A note on the irrationality of $ζ_2(5)$

[2505.05005] A note on the irrationality of $ζ_2(5)$ Skip to main content Search Submit Donate Log in Mathematics > Number Theory arXiv:2505.05005 (math) [Submitted on 8 May 2025 ( v1 ), last revised 26 May 2026 (this version, v2)] Title: A note on the irrationality of $ζ_2(5)$ Authors: Li Lai , Johannes Sprang , Wadim Zudilin View PDF HTML (experimental) Abstract: In a spirit of Apéry's proof of the irrationality of $\zeta(3)$, we construct a sequence $p_n/q_n$ of rational approximations to the $2$-adic zeta value $\zeta_2(5)$ which satisfy $0 < |\zeta_2(5)-p_n/q_n|_2 < \max\{|p_n|,|q_n|\}^{-1-\delta}$ for an explicit constant $\delta>0$. This leads to a new proof of the irrationality of $\zeta_2(5)$, 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 $\mu(\zeta_2(5)) \le (16\log2)/(8\log2-5) = 20.342\dots$. Comments: 2^2 x 5 pages Subjects: Number Theory (math.NT) ; Algebraic Geometry (math.AG); Classical Analysis and ODEs (math.CA); Combinatorics (math.CO) MSC classes: 11J72, 11J82, 11M06, 33C20 Report number: MPIM-Bonn-2025 Cite as: arXiv:2505.05005 [math.NT] (or arXiv:2505.05005v2 [math.NT] for this version) https://doi.org/10.48550/arXiv.2505.05005 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Intern. Math. Research Notices, Volume 2026 (August 2026), Issue 16, rnag180, 19 pages Related DOI : https://doi.org/10.1093/imrn/rnag180 Focus to learn more DOI(s) linking to related resources Submission history From: Wadim Zudilin [ view email ] [v1] Thu, 8 May 2025 07:22:25 UTC (18 KB) [v2] Tue, 26 May 2026 19:32:56 UTC (18 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2025-05 Change to browse by: math math.AG math.CA math.CO References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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