Topics in the Analytic Theory of $L$-functions

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

The investigator and his colleagues study several problems in the analytic theory of L-functions. The long term goal of these projects is to gain an improved understanding on zeros of L-functions, and on the distribution of values of L-functions and related number theoretic objects. In collaboration with Zeev Rudnick, the investigator is developing methods to obtain lower bounds in many families of L-functions. These lower bounds are of the same order of magnitude as the conjectured asymptotic formulae of Keating and Snaith. In related work, the investigator is developing a new method for obtaining extreme values of L-functions, and the emerging ideas seem to be of wide applicability. The investigator will also work on finding new asymptotic formulae for certain moments of quadratic twists of L-functions. Lastly, with Andrew Granville, the investigator will work on questions from multilplicative number theory, especially related to character sums. The investigator's area of research is analytic number theory. The central objects of study in number theory are L-functions. These are certain analytic functions built out of arithmetically interesting objects such as prime numbers. Understanding the zeros and values of L-functions is one of the major themes of modern mathematics. Apart from the intrinsic interest in these questions, progress in this area has practical implications for cryptography and coding theory.