On the irrationality of certain $p$-adic zeta values
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866910973331243008 |
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| author | Lai, Li Lupu, Cezar Sprang, Johannes |
| author_facet | Lai, Li Lupu, Cezar Sprang, Johannes |
| contents | A famous theorem of Zudilin states that at least one of the Riemann zeta values $ζ(5), ζ(7), ζ(9), ζ(11)$ is irrational. In this paper, we establish the $p$-adic analogue of Zudilin's theorem. As a weaker form of our result, it is proved that for any prime number $p \geqslant 5$ there exists an odd integer $i$ in the interval $[3,p+p/\log p+5]$ such that the $p$-adic zeta value $ζ_p(i)$ is irrational. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_23088 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the irrationality of certain $p$-adic zeta values Lai, Li Lupu, Cezar Sprang, Johannes Number Theory 11J72 (primary), 11M06, 33C20 (secondary) A famous theorem of Zudilin states that at least one of the Riemann zeta values $ζ(5), ζ(7), ζ(9), ζ(11)$ is irrational. In this paper, we establish the $p$-adic analogue of Zudilin's theorem. As a weaker form of our result, it is proved that for any prime number $p \geqslant 5$ there exists an odd integer $i$ in the interval $[3,p+p/\log p+5]$ such that the $p$-adic zeta value $ζ_p(i)$ is irrational. |
| title | On the irrationality of certain $p$-adic zeta values |
| topic | Number Theory 11J72 (primary), 11M06, 33C20 (secondary) |
| url | https://arxiv.org/abs/2505.23088 |