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

Abstract

In this short note, we study the derivatives of all orders for the random field $$X_T(h)=\sum _{p\le T}\frac{\text{Re}(U_p{p}^{-ih})}{p^{1/2}},h\in [0,1],$$ 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.

4 pages, 0 figures