{"paper":{"title":"Two $q$-analogues of Euler's formula $\\zeta(2)=\\pi^2/6$","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":"2018-02-05T15:52:27Z","abstract_excerpt":"It is well known that $\\zeta(2)=\\pi^2/6$ as discovered by Euler. In this paper we present the following two $q$-analogues of this celebrated formula: $$\\sum_{k=0}^\\infty\\frac{q^k(1+q^{2k+1})}{(1-q^{2k+1})^2}=\\prod_{n=1}^\\infty\\frac{(1-q^{2n})^4}{(1-q^{2n-1})^4}$$ and $$\\sum_{k=0}^\\infty\\frac{q^{2k-\\lfloor(-1)^kk/2\\rfloor}}{(1-q^{2k+1})^2} =\\prod_{n=1}^\\infty\\frac{(1-q^{2n})^2(1-q^{4n})^2}{(1-q^{2n-1})^2(1-q^{4n-2})^2},$$ where $q$ is any complex number with $|q|<1$. We also give a $q$-analogue of the identity $\\zeta(4)=\\pi^4/90$, and pose a problem on $q$-analogues of Euler's formula for $\\zet"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.01473","kind":"arxiv","version":6},"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/1802.01473/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"}