Shifted moments over the unitary ensemble

Chandee, Vorrapan

doi:10.1090/proc/13409

Shifted moments over the unitary ensemble
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Proc. Amer. Math. Soc. 145 (2017), 2391-2405 Request permission

Abstract:

In 2000, Keating and Snaith suggested that the value distribution of the Riemann zeta function $\zeta (1/2 + it)$ is related to that of the characteristic polynomials of random unitary matrices, $\Lambda _U(\theta ) = \prod _{n=1}^N (1 - e^{i(\theta _n + \theta )}) ,$ with respect to the circular unitary ensemble. They derived the conjecture for the moment of the Riemann zeta function through computing the exact formula for the moments of the characteristic polynomials. In this paper, we compute the shifted moments of the characteristic polynomials of random unitary matrices and express them in a determinant form. When shifts are the roots of unity, we can obtain a precise formula, and this also leads to a new formula analogous to the Selberg’s identity applied in Keating and Snaith’s computation.

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