Large deviations and continuity estimates for the derivative of a random model of $\log |\zeta|$ on the critical line

TL;DR

This paper investigates the derivative of a random model for the logarithm of the Riemann zeta function on the critical line, providing large deviation and continuity estimates that advance understanding of its maximum behavior.

Contribution

It introduces large deviation and continuity estimates for the derivative of a random model of |, aiding in the analysis of its maximum and tightness properties.

Findings

Maxima of the field are close to maxima over a discrete set with high probability

Provides bounds on the difference between the maximum over the entire interval and a discrete subset

Establishes probabilistic estimates for the derivative of the random field

Abstract

In this paper, we study the random field \begin{equation*} X(h) \circeq \sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}}, \quad h\in [0,1], \end{equation*} where $(U_p, \, p ~\text{primes})$ is an i.i.d. sequence of uniform random variables on the unit circle in $\mathbb{C}$. Harper (2013) showed that $(X(h), \, h\in (0,1))$ is a good model for the large values of $(\log |\zeta(\frac{1}{2} + i (T + h))|, \, h\in [0,1])$ when $T$ is large, if we assume the Riemann hypothesis. The asymptotics of the maximum were found in Arguin, Belius & Harper (2017) up to the second order, but the tightness of the recentered maximum is still an open problem. As a first step, we provide large deviation estimates and continuity estimates for the field's derivative $X'(h)$. The main result shows that, with probability arbitrarily close to $1$, \begin{equation*} \max_{h\in [0,1]} X(h) - \max_{h\in \mathcal{S}} X(h) = O(1), \end{equation*} where $\mathcal{S}$ a discrete set containing $O(\log T \sqrt{\log \log T})$ points.