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15 February 2007 A hybrid Euler-Hadamard product for the Riemann zeta function
S. M. Gonek, C. P. Hughes, J. P. Keating
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Duke Math. J. 136(3): 507-549 (15 February 2007). DOI: 10.1215/S0012-7094-07-13634-2

Abstract

We use a smoothed version of the explicit formula to find an accurate pointwise approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of the zeta function which involves the primes in a natural way. We then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line. For the second and fourth moments, we establish all of the steps in our approach rigorously. This calculation illuminates recent conjectures for these moments based on connections with random matrix theory

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S. M. Gonek. C. P. Hughes. J. P. Keating. "A hybrid Euler-Hadamard product for the Riemann zeta function." Duke Math. J. 136 (3) 507 - 549, 15 February 2007. https://doi.org/10.1215/S0012-7094-07-13634-2

Information

Published: 15 February 2007
First available in Project Euclid: 29 January 2007

zbMATH: 1171.11049
MathSciNet: MR2309173
Digital Object Identifier: 10.1215/S0012-7094-07-13634-2

Subjects:
Primary: 11M26
Secondary: 11M06 , 15A52

Rights: Copyright © 2007 Duke University Press

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Vol.136 • No. 3 • 15 February 2007
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