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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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

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

First available in Project Euclid: 29 January 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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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