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

## TL;DR

This paper analyzes the moments of the Riemann zeta function on short intervals along the critical line, revealing phase transitions and differences between mesoscopic and macroscopic scales, with implications for understanding zeta correlations.

## Contribution

It extends the understanding of zeta moments on short intervals, proving phase transitions and interval-dependent behaviors, and generalizes previous results with unconditional proofs.

## Key findings

- Moments exhibit phase transition at critical exponent _ heta(eta)
- Different behavior of moments between mesoscopic and macroscopic intervals
- Maximal size of zeta on short intervals quantified as (\u2212 T)^{m( heta)+o(1)}

## Abstract

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 shows 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. (2019) for $\theta = 0$. The proofs are unconditional, except for the upper bounds when $\theta > 3$, where the Riemann hypothesis is assumed.