Integrals of characteristic polynomials of unitary matrices, and applications to the Riemann zeta function

Friday, March 26, 2010 - 10:00am - 10:35am
EE/CS 3-180
Donald Richards (The Pennsylvania State University)
In recent research on the Riemann zeta function and the Riemann Hypothesis, it is important to calculate certain integrals involving the characteristic functions of N × N unitary matrices and to develop asymptotic expansions of these integrals as N → ∞. In this talk, I will derive exact formulas for several of these integrals, verify that the leading coefficients in their asymptotic expansions are non-zero, 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 calculations are related to mathematical statistics and to the hypergeometric functions of Hermitian matrix argument.