## 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 $\mathrm{o}(\mathit{T})$,

$${\int}_{-{log}^{\mathit{\theta}}\mathit{T}}^{{log}^{\mathit{\theta}}\mathit{T}}{\left|\mathit{\zeta}\right(\frac{1}{2}+\mathrm{i}\mathit{t}+\mathrm{i}\mathit{h}\left)\right|}^{\mathit{\beta}}\phantom{\rule{0.1667em}{0ex}}\mathrm{d}\mathit{h}={(log\mathit{T})}^{{\mathit{f}}_{\mathit{\theta}}(\mathit{\beta})+\mathrm{o}(1)},$$

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.

## Funding Statement

L.-P. A. is supported in part by NSF Grant DMS-1513441 and by NSF CAREER DMS-1653602. F. O. is supported by postdoctoral fellowships from the NSERC (PDF) and the FRQNT (B3X). M. R. acknowledges support of a Sloan fellowship and NSF Grant DMS-1902063.

## Citation

Louis-Pierre Arguin. Frédéric Ouimet. Maksym Radziwiłł. "Moments of the Riemann zeta function on short intervals of the critical line." Ann. Probab. 49 (6) 3106 - 3141, November 2021. https://doi.org/10.1214/21-AOP1524

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