The modified Mellin transform of powers of the zeta-function

Abstract

The modified Mellin transform ${\mathcal{Z}}_k(s)={\int }_1^{\infty }|\zeta (1/2+ix){|}^{2k}{x}^{-s}\mathrm{d}x(k=1,2,...)$ is investigated. Analytic continuation and mean square estimates of ${\mathcal{Z}}_k(s)$ are discussed, as well as connections with power moments of $|\zeta (1/2+ix)|$, with the special emphasis on the cases $k=1,2$.

20 pages