Integrals of Characteristic Polynomials of Unitary Matrices, and Applications to the Riemann Zeta Function

Stochastics Seminar
Thursday, October 20, 2011 - 15:05
1 hour (actually 50 minutes)
Skiles 006
Penn State, Department of Statistics
In work on the Riemann zeta function, it is of interest to evaluate certain integrals involving the characteristic polynomials of N x N unitary matrices and to derive asymptotic expansions of these integrals as N -> \infty. In this talk, I will obtain exact formulas for several of these integrals, and relate these results to conjectures about the distribution of the zeros of the Riemann zeta function on the critical line. I will also explain how these results are related to multivariate statistical analysis and to the hypergeometric functions of Hermitian matrix argument.