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

Abstract

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.