  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.
