Paper detail

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

We show that as $T\to \infty$, for all $t\in [T,2T]$ outside of a set of measure $\mathrm{o}(T)$, $$ \int_{-(\log T)^θ}^{(\log T)^θ} |ζ(\tfrac 12 + \mathrm{i} t + \mathrm{i} h)|^β \mathrm{d} h = (\log T)^{f_θ(β) + \mathrm{o}(1)}, $$ for some explicit exponent $f_θ(β)$, where $θ> -1$ and $β> 0$. This proves an extended version of a conjecture of Fyodorov and Keating (2014). In particular, it shows that, for all $θ> -1$, the moments exhibit a phase transition at a critical exponent $β_c(θ)$, below which $f_θ(β)$ is quadratic and above which $f_θ(β)$ is linear. The form of the exponent $f_θ$ also differs between mesoscopic intervals ($-1<θ<0$) and macroscopic intervals ($θ>0$), 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 $\mathrm{o}(T)$, $$ \max_{|h| \leq (\log T)^θ} |ζ(\tfrac{1}{2} + \mathrm{i} t + \mathrm{i} h)| = (\log T)^{m(θ) + \mathrm{o}(1)}, $$ for some explicit $m(θ)$. This generalizes earlier results of Najnudel (2018) and Arguin et al. (2019) for $θ= 0$. The proofs are unconditional, except for the upper bounds when $θ> 3$, where the Riemann hypothesis is assumed.