Clay Mathematics Institute

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The circular unitary ensemble: microscopic scale, ratios and the Riemann zeta function

Ashkan Nikeghbali (Universität Zürich)

Abstract:  We provide some convergence results for the characteristic polynomial of random unitary matrices on the microscopic scale and as a consequence establish several facts about ratios of characteristic polynomials and correlations for the log of the characteristic polynomial, thus answering some open problems on that scale. These answers exhibit a natural random analytic function. We shall also give evidence about the connection between this object and the Riemann zeta function and its logarithmic derivative.