A series involving a product of four consecutive harmonic numbers

TL;DR

This paper evaluates a complex series involving four consecutive harmonic numbers using elementary techniques, expressing it in terms of zeta values, and discusses its implications for conjectures on irrationality and linear independence of zeta values.

Contribution

It provides a closed-form evaluation of a specific harmonic series in terms of zeta values, challenging existing conjectures and connecting to irrationality problems.

Findings

Derived a closed-form expression involving zeta values.

Identified the series as a potential counterexample to a conjecture.

Linked the series evaluation to conjectures on irrationality and linear independence.

Abstract

In correspondence with Goldbach, Euler began investigating series of the form $\sum_{k \geq 1} k^{-m}\left(1 + 2^{-n} + \cdots + k^{-n}\right)$, which are known today as Euler sums. For the case where $n=1$ and $m \geq 2$, Euler was able to obtain a closed form in terms of zeta values. We use elementary techniques in the spirit of Euler to evaluate the series $\sum_{k \geq 1} \frac{H_k H_{k+1} H_{k+2} H_{k+3}}{k(k+1)(k+2)(k+3)},$ where $H_k := 1 + \frac{1}{2} + \cdots + \frac{1}{k}$ is the $k$th harmonic number, in terms of zeta values. The closed form is a potential counterexample to a conjecture of Furdui and S\^int\u{a}m\u{a}rian. We relate this problem to conjectures regarding irrationality and $\mathbb{Q}$-linear independence of zeta values.