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

Louis-Pierre Arguin, Frédéric Ouimet, Maksym Radziwiłł

Published 2019-01-13Version 1

We show that as $T\to \infty$, for all $t\in [T,2T]$ outside of a set of measure $\mathrm{o}(T)$, $$ \int_{-(\log T)^{\theta}}^{(\log T)^{\theta}} |\zeta(\tfrac 12 + \mathrm{i} t + \mathrm{i} h)|^{\beta} \mathrm{d} h = (\log T)^{f_{\theta}(\beta) + \mathrm{o}(1)}, $$ for some explicit exponent $f_{\theta}(\beta)$, where $\theta > -1$ and $\beta > 0$. This proves an extended version of a conjecture of Fyodorov and Keating (2014). In particular, it show that, for all $\theta > -1$, the moments exhibit a phase transition at a critical exponent $\beta_c(\theta)$, below which $f_\theta(\beta)$ is quadratic and above which $f_\theta(\beta)$ is linear. The form of the exponent $f_\theta$ also differs between mesoscopic intervals ($-1<\theta<0$) and macroscopic intervals ($\theta>0$), a phenomenon that stems from an approximate tree structure for the correlations of zeta. We also prove that, for all $t\in [T,2T]$ outside a set of measure $\mathrm{o}(T)$, $$ \max_{|h| \leq (\log T)^{\theta}} |\zeta(\tfrac{1}{2} + \mathrm{i} t + \mathrm{i} h)| = (\log T)^{m(\theta) + \mathrm{o}(1)}, $$ for some explicit $m(\theta)$. This generalizes earlier results of Najnudel (2018) and Arguin et al. (2018) for $\theta = 0$. The proofs are unconditional, except for the upper bounds when $\theta > 3$, where the Riemann hypothesis is assumed.