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

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 $X^{\prime }(h)$. 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.

7 pages, 0 figure

References in corpus (2)

• A note on the maximum of the Riemann zeta function, and log-correlated random variables
• Poisson-Dirichlet statistics for the extremes of a randomized Riemann zeta function

Cited by in corpus (1)

• Moments of the Riemann zeta function on short intervals of the critical line