On sums of integrals of powers of the zeta-function in short intervals

Abstract

The modified Mellin transform ${\mathcal{Z}}_k(s)={\int }_1^{\infty }|\zeta (\frac{1}{2}+ix{|}^{2k}{x}^{-s}\mathrm{d}x$ ($k\ge 1$ is a fixed integer, $s=\sigma +it$) is used to obtain estimates for $$\sum _{r=1}^R{\int }_{t_r-G}^{t_r+G}|\zeta (1/2+it){|}^{2k}\mathrm{d}t(T<t_1<>...<t_R<2T),$$ where $t_{r+1} - t_r \ge G (r =1,..., R-1), T^ε\le G \le T^{1-ε$. These results can be used to derive bounds for the moments of $}|ζ(1/2+it)|$.

14 pages