On the moments of the Riemann zeta-function in short intervals

TL;DR

Under the Riemann Hypothesis, the paper establishes an upper bound for the moments of the Riemann zeta-function over short intervals, extending previous results on its size and distribution.

Contribution

It proves a new upper bound for the moments of  in short intervals, utilizing recent methods for counting large values of .

Findings

Bound for  moments in short intervals under RH

Extension of Soundararajan's large value counting method

Refined estimates involving  in short intervals

Abstract

Assuming the Riemann Hypothesis it is proved that, for fixed $k>0$ and $H = T^\theta$ with fixed $0<\theta \le 1$, $$ \int_T^{T+H}|\zeta(1/2+it)|^{2k} dt \ll H(\log T)^{k^2(1+O(1/\log_3T))}, $$ where $\log_jT = \log(\log_{j-1}T)$. The proof is based on the recent method of K. Soundararajan for counting the occurrence of large values of $\log|\zeta(1/2+it)|$, who proved that $$ \int_0^{T}|\zeta(1/2+it)|^{2k} dt \ll_\epsilon T(\log T)^{k^2+\epsilon}. $$