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arXiv · 1609.05658

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

M. A. Shpot·R. B. Paris
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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.

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M. A. Shpot, R. B. Paris. 2016-09-19. Integrals of products of Hurwitz zeta functions via Feynman parametrization and two double sums of Riemann zeta functions. https://arxiv.org/abs/1609.05658

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