{"paper":{"title":"A new series for $\\pi^3$ and related congruences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2010-09-27T19:56:59Z","abstract_excerpt":"Let $H_n^{(2)}$ denote the second-order harmonic number $\\sum_{0<k\\le n}1/k^2$ for $n=0,1,2,\\ldots$. In this paper we obtain the following identity: $$\\sum_{k=1}^\\infty\\frac{2^kH_{k-1}^{(2)}}{k\\binom{2k}k}=\\frac{\\pi^3}{48}.$$ We explain how we found the series and develop related congruences involving Bernoulli or Euler numbers; for example, it is shown that $$\\sum_{k=1}^{p-1}\\frac{\\binom{2k}k}{2^k}H_k^{(2)}\\equiv-E_{p-3}\\pmod{p}$$ for any prime $p>3$, where $E_0,E_1,E_2,\\ldots$ are Euler numbers. Motivated by the Amdeberhan-Zeilberger identity $\\sum_{k=1}^\\infty(21k-8)/(k^3\\binom{2k}k^3)=\\pi^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.5375","kind":"arxiv","version":8},"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"}