Relations among the Riemann zeta and Hurwitz zeta functions as well as their products

TL;DR

This paper explores new relations among the Riemann and Hurwitz zeta functions, providing simpler derivations of known results and laying groundwork for a novel approach toward proving the Lindelöf hypothesis, including analysis of the modified Hurwitz zeta function.

Contribution

It introduces new relations among zeta functions, offers a simplified derivation of key results, and proposes a new approach toward the Lindelöf hypothesis involving the modified Hurwitz zeta function.

Findings

Derived relations among zeta functions and their products.

Simplified derivation of important existing results.

Foundation for a new approach to the Lindelöf hypothesis.

Abstract

Several relations are obtained among the Riemann zeta and Hurwitz zeta functions, as well as their products. A particular case of these relations give rise to a simple re-derivation if the important results of [11]. Also, a relation derived here provides the starting point of a novel approach which in a series of companion papers yields a formal proof of the Lindel\"{o}f hypothesis. Some of the above relations motivate the need for analysing the large $\alpha$ behaviour of the modified Hurwitz zeta function $\zeta_1(s,\alpha)$, $s\in \mathbf{C}$, $\alpha\in \mathbf{R}$, which is also presented here.