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

Abstract

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

9 pages