Duke Mathematical Journal

A hybrid Euler-Hadamard product for the Riemann zeta function

S. M. Gonek, C. P. Hughes, and J. P. Keating

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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

Article information

Source
Duke Math. J., Volume 136, Number 3 (2007), 507-549.

Dates
First available in Project Euclid: 29 January 2007

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1170084897

Digital Object Identifier
doi:10.1215/S0012-7094-07-13634-2

Mathematical Reviews number (MathSciNet)
MR2309173

Zentralblatt MATH identifier
1171.11049

Subjects
Primary: 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
Secondary: 11M06: $\zeta (s)$ and $L(s, \chi)$ 15A52

Citation

Gonek, S. M.; Hughes, C. P.; Keating, J. P. A hybrid Euler-Hadamard product for the Riemann zeta function. Duke Math. J. 136 (2007), no. 3, 507--549. doi:10.1215/S0012-7094-07-13634-2. https://projecteuclid.org/euclid.dmj/1170084897


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