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

Abstract

Assuming the Riemann Hypothesis it is proved that, for fixed $k>0$ and $H = T^θ$ with fixed $0<θ\le 1$, $$ \int_T^{T+H}|ζ(1/2+it)|^{2k} dt \ll H(\log T)^{k^2(1+O(1/\log_3T))}, $$ 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 $\log|ζ(1/2+it)|$, who proved that $$ \int_0^{T}|ζ(1/2+it)|^{2k} dt \ll_εT(\log T)^{k^2+ε}. $$