The evaluation of character Euler double sums

Abstract

Euler considered sums of the form ∞ m=1 1 m s m−1 n=1 1 n t .Here natural generalizations of these sums namely [p, q] := [p, q](s, t) = ∞ m=1 χ p (m) m s m−1 n=1 χ q (n) n t ,are investigated, where χ p and χ q are characters, and s and t are positive integers. The cases when p and q are either 1, 2a, 2b or −4 are examined in detail, and closedform expressions are found for t = 1 and general s in terms of the Riemann zeta function and the Catalan zeta function-the Dirichlet series L −4 (s) = 1 −s − 3 −s + 5 −s − 7 −s + · · · . Some results for arbitrary p and q are obtained as well.In Memoriam: Between the submission and acceptance of this report we greatly regret that our esteemed colleague John Boersma passed away. This paper is dedicated to his memory.

Integrals of logarithmic and hypergeometric functions

“…Some results for sums of alternating harmonic numbers may be seen in the works [1], [2], [5], [6], [7], [9], [10], [12], [15], [16], [17], [20], [21], [19], [22], [23], [27], [28], [29] and [30] and references therein.…”

Polylogarithmic Connections With Euler Sums

“…Some results for sums of alternating harmonic numbers may be seen in the works of [2], [4], [5], [6], [7], [8], [10], [11], [13], [14], [15], [17], [18], [22], [23], [24] and [25] and references therein.…”

On Summation of Subseries of the Riemann Zeta Function

“…Using the contour integral, Zucker and Robertson [7] evaluated some Dirichlet L-series exactly, which are alternating series of (1). The main results are summarized in the Appendix A of [1]. In this paper, we present another method to study ζ(a, b, s) in closed form.…”