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Shifted moments over the unitary ensemble


Author: Vorrapan Chandee
Journal: Proc. Amer. Math. Soc. 145 (2017), 2391-2405
MSC (2010): Primary 11M06, 15B52
DOI: https://doi.org/10.1090/proc/13409
Published electronically: December 15, 2016
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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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Vorrapan Chandee
Affiliation: Department of Mathematics, Burapha University, 169 Long-Hard Bangsaen Road, Saensuk, Mueng, Chonburi, Thailand, 20131
Email: vorrapan@buu.ac.th

DOI: https://doi.org/10.1090/proc/13409
Keywords: Random matrice, circular unitary ensemble, shifted moments, the Riemann zeta function, Selberg's formula
Received by editor(s): June 7, 2016
Received by editor(s) in revised form: August 2, 2016
Published electronically: December 15, 2016
Communicated by: Ken Ono
Article copyright: © Copyright 2016 American Mathematical Society