On the Mellin transforms of powers of Hardy's function

Details On the Mellin transforms of powers of Hardy's function On the Mellin transforms of powers of Hardy's function

2010-01-12

arxiv.org/abs/1001.1824v3

Hardy-Ramanujan Journal 33(2010), 32-58

Mathematics

Number Theory

26 pages

Scientific paper

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.

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