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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

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Volume 27, Issue 3


A hybrid Euler–Hadamard product and moments of ζ'(ρ)

H. M. Bui / S. M. Gonek / M. B. Milinovich
Published Online: 2013-07-05 | DOI: https://doi.org/10.1515/forum-2013-6011


Keating and Snaith modeled the Riemann zeta-function ζ(s) by characteristic polynomials of random N×N unitary matrices, and used this to conjecture the asymptotic main term for the 2k-th moment of ζ(1/2+it) 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 ζ(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 ζ(s), incorporating the arithmetical factor in a natural way.

Keywords: Riemann zeta-function; moments; hybrid Euler–Hadamard product

MSC: 11M26; 11M06; 11M50; 15B52

About the article

Received: 2013-01-15

Published Online: 2013-07-05

Published in Print: 2015-05-01

Funding Source: University of Zurich

Award identifier / Grant number: Forschungskredit

Funding Source: National Science Foundation

Award identifier / Grant number: DMS-0653809

Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1200582

Funding Source: NSA

Award identifier / Grant number: Young Investigator Grant H98230-13-1-0217

Funding Source: AMS-Simons

Award identifier / Grant number: Travel Grant

Citation Information: Forum Mathematicum, Volume 27, Issue 3, Pages 1799–1828, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2013-6011.

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