La relation entre $ζ(4n-1)$, $ζ(2p)$ et $ζ(4n-1-2p)$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914730177724416 |
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| author | Kabongo, Mundankulu |
| author_facet | Kabongo, Mundankulu |
| contents | The functional relation of the Riemann zêta function provides us with neither the nature nor the expression of zêta at positive odd numbers. From the function $F(z)=\frac{z^{-2n}}{e^z-1}$, we find a functional relation involving $ζ(4n- 1)$, $ζ(2p)$ and $ζ(4n-1-2p)$. It is given by: \begin{equation} ζ(4n-1)=\frac{1}{2n-1}\sum_{p=1}^{2n-2}ζ(2p)ζ(4n-1-2p). \end{equation} $n=2, 3, 4, 5, 6, ...$ From this formula we introduce a new approach to study the nature of $ζ$ on these integers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17997 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | La relation entre $ζ(4n-1)$, $ζ(2p)$ et $ζ(4n-1-2p)$ Kabongo, Mundankulu General Mathematics The functional relation of the Riemann zêta function provides us with neither the nature nor the expression of zêta at positive odd numbers. From the function $F(z)=\frac{z^{-2n}}{e^z-1}$, we find a functional relation involving $ζ(4n- 1)$, $ζ(2p)$ and $ζ(4n-1-2p)$. It is given by: \begin{equation} ζ(4n-1)=\frac{1}{2n-1}\sum_{p=1}^{2n-2}ζ(2p)ζ(4n-1-2p). \end{equation} $n=2, 3, 4, 5, 6, ...$ From this formula we introduce a new approach to study the nature of $ζ$ on these integers. |
| title | La relation entre $ζ(4n-1)$, $ζ(2p)$ et $ζ(4n-1-2p)$ |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2403.17997 |