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

arXiv eprints
 Pub Date:
 September 2016
 arXiv:
 arXiv:1609.05658
 Bibcode:
 2016arXiv160905658S
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 11M35;
 33B15;
 33E20
 EPrint:
 14 pages 0 figures