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

Abstract

We show that as $\mathit{T}\to \infty$, for all $\mathit{t}\in [\mathit{T},2\mathit{T}]$ outside of a set of measure $\mathrm{o}(\mathit{T})$,

$${\int }_{-{log}^{\mathit{\theta }}\mathit{T}}^{{log}^{\mathit{\theta }}\mathit{T}}{\left|\mathit{\zeta }\right(\frac{1}{2}+\mathrm{i}\mathit{t}+\mathrm{i}\mathit{h}\left)\right|}^{\mathit{\beta }}\mathrm{d}\mathit{h}={(log\mathit{T})}^{{\mathit{f}}_{\mathit{\theta }}(\mathit{\beta })+\mathrm{o}(1)},$$

for some explicit exponent ${\mathit{f}}_{\mathit{\theta }}(\mathit{\beta })$, where $\mathit{\theta }>-1$ and $\mathit{\beta }>0$. This proves an extended version of a conjecture of Fyodorov and Keating (Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014) 20120503, 32). In particular, it shows that, for all $\mathit{\theta }>-1$, the moments exhibit a phase transition at a critical exponent ${\mathit{\beta }}_{\mathit{c}}(\mathit{\theta })$, below which ${\mathit{f}}_{\mathit{\theta }}(\mathit{\beta })$ is quadratic and above which ${\mathit{f}}_{\mathit{\theta }}(\mathit{\beta })$ is linear. The form of the exponent ${\mathit{f}}_{\mathit{\theta }}$ also differs between mesoscopic intervals ($-1<\mathit{\theta }<0$) and macroscopic intervals ($\mathit{\theta }>0$), a phenomenon that stems from an approximate tree structure for the correlations of zeta. We also prove that, for all $\mathit{t}\in [\mathit{T},2\mathit{T}]$ outside a set of measure $\mathrm{o}(\mathit{T})$,

$$\underset{|\mathit{h}|\le {log}^{\mathit{\theta }}\mathit{T}}{max}\left|\mathit{\zeta }\right(\frac{1}{2}+\mathrm{i}\mathit{t}+\mathrm{i}\mathit{h}\left)\right|={(log\mathit{T})}^{\mathit{m}(\mathit{\theta })+\mathrm{o}(1)},$$

for some explicit $\mathit{m}(\mathit{\theta })$. This generalizes earlier results of Najnudel (Probab. Theory Related Fields 172 (2018) 387–452) and Arguin et al. (Comm. Pure Appl. Math. 72 (2019) 500–535) for $\mathit{\theta }=0$. The proofs are unconditional, except for the upper bounds when $\mathit{\theta }>3$, where the Riemann hypothesis is assumed.