标题： 一个涉及四个连续调和数乘积的级数 显示英文标题

标题： A series involving a product of four consecutive harmonic numbers

摘要： 与哥德巴赫通信时，欧拉开始研究形式为 $\sum_{k \geq 1} k^{-m}\left(1 + 2^{-n} + \cdots + k^{-n}\right)$的级数，这些级数如今被称为欧拉和。 当 $n=1$且 $m \geq 2$时，欧拉能够以zeta值的形式得到闭式表达式。 我们使用欧拉精神的初等技巧，以zeta值的形式计算级数 $\sum_{k \geq 1} \frac{H_k H_{k+1} H_{k+2} H_{k+3}}{k(k+1)(k+2)(k+3)},$，其中 $H_k := 1 + \frac{1}{2} + \cdots + \frac{1}{k}$是第 $k$个调和数。 闭式表达式可能是Furdui和Sîntămărian的一个猜想的反例。 我们将这个问题与关于zeta值的无理性和 $\mathbb{Q}$-线性无关性的猜想联系起来。 显示英文摘要

摘要： 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.