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A hybrid Euler-Hadamard product for the Riemann zeta function

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A hybrid Euler-Hadamard product for the Riemann zeta function. / Hughes, C.P.; Keating, J.P.; Gonek, S.M.

In: Duke Mathematical Journal, Vol. 136, No. 3, 2007, p. 507-549.

Research output: Contribution to journalArticlepeer-review

Harvard

Hughes, CP, Keating, JP & Gonek, SM 2007, 'A hybrid Euler-Hadamard product for the Riemann zeta function', Duke Mathematical Journal, vol. 136, no. 3, pp. 507-549. https://doi.org/10.1215/S0012-7094-07-13634-2

APA

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

Vancouver

Hughes CP, Keating JP, Gonek SM. A hybrid Euler-Hadamard product for the Riemann zeta function. Duke Mathematical Journal. 2007;136(3):507-549. https://doi.org/10.1215/S0012-7094-07-13634-2

Author

Hughes, C.P. ; Keating, J.P. ; Gonek, S.M. / A hybrid Euler-Hadamard product for the Riemann zeta function. In: Duke Mathematical Journal. 2007 ; Vol. 136, No. 3. pp. 507-549.

Bibtex - Download

@article{8e81248470a84921a6d20d47b18841c4,
title = "A hybrid Euler-Hadamard product for the Riemann zeta function",
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",
author = "C.P. Hughes and J.P. Keating and S.M. Gonek",
year = "2007",
doi = "10.1215/S0012-7094-07-13634-2",
language = "English",
volume = "136",
pages = "507--549",
journal = "Duke Mathematical Journal",
issn = "0012-7094",
publisher = "Duke University Press",
number = "3",

}

RIS (suitable for import to EndNote) - Download

TY - JOUR

T1 - A hybrid Euler-Hadamard product for the Riemann zeta function

AU - Hughes, C.P.

AU - Keating, J.P.

AU - Gonek, S.M.

PY - 2007

Y1 - 2007

N2 - 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

AB - 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

U2 - 10.1215/S0012-7094-07-13634-2

DO - 10.1215/S0012-7094-07-13634-2

M3 - Article

VL - 136

SP - 507

EP - 549

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 3

ER -