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

Abstract

We show that as $T\to \infty$, for all $t\in [T,2T]$ outside of a set of measure $\mathrm{o}(T)$, $${\int }_{-(\log T{)}^{\theta }}^{(\log T{)}^{\theta }}|\zeta (\frac{1}{2}+\mathrm{i}t+\mathrm{i}h){|}^{\beta }\mathrm{d}h=(\log T{)}^{f_{\theta }(\beta )+\mathrm{o}(1)},$$ 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 $\mathrm{o}(T)$, $$\underset{|h|\le (\log T{)}^{\theta }}{\max }|\zeta (\frac{1}{2}+\mathrm{i}t+\mathrm{i}h)|=(\log T{)}^{m(\theta )+\mathrm{o}(1)},$$ 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.

35 pages, 2 figures; Final version accepted for publication in Annals of Probability

References in corpus (9)

• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• The Fyodorov-Bouchaud formula and Liouville conformal field theory
• On the partition function of the Riemann zeta function, and the Fyodorov--Hiary--Keating conjecture
• The Fyodorov-Hiary-Keating Conjecture. I
• A note on the maximum of the Riemann zeta function, and log-correlated random variables
• On the moments of the moments of the characteristic polynomials of random unitary matrices
• Extreme values of CUE characteristic polynomials: a numerical study
• Poisson-Dirichlet statistics for the extremes of a randomized Riemann zeta function
• Large deviations and continuity estimates for the derivative of a random model of $\log |\zeta |$ on the critical line

Cited by in corpus (11)

• On the partition function of the Riemann zeta function, and the Fyodorov--Hiary--Keating conjecture
• The Fyodorov-Hiary-Keating Conjecture. I
• Critical Gaussian multiplicative chaos: a review
• Statistics of extremes in eigenvalue-counting staircases
• On the moments of the moments of $\zeta (1/2+it)$
• The Riemann zeta function in short intervals [after Najnudel, and Arguin, Belius, Bourgade, Radziwiłł, and Soundararajan]
• A model problem for multiplicative chaos in number theory
• Maxima of log-correlated fields: some recent developments
• Constants of de Bruijn-Newman type in analytic number theory and statistical physics
• Evidence of Random Matrix Corrections for the Large Deviations of Selberg's Central Limit Theorem
• Large values of $L$-functions on $1$-line