The Annals of Applied Probability

Maxima of a randomized Riemann zeta function, and branching random walks

Louis-Pierre Arguin, David Belius, and Adam J. Harper

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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.

Article information

Ann. Appl. Probab., Volume 27, Number 1 (2017), 178-215.

Received: June 2015
Revised: April 2016
First available in Project Euclid: 6 March 2017

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Zentralblatt MATH identifier

Primary: 60G70: Extreme value theory; extremal processes 11M06: $\zeta (s)$ and $L(s, \chi)$

Extreme value theory Riemann zeta function branching random walk


Arguin, Louis-Pierre; Belius, David; Harper, Adam J. Maxima of a randomized Riemann zeta function, and branching random walks. Ann. Appl. Probab. 27 (2017), no. 1, 178--215. doi:10.1214/16-AAP1201.

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