{"paper":{"title":"On Ramanujan's formula for $\\zeta(1/2)$ and $\\zeta(2m+1)$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Anushree Gupta, Bibekananda Maji","submitted_at":"2021-06-09T04:09:40Z","abstract_excerpt":"Page 332 of Ramanujan's Lost Notebook contains a compelling identity for $\\zeta(1/2)$, which has been studied by many mathematicians over the years. On the same page, Ramanujan also recorded the series, \\begin{align*} \\frac{1^r}{\\exp(1^s x) - 1} + \\frac{2^r}{\\exp(2^s x) - 1} + \\frac{3^r}{\\exp(3^s x) - 1} + \\cdots, \\end{align*} where $s$ is a positive integer and $r-s$ is any even integer. Unfortunately, Ramanujan doesn't give any formula for it. This series was rediscovered by Kanemitsu, Tanigawa, and Yoshimoto, although they studied it only when $r-s$ is a negative even integer. Recently, Dix"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.04797","kind":"arxiv","version":1},"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/2106.04797/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"}