The Mellin transform of the square of Riemann's zeta-function

Abstract

Let ${\cal Z}_1(s) = \int_1^\infty |\zeta({1\over2}+ix)|^2x^{-s}{\rm d}x (\sigma = \Re s > 1)$. A result concerning analytic continuation of ${\cal Z}_1(s)$ to $\bf C$ is proved, and also a result relating the order of ${\cal Z}_1(\sigma + it) (1/2 \le \sigma \le 1, t\ge t_0)$ to the order of ${\cal Z}_1({1\over2}+it)$.