{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/at-least-two-of-z-5-z-7-ldots-z-35-are","title":"At least two of $ζ(5),ζ(7),\\ldots,ζ(35)$ are irrational","arxiv_id":"2103.00904","date":"2021-03-01","proceeding":null,"authors":["Li Lai","Li Zhou"],"abstract":"Let $\\zeta(s)$ be the Riemann zeta function. We prove the statement in the title, which improves a recent result of Rivoal and Zudilin by lowering $69$ to $35$. We also prove that at least one of $\\beta(2),\\beta(4),\\ldots,\\beta(10)$ is irrational, where $\\beta(s) = L(s,\\chi_4)$ and $\\chi_4$ is the Dirichlet character with conductor $4$.","url_abs":"https://arxiv.org/abs/2103.00904v2","url_pdf":"https://arxiv.org/pdf/2103.00904v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"at-least-two-of-z-5-z-7-ldots-z-35-are","repo_url":"https://github.com/lzhou-xyz/zeta35","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}