# Integrals of products of Hurwitz zeta functions via Feynman parametrization and two double sums of Riemann zeta functions by M. A. Shpot, R. B. Paris

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\limits_{0}^{1} \zeta_{1}(a,x) \zeta_{1}(b,x) \mathrm{d}x

and

\int\limits_{0}^{1} \zeta_{1}(a,x) \zeta_{1}(b,1-x) \mathrm{d}x

when $$\Re (a,b) \gt 1$$. These integrals have been investigated recently in [23]; 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.

The paper is available here.