Analytic Theory of L-functions

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This research project centers on the connections between number theory and random matrix theory. In particular, the study of prime numbers and divisors is now influenced by the work of physicists in modeling high energy systems by understanding the statistics of random matrix theory. This project explores the relationship between these apparently independent areas of research. The work studies the analytic theory of L-functions, which are fundamental objects in number theory that encode arithmetic information. Examples include the Riemann zeta-function, which encodes information about prime numbers, Dirichlet L-functions, which encode information about the equidistribution of primes in arithmetic progressions, and the L-functions associated with modular forms, which encode the equidistribution of more complex sequences, including rational points on elliptic curves. The investigator aims to develop a theoretical framework to explain by number theoretic means the statistical behavior of the values and zeros of such L-functions.The investigator and collaborators plan work in a variety of projects, each involved with some aspect of L-functions. One project begins with a novel approach to understanding moments of the Riemann zeta-function through a study of convolutions of correlations of shifted divisor functions. This research incorporates a multidimensional discrete analogue of the Hardy-Littlewood Circle method. Another project is to improve bounds on the proportion of zeros of the Riemann zeta function on the critical line. A third project involves work related to an approach to the Riemann Hypothesis as a mollification problem; it is to understand an exact formula for the second moment of the zeta-function multiplied by a specific long Dirichlet polynomial. A fourth project is to prove that at least 60% of the zeros of Dirichlet L-functions are on the critical line.