On some relations for Mellin transforms of Hardy's function

Some relations involving the Mellin and Laplace transforms of powers of the classical Hardy function $Z(t) := ζ(1/2+it)\bigl(χ(1/2+it)\bigr)^{-1/2}, \quad ζ(s) = χ(s)ζ(1-s)$ are obtained. In particular, we discuss some mean square identities and their consequences.

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