The modified Mellin transform of powers of the zeta-function

Abstract

The modified Mellin transform ${\cal Z}_k(s) = \int_1^\infty |\zeta(1/2+ix)|^{2k}x^{-s}{\rm d} x (k = 1,2,...)$ is investigated. Analytic continuation and mean square estimates of ${\cal Z}_k(s) $ are discussed, as well as connections with power moments of $|\zeta(1/2+ix)|$, with the special emphasis on the cases $k = 1,2$.