{"paper":{"title":"Rapidly converging formulae for $\\zeta(4k\\pm 1)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Blake Wilkerson, Shubho Banerjee","submitted_at":"2018-03-09T18:50:20Z","abstract_excerpt":"We provide rapidly converging formulae for the Riemann zeta function at odd integers using the Lambert series $\\mathscr{L}_q(s) = \\sum_{n=1}^\\infty n^{s} q^{n}/(1-q^n)$, $s=-(4k\\pm 1)$. Our main formula for $\\zeta(4k-1)$ converges at rate of about $e^{-\\sqrt{15}\\pi}$ per term, and the formula for $\\zeta(4k+1)$, at the rate of $e^{-4\\pi}$ per term. For example, the first order approximation yields $\\zeta(3)\\approx\\frac{\\pi ^3 \\sqrt{15}}{100} +e^{-\\sqrt{15} \\pi }\\left[\\frac{9}{4}+\\frac{4}{\\sqrt{15}}\\sinh (\\frac{\\sqrt{15} \\pi }{2})\\right]$ which has an error only of order $10^{-10}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.03291","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":""},"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"}