# Discretization of the maximum for the derivatives of a random model of $\log|\zeta|$ on the critical line

## TL;DR

This paper investigates the maximum values of derivatives of a random model related to the logarithm of the Riemann zeta function on the critical line, revealing precise scaling behaviors and simplifying previous proofs.

## Contribution

It extends previous results by analyzing the derivatives of the model, establishing their maximum's scale, and providing a simpler proof approach.

## Key findings

- Maximum of derivatives varies on a specific logarithmic scale
- Results improve and extend previous bounds from 2019
- Progress towards the open problem of maximum tightness

## Abstract

In this short note, we study the derivatives of all orders for the random field $$ X_T(h) = \sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}}, \quad h\in [0,1], $$ where $(U_p, \, p ~\text{primes})$ is an i.i.d. sequence of uniform random variables on the unit circle in $\mathbb{C}$. We show that the maximum of $X_{T}$, and more generally the maximum of its $j$-th derivative, varies on a $(\log T)^{-\frac{1}{2}(j+2)}$ scale, which improves and extends the main result in Arguin & Ouimet (2019) and makes further progress towards the open problem of the tightness of the recentered maximum of $X_{T}$. Our proof is also much simpler and shorter.