Small improvements on the Ball-Rivoal theorem and its $p$-adic variant
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929691506507776 |
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| author | Lai, Li |
| author_facet | Lai, Li |
| contents | We prove that the dimension of the $\mathbb{Q}$-linear span of $1,ζ(3),ζ(5),\ldots,ζ(s-1)$ is at least $(1.119 \cdot \log s)/(1+\log 2)$ for any sufficiently large even integer $s$. This slightly refines a well-known result of Rivoal (2000) or Ball-Rivoal (2001). Quite unexpectedly, the proof only involves inserting the arithmetic observation of Zudilin (2001) into the original proof of Ball-Rivoal. Although this result is covered by a recent development of Fischler (2021+), our proof has the advantages of being simple and providing explicit non-vanishing small linear forms in $1$ and odd zeta values.
Moreover, we establish the $p$-adic variant: for any prime number $p$, the dimension of the $\mathbb{Q}$-linear span of $1,ζ_p(3),ζ_p(5),\ldots,ζ_p(s-1)$ is at least $(1.119 \cdot \log s)/(1+\log 2)$ for any sufficiently large even integer $s$. This is new, it slightly refines a result of Sprang (2020). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14236 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Small improvements on the Ball-Rivoal theorem and its $p$-adic variant Lai, Li Number Theory 11J72, 11M06, 33C20 We prove that the dimension of the $\mathbb{Q}$-linear span of $1,ζ(3),ζ(5),\ldots,ζ(s-1)$ is at least $(1.119 \cdot \log s)/(1+\log 2)$ for any sufficiently large even integer $s$. This slightly refines a well-known result of Rivoal (2000) or Ball-Rivoal (2001). Quite unexpectedly, the proof only involves inserting the arithmetic observation of Zudilin (2001) into the original proof of Ball-Rivoal. Although this result is covered by a recent development of Fischler (2021+), our proof has the advantages of being simple and providing explicit non-vanishing small linear forms in $1$ and odd zeta values. Moreover, we establish the $p$-adic variant: for any prime number $p$, the dimension of the $\mathbb{Q}$-linear span of $1,ζ_p(3),ζ_p(5),\ldots,ζ_p(s-1)$ is at least $(1.119 \cdot \log s)/(1+\log 2)$ for any sufficiently large even integer $s$. This is new, it slightly refines a result of Sprang (2020). |
| title | Small improvements on the Ball-Rivoal theorem and its $p$-adic variant |
| topic | Number Theory 11J72, 11M06, 33C20 |
| url | https://arxiv.org/abs/2407.14236 |