Topics in the Analytic Theory of $L$-functions

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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. Incollaboration with Zeev Rudnick, the investigator is developing methodsto obtain lower bounds in many families of L-functions. These lower bounds are of the same order of magnitude as the conjecturedasymptotic formulae of Keating and Snaith. In related work, theinvestigator is developing a new method for obtaining extreme values ofL-functions, and the emerging ideas seem to be of wide applicability.The investigator will also work on finding new asymptotic formulaefor certain moments of quadratic twists of L-functions.Lastly, with Andrew Granville, the investigator will work onquestions from multilplicative number theory, especiallyrelated to character sums. The investigator's area of research is analytic number theory. Thecentral objects of study in number theory are L-functions. Theseare 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.