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

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.

Leave a Reply

Your email address will not be published. Required fields are marked *