On the Mellin transforms of powers of Hardy's function

Abstract

Various properties of the Mellin transform function $$ {\cal M}_k(s) := \int_1^\infty Z^k(x)x^{-s}dx $$ are investigated, where $$ Z(t) := \zeta(1/2+it){\bigl(\chi(1/2+it)\bigr)}^{-1/2}, \quad \zeta(s) = \chi(s)\zeta(1-s) $$ is Hardy's function and $\zeta(s)$ is Riemann's zeta-function. Connections with power moments of $|\zeta(1/2+it)|$ are established, and natural boundaries of ${\cal M}_k(s)$ are discussed.