  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
$$\int_0^1\zeta_1(a,x)\zeta_1(b,x)\,dx\quad\mbox{and}\quad
\int_0^1\zeta_1(a,x)\zeta_1(b,1-x)\,dx$$ 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.
