Analytic number theory and random matrix theory

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My research will focus on L-functions which connect with many number theoretic objects. I am interested in exploring the moments in several families of L-functions at a very detailed level. The moments are key to understanding the value distribution of L-functions, and reveal glimpses of a rich tapestry that underlies all of number theory. The interconnectedness of seemingly disparate number theoretic objects becomes apparent when looking at them from a statistical point of view. I will also study identities and algorithms for L-functions, and connections with random matrix statistics. Florea has studied moments at the critical point of L-functions over function fields associated to hyperelliptic curves. For the first moment, she found an extra term of size cube root the main term. I will investigate whether a similar term exists in the number field setting, in the family of quadratic Dirichlet L-functions. A related problem concerns the 10-th and higher moments of the L-functions associated to elliptic curves over finite fields. Here the traces of all the Hecke operators for the full modular group. Do the Hecke operators enter into the higher moments in the case of quadratic Dirichlet L-functions over number fields? The theory of multiple Dirichlet series produces the sharpest known remainder terms for the moments in cases where the method yields proofs. An interesting feature of the multiple Dirichlet series approach is that it predicts extra lower terms in the moments, starting with the cubic moment of quadratic Dirichlet L-functions, and with the second moment for quadratic twists of an elliptic curve L-function. With my grad student Kumar, we plan to examine the second moment of quadratic twists of an elliptic curve L-functions for such terms. With colleagues, I have studied the moment of the logarithmic derivative of characteristic polynomials of unitary matrices and its connection to the same statistic for the Riemann zeta function. We have uncovered remarkable identities whose combinatorics we would like to better understand. With Peter Sarnak, we have been studying the potential and the limitations in the Weil explicit formula for computing the zeros of L-functions. We model the problem in a random matrix theory setting and ask related questions. Given the first k moments of the eigenvalues of a matrix A, what can be determined about the location of the eigenvalues of the matrix? We believe, by exploiting the highly non-convex geometry inherent in our problem, that we can improve the complexity of certain algorithms in number theory and we plan to investigate this both in the random matrix setting and in the L-function setting. My research will contribute to the field of number theory, providing fundamental and important advances in knowledge, and also resulting in training of highly qualified personnel.