Integrals of products of Hurwitz zeta functions via Feynman parametrization and two double sums of Riemann zeta functions

We consider two integrals over $x\in [0,1]$ involving products of the function ${\zeta }_1(a,x)\equiv \zeta (a,x)-x^{-a}$, where $\zeta (a,x)$ is the Hurwitz zeta function, given by $${\int }_0^1{\zeta }_1(a,x){\zeta }_1(b,x)dx\text{quadand}{\int }_0^1{\zeta }_1(a,x){\zeta }_1(b,1-x)dx$$ when $\mathrm{\Re }(a,b)>1$. These integrals have been investigated recently in \cite{SCP}; here we provide an alternative derivation by application of Feynman parametrization. We also discuss a moment integral and the evaluation of two doubly infinite sums containing the Riemann zeta function $\zeta (x)$ and two free parameters $a$ and $b$. The limiting forms of these sums when $a+b$ takes on integer values are considered.