Beukers-like proofs of irrationality for $ζ{(2)}$ and $ζ{(3)}$

[1308.2720] Beukers-like proofs of irrationality for $ζ{(2)}$ and $ζ{(3)}$ Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:1308.2720 (math) [Submitted on 12 Aug 2013 ( v1 ), last revised 9 Apr 2026 (this version, v3)] Title: Beukers-like proofs of irrationality for $ζ{(2)}$ and $ζ{(3)}$ Authors: F. M. S. Lima View a PDF of the paper titled Beukers-like proofs of irrationality for $\zeta{(2)}$ and $\zeta{(3)}$, by F. M. S. Lima View PDF HTML (experimental) Abstract: In this note, I develop step-by-step proofs of irrationality for $\,\zeta{(2)}\,$ and $\,\zeta{(3)}$. Though the proofs follow closely those based upon unit-square integrals proposed originally by Beukers, I introduce some modifications which certainly will be useful for those interested in understanding this kind of proof and/or trying to extend it to higher zeta values, Catalan's constant, or other related numbers. Comments: A few corrections in the text and footnotes. New footnotes. Two additional references. Elsevier document class updated. 13 pages, no figures Subjects: Number Theory (math.NT) MSC classes: 11J72, 11J82, 11M06 Cite as: arXiv:1308.2720 [math.NT] (or arXiv:1308.2720v3 [math.NT] for this version) https://doi.org/10.48550/arXiv.1308.2720 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Fabio M. S. Lima [ view email ] [v1] Mon, 12 Aug 2013 23:50:12 UTC (20 KB) [v2] Fri, 13 May 2016 21:01:44 UTC (20 KB) [v3] Thu, 9 Apr 2026 01:43:32 UTC (20 KB) Full-text links: Access Paper: View a PDF of the paper titled Beukers-like proofs of irrationality for $\zeta{(2)}$ and $\zeta{(3)}$, by F. M. S. Lima View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2013-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: 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) Major funding support from

Related documents

• A new proof of the irrationality of $e$#1051589
• Explaining Evidence Denial as Motivated Pragmatically Rational Epistemic Irrationality#767273
• Catalan's constant is irrational#927041
• The Evil God Challenge: Two Significant Asymmetries#71237
• On the arithmetic of Tate-Shafarevich groups via Kolyvagin's conjecture#1109355