On the Mellin transforms of powers of Hardy's function

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

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