## Matti Jutila - An estimate for the Mellin transform of powers of Hardy's function.

hrj:169 - Hardy-Ramanujan Journal, January 1, 2010, Volume 33 - 2010 - https://doi.org/10.46298/hrj.2010.169
An estimate for the Mellin transform of powers of Hardy's function.

Authors: Matti Jutila

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.

Volume: Volume 33 - 2010
Published on: January 1, 2010
Imported on: March 3, 2015
Keywords: Hardy's function,Riemann's zeta-function, Mellin transform,[MATH] Mathematics [math]