Explicit Evaluations of Euler Sums Involving Harmonic Numbers with Rational Arguments
General Mathematics
2026-01-14 v2
Abstract
This study presents explicit evaluations of the series \begin{equation*} \sum_{k=1}^\infty \frac{H_{k/n}^{(p)}}{k^q} \quad \text{and} \quad \sum_{k=1}^\infty \frac{(-1)^k H_{k/2n}^{(p)}}{k^q}, \quad p,q,n \in \mathbb{Z}_{\ge 1},\; q \ne 1, \end{equation*} for odd values of . These explicit evaluations are expressed in terms of the Riemann zeta function and the Hurwitz zeta function.
Cite
@article{arxiv.2601.06895,
title = {Explicit Evaluations of Euler Sums Involving Harmonic Numbers with Rational Arguments},
author = {Ali Olaikhan},
journal= {arXiv preprint arXiv:2601.06895},
year = {2026}
}