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 $\zeta 1(s,\alpha )$, $s\in C$, $\alpha \in R$, which is also presented here.