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A hybrid Euler–Hadamard product and moments of $\zeta (\rho)$

Bui, H M; Gonek, S M; Milinovich, M B (2015). A hybrid Euler–Hadamard product and moments of $\zeta (\rho)$. Forum Mathematicum, 27(3):1799-1828.

Abstract

Keating and Snaith modeled the Riemann zeta-function $\zeta (s)$ by characteristic polynomials of random N x N unitary matrices, and used this to conjecture the asymptotic main term for the 2k-th moment of $\zeta (\rho)$.1=2 C i t / when k > -1=2. However, an arithmetical factor, widely believed to be part of the leading term coefficient, had to be inserted in an ad hoc manner. Gonek, Hughes and Keating later developed a hybrid formula for $\zeta (s)$ that combines a truncation of its Euler product with a product over its zeros. Using it, they recovered the moment conjecture of Keating and Snaith in a way that naturally includes the arithmetical factor. Here we use the hybrid formula to recover a conjecture of Hughes, Keating and O’Connell concerning the discrete moments of the derivative of the Riemann zeta-function averaged over the zeros of $\zeta (s)$, incorporating the arithmetical factor in a natural way.

Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:National licences > 142-005
Dewey Decimal Classification:Unspecified
Scopus Subject Areas:Physical Sciences > General Mathematics
Physical Sciences > Applied Mathematics
Language:English
Date:1 January 2015
Deposited On:06 Aug 2019 14:45
Last Modified:21 Oct 2024 01:40
Publisher:De Gruyter
ISSN:0933-7741
OA Status:Green
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1515/forum-2013-6011
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