Explicit Evaluations of Euler Sums Involving Harmonic Numbers with Rational Arguments
Ali Olaikhan
TL;DR
The paper addresses explicit Euler-type sums involving generalized harmonic numbers with rational arguments, focusing on $\sum_{k=1}^\infty \frac{H_{k/n}^{(p)}}{k^q}$ and the alternating variant for odd $p+q$ with $q\neq 1$. It develops integral representations via polylogarithms and differentiations to define $T(p,q,n)$ and $S(p,q,n)$, enabling closed forms in terms of the Riemann zeta function $\zeta$ and the Hurwitz zeta function $\zeta(s,t)$. The main results express $A(p,q,n)=\sum_{k=1}^\infty \frac{H_{k/n}^{(p)}}{k^q}$ for odd $p+q$ as $\zeta(p)\zeta(q)-\frac{(-n)^{p-1}}{(p-1)!}\{T(p,q-1,n)+(p-1)T(p-1,q,n)\}$ and define $B(p,q,n)=2^{1-q}A(p,q,n)-A(p,q,2n)$, with all terms reducible to $\zeta$ and $\zeta(s,t)$; these results are presented as exact formulas rather than numeric approximations. The work highlights that these evaluations yield new representations not readily obtainable from standard computer algebra systems and provides conversion rules and explicit examples to express results purely with Hurwitz zeta. The approach broadens the toolkit for analyzing Euler sums with rational arguments and offers practical formulas for applications in analytic number theory.
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 $p+q$. These explicit evaluations are expressed in terms of the Riemann zeta function and the Hurwitz zeta function.