On some relations for Mellin transforms of Hardy's function

TL;DR

This paper explores relations between Mellin and Laplace transforms of Hardy's function, deriving mean square identities and their implications for understanding the function's properties.

Contribution

It introduces new relations involving Mellin and Laplace transforms of Hardy's function, including mean square identities and their consequences.

Findings

Derived new relations between Mellin and Laplace transforms of Hardy's function

Established mean square identities for Hardy's function

Discussed implications of these identities for Hardy's function analysis

Abstract

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