On the Mellin transforms of powers of Hardy's function

Aleksandar Ivić

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arXiv · 1001.1824

On the Mellin transforms of powers of Hardy's function

Aleksandar Ivić

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Abstract

Various properties of the Mellin transform function $$ {\cal M}_k(s) := \int_1^\infty Z^k(x)x^{-s}dx $$ are investigated, where $$ Z(t) := ζ(1/2+it){\bigl(χ(1/2+it)\bigr)}^{-1/2}, \quad ζ(s) = χ(s)ζ(1-s) $$ 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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Aleksandar Ivić. 2010-11-11. On the Mellin transforms of powers of Hardy's function. https://arxiv.org/abs/1001.1824

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