Open Access
February 2017 Maxima of a randomized Riemann zeta function, and branching random walks
Louis-Pierre Arguin, David Belius, Adam J. Harper
Ann. Appl. Probab. 27(1): 178-215 (February 2017). DOI: 10.1214/16-AAP1201

Abstract

A recent conjecture of Fyodorov–Hiary–Keating states that the maximum of the absolute value of the Riemann zeta function on a typical bounded interval of the critical line is $\exp \{\log \log T-\frac{3}{4}\log \log \log T+O(1)\}$, for an interval at (large) height $T$. In this paper, we verify the first two terms in the exponential for a model of the zeta function, which is essentially a randomized Euler product. The critical element of the proof is the identification of an approximate tree structure, present also in the actual zeta function, which allows us to relate the maximum to that of a branching random walk.

Citation

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Louis-Pierre Arguin. David Belius. Adam J. Harper. "Maxima of a randomized Riemann zeta function, and branching random walks." Ann. Appl. Probab. 27 (1) 178 - 215, February 2017. https://doi.org/10.1214/16-AAP1201

Information

Received: 1 June 2015; Revised: 1 April 2016; Published: February 2017
First available in Project Euclid: 6 March 2017

zbMATH: 1362.60050
MathSciNet: MR3619786
Digital Object Identifier: 10.1214/16-AAP1201

Subjects:
Primary: 11M06 , 60G70

Keywords: Branching random walk , Extreme value theory , Riemann zeta function

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.27 • No. 1 • February 2017
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