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 $$\int_0^1\zeta_1(a,x)\zeta_1(b,x)\,dx\quad\mbox{and}\quad \int_0^1\zeta_1(a,x)\zeta_1(b,1-x)\,dx$$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2016
- DOI:
- 10.48550/arXiv.1609.05658
- arXiv:
- arXiv:1609.05658
- Bibcode:
- 2016arXiv160905658S
- Keywords:
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- Mathematics - Classical Analysis and ODEs;
- 11M35;
- 33B15;
- 33E20
- E-Print:
- 14 pages 0 figures