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

TL;DR

This paper verifies the first two terms of the conjectured maximum of a model of the Riemann zeta function on the critical line by relating it to a branching random walk through an approximate tree structure.

Contribution

It introduces a randomized Euler product model of the zeta function and establishes a connection to branching random walks, verifying key conjectured asymptotics.

Findings

Confirmed the first two terms of the maximum's asymptotic expansion.

Identified an approximate tree structure in the model similar to the zeta function.

Linked the maximum of the model to the maximum of a branching random walk.

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.