{"paper":{"title":"A note on the irrationality of $\\zeta_2(5)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.CA","math.CO"],"primary_cat":"math.NT","authors_text":"Johannes Sprang, Li Lai, Wadim Zudilin","submitted_at":"2025-05-08T07:22:25Z","abstract_excerpt":"In a spirit of Ap\\'ery'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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.05005","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.05005/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}