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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.22421 |
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| _version_ | 1866911704634359808 |
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| author | Deng, Junfa Yang, Yunyun Zhang, Hao |
| author_facet | Deng, Junfa Yang, Yunyun Zhang, Hao |
| contents | The evaluation of the Riemann zeta function at negative integers is a classical result typically obtained through analytic continuation or contour integration. In this paper, we present a novel and concise derivation of these special values by employing the theory of Cesàro limit of distributions, a generalized limit concept developed by Estrada, Kanwal, and Fulling. We use this tool to give a quick proof of the result that \[ ζ(-n)=-\frac{B_{n+1}}{n+1}, \] for $n\in\mathbb{N}^+.$ We also give a short discussion on $ζ^{\prime }(α)$ and compute the value of $ζ^{\prime}(0)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_22421 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A quick distributional way to reproduce some results of the Riemann zeta function Deng, Junfa Yang, Yunyun Zhang, Hao Number Theory Functional Analysis 11M06, 46F05 The evaluation of the Riemann zeta function at negative integers is a classical result typically obtained through analytic continuation or contour integration. In this paper, we present a novel and concise derivation of these special values by employing the theory of Cesàro limit of distributions, a generalized limit concept developed by Estrada, Kanwal, and Fulling. We use this tool to give a quick proof of the result that \[ ζ(-n)=-\frac{B_{n+1}}{n+1}, \] for $n\in\mathbb{N}^+.$ We also give a short discussion on $ζ^{\prime }(α)$ and compute the value of $ζ^{\prime}(0)$. |
| title | A quick distributional way to reproduce some results of the Riemann zeta function |
| topic | Number Theory Functional Analysis 11M06, 46F05 |
| url | https://arxiv.org/abs/2605.22421 |