A hybrid Euler-Hadamrd product for the Riemann zeta-function

SM Gonek, CP Hughes, JP Keating

Research output: Contribution to journalArticle (Academic Journal)peer-review

45 Citations (Scopus)


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
Translated title of the contributionA hybrid Euler-Hadamrd product for the Riemann zeta-function
Original languageEnglish
Pages (from-to)507 - 549
Number of pages43
JournalDuke Mathematical Journal
Volume136 (3)
Publication statusPublished - Feb 2007

Bibliographical note

Publisher: Duke University Press

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