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

We consider two integrals over $xin [0,1]$ 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 $$int_0^1zeta_1(a,x)zeta_1(b,x),dxquadmbox{and}quad int_0^1zeta_1(a,x)zeta_1(b,1-x),dx$$ when $Re (a,b)>1$. 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.