Mathematica Eterna
Open Access

Abstract

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

M.A. Shpot and R.B. Paris

We consider two integrals over x ∈ [0, 1] involving products of the function ζ1(a, x) ≡ ζ(a, x) − x −a , where ζ(a, x) is the Hurwitz zeta function, given by Z 1 0 ζ1(a, x)ζ1(b, x) dx and Z 1 0 ζ1(a, x)ζ1(b, 1 − x) dx when ℜ(a, b) > 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 ζ(x) and two free parameters a and b. The limiting forms of these sums when a + b takes on integer values are considered.