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 $H=T^{\theta }$ with fixed $0<\theta \le 1$, $${\int }_T^{T+H}|\zeta (1/2+it){|}^{2k}dt\ll H(\log T{)}^{k^2(1+O(1/{\log }_{3}T))},$$ where ${\log }_{j}T=\log ({\log }_{j-1}T)$. 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 $${\int }_0^T|\zeta (1/2+it){|}^{2k}dt{\ll }_{\epsilon }T(\log T{)}^{k^2+\epsilon }.$$

10 pages

Cited by in corpus (1)

• The 4.36-th moment of the Riemann zeta-function