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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2507.19502 |
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| _version_ | 1866912502133030912 |
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| author | Chen, Wilson J. Nguyen, Vincent |
| author_facet | Chen, Wilson J. Nguyen, Vincent |
| contents | 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 Sîntămărian. We relate this problem to conjectures regarding irrationality and $\mathbb{Q}$-linear independence of zeta values. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19502 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A series involving a product of four consecutive harmonic numbers Chen, Wilson J. Nguyen, Vincent Number Theory 40A05, 40C99, 11J72 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 Sîntămărian. We relate this problem to conjectures regarding irrationality and $\mathbb{Q}$-linear independence of zeta values. |
| title | A series involving a product of four consecutive harmonic numbers |
| topic | Number Theory 40A05, 40C99, 11J72 |
| url | https://arxiv.org/abs/2507.19502 |