Many $p$-adic odd zeta values are irrational
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929716181598208 |
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| author | Lai, Li Sprang, Johannes |
| author_facet | Lai, Li Sprang, Johannes |
| contents | For any prime $p$ and $\varepsilon>0$ we prove that for any sufficiently large positive odd integer $s$ at least $(c_p-\varepsilon) \sqrt{\frac{s}{\log s}}$ of the $p$-adic zeta values $ζ_p(3),ζ_p(5),\dots,ζ_p(s)$ are irrational. The constant $c_p$ is positive and does only depend on $p$. This result establishes a $p$-adic version of the elimination technique used by Fischler--Sprang--Zudilin and Lai--Yu to prove a similar result on classical zeta values. The main difficulty consists in proving the non-vanishing of the resulting linear forms. We overcome this problem by using a new irrationality criterion. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_10393 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Many $p$-adic odd zeta values are irrational Lai, Li Sprang, Johannes Number Theory 11J72 (Primary) 11F85, 11M06 (Secondary) For any prime $p$ and $\varepsilon>0$ we prove that for any sufficiently large positive odd integer $s$ at least $(c_p-\varepsilon) \sqrt{\frac{s}{\log s}}$ of the $p$-adic zeta values $ζ_p(3),ζ_p(5),\dots,ζ_p(s)$ are irrational. The constant $c_p$ is positive and does only depend on $p$. This result establishes a $p$-adic version of the elimination technique used by Fischler--Sprang--Zudilin and Lai--Yu to prove a similar result on classical zeta values. The main difficulty consists in proving the non-vanishing of the resulting linear forms. We overcome this problem by using a new irrationality criterion. |
| title | Many $p$-adic odd zeta values are irrational |
| topic | Number Theory 11J72 (Primary) 11F85, 11M06 (Secondary) |
| url | https://arxiv.org/abs/2306.10393 |