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

Abstract

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 $$\int_0^1ζ_1(a,x)ζ_1(b,x)\,dx\quad\mbox{and}\quad \int_0^1ζ_1(a,x)ζ_1(b,1-x)\,dx$$ 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.

14 pages 0 figures

References in corpus (1)

• Special values and integral representations for the Hurwitz-type Euler zeta functions

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• Determining the Indeterminate: On the Evaluation of Integrals that connect Riemann's, Hurwitz' and Dirichlet's Zeta, Eta and Beta functions