Electronic Communications in Probability

Poisson-Dirichlet statistics for the extremes of a randomized Riemann zeta function

Frédéric Ouimet

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In [4], the authors prove the convergence of the two-overlap distribution at low temperature for a randomized Riemann zeta function on the critical line. We extend their results to prove the Ghirlanda-Guerra identities. As a consequence, we find the joint law of the overlaps under the limiting mean Gibbs measure in terms of Poisson-Dirichlet variables. It is expected that we can adapt the approach to prove the same result for the Riemann zeta function itself.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 46, 15 pp.

Received: 7 February 2018
Accepted: 18 July 2018
First available in Project Euclid: 27 July 2018

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Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$ 60F05: Central limit and other weak theorems 60G60: Random fields 60G70: Extreme value theory; extremal processes

extreme value theory Riemann zeta function Ghirlanda-Guerra identities Gibbs measure Poisson-Dirichlet variable ultrametricity spin glasses

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Ouimet, Frédéric. Poisson-Dirichlet statistics for the extremes of a randomized Riemann zeta function. Electron. Commun. Probab. 23 (2018), paper no. 46, 15 pp. doi:10.1214/18-ECP154. https://projecteuclid.org/euclid.ecp/1532657018

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