some families are connected to other families, a property that has been detected through careful evaluation of certain mean values; for example, Motohashi's celebrated exact formula connecting the fourth moment of the Riemann zeta function to the Hecke-Maass L-functions associated to the full modular group. The PI intends to investigate certain higher moment problems using techniques from analytic number theory to gain insight into higher degree L-functions. Furthermore, the PI seeks to develop new formulas connecting families of L-functions. The Riemann zeta function is the basic example of an L-function. The distribution of the prime numbers, the fundamental particles of the integers, are intertwined with properties of the Riemann zeta function, as predicted by the Riemann Hypothesis. The Riemann zeta function itself is just one part of a complex web of L-functions that themselves interact in mysterious ways. This project aims to further the understanding of the Riemann zeta function and its generalizations, especially by finding new links between different kinds of L-functions.