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

Abstract

The modified Mellin transform ${\cal Z}_k(s) = \int_1^\infty |\zeta({1\over2}+ix|^{2k}x^{-s}{\rm d} x$ ($k\ge1$ is a fixed integer, $s = \sigma + it$) is used to obtain estimates for $$ \sum_{r=1}^R\int_{t_r-G}^{t_r+G}|\zeta(1/2+it)|^{2k}{\rm d} t\quad(T < t_1 < >... < t_R < 2T), $$ where $t_{r+1} - t_r \ge G (r =1,..., R-1), T^\epsilon \le G \le T^{1-\epsilon$. These results can be used to derive bounds for the moments of $}|\zeta(1/2+it)|$.