A note on the irrationality of $ζ_2(5)$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913164804751360 |
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| author | Lai, Li Sprang, Johannes Zudilin, Wadim |
| author_facet | Lai, Li Sprang, Johannes Zudilin, Wadim |
| contents | In a spirit of Apéry's proof of the irrationality of $ζ(3)$, we construct a sequence $p_n/q_n$ of rational approximations to the $2$-adic zeta value $ζ_2(5)$ which satisfy $0 < |ζ_2(5)-p_n/q_n|_2 < \max\{|p_n|,|q_n|\}^{-1-δ}$ for an explicit constant $δ>0$. This leads to a new proof of the irrationality of $ζ_2(5)$, the result established recently by Calegari, Dimitrov and Tang using a different method. Furthermore, our approximations allow us to obtain an upper bound for the irrationality measure of this $2$-adic quantity; namely, we show that $μ(ζ_2(5)) \le (16\log2)/(8\log2-5) = 20.342\dots$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05005 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on the irrationality of $ζ_2(5)$ Lai, Li Sprang, Johannes Zudilin, Wadim Number Theory Algebraic Geometry Classical Analysis and ODEs Combinatorics 11J72, 11J82, 11M06, 33C20 In a spirit of Apéry's proof of the irrationality of $ζ(3)$, we construct a sequence $p_n/q_n$ of rational approximations to the $2$-adic zeta value $ζ_2(5)$ which satisfy $0 < |ζ_2(5)-p_n/q_n|_2 < \max\{|p_n|,|q_n|\}^{-1-δ}$ for an explicit constant $δ>0$. This leads to a new proof of the irrationality of $ζ_2(5)$, the result established recently by Calegari, Dimitrov and Tang using a different method. Furthermore, our approximations allow us to obtain an upper bound for the irrationality measure of this $2$-adic quantity; namely, we show that $μ(ζ_2(5)) \le (16\log2)/(8\log2-5) = 20.342\dots$. |
| title | A note on the irrationality of $ζ_2(5)$ |
| topic | Number Theory Algebraic Geometry Classical Analysis and ODEs Combinatorics 11J72, 11J82, 11M06, 33C20 |
| url | https://arxiv.org/abs/2505.05005 |