Relations among the Riemann Zeta and Hurwitz Zeta Functions, as Well as Their Products

Abstract

In this paper, 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 of the important results of Katsurada and Matsumoto on the mean square of the Hurwitz zeta function. 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 Lindelöf hypothesis. Some of the above relations motivate the need for analysing the large α behaviour of the modified Hurwitz zeta function ζ 1 ( s , α ) , s ∈ C , α ∈ ( 0 , ∞ ) , which is also presented here.