Shifted moments over the unitary ensemble
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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.References
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Additional Information
- Vorrapan Chandee
- Affiliation: Department of Mathematics, Burapha University, 169 Long-Hard Bangsaen Road, Saensuk, Mueng, Chonburi, Thailand, 20131
- Email: vorrapan@buu.ac.th
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 2391-2405
- MSC (2010): Primary 11M06, 15B52
- DOI: https://doi.org/10.1090/proc/13409
- MathSciNet review: 3626498