On the Mellin transforms of powers of Hardy's function

TL;DR

This paper investigates the Mellin transforms of powers of Hardy's function, exploring their properties, connections to zeta function moments, and discussing natural boundaries, thereby advancing understanding of the analytic behavior of these transforms.

Contribution

It introduces new analyses of Mellin transforms of Hardy's function powers, revealing their properties, links to zeta moments, and identifying natural boundaries.

Findings

Properties of Mellin transforms ${ m f M}_k(s)$ are characterized.

Connections between ${ m f M}_k(s)$ and moments of $| ext{zeta}(1/2+it)|$ are established.

Natural boundaries of ${ m f M}_k(s)$ are discussed.

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.