On a Formula of Mellin and Its Application to the Study of the Riemann Zeta-function (with an erratum added 03/11/13)

Abstract

In this paper, we reconsider a formula of Mellin. We present a formula which relates the sum of two positive real numbers $m, n$ to their product $mn$. We apply this formula to derivation of a relationship involving the Hurwitz zeta-function. Then we define a series function (stemming from the proved relationship) and discuss an analogy in regard to the Lindel\"{o}f hypothesis. Finally, a proof of the Lindel\"{o}f hypothesis of the Riemann zeta-function is deduced from this analogy.