An estimate for the Mellin transform of powers of Hardy's function.

Abstract

We show that a certain modified Mellin transform $\mathcal M(s)$ of Hardy's function is an entire function. There are reasons to connect $\mathcal M(s)$ with the function $\zeta(2s-1/2)$, and then the orders of $\mathcal M(s)$ and $\zeta(s)$ should be comparable on the critical line. Indeed, an estimate for $\mathcal M(s)$ is proved which in the particular case of the critical line coincides with the classical estimate of the zeta-function.