{"paper":{"title":"On convergence of the Flint Hills series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CA","authors_text":"Max A. Alekseyev","submitted_at":"2011-04-27T09:50:10Z","abstract_excerpt":"It is not known whether the Flint Hills series $\\sum_{n=1}^{\\infty} \\frac{1}{n^3\\cdot\\sin(n)^2}$ converges. We show that this question is closely related to the irrationality measure of $\\pi$, denoted $\\mu(\\pi)$. In particular, convergence of the Flint Hills series would imply $\\mu(\\pi) \\leq 2.5$ which is much stronger than the best currently known upper bound $\\mu(\\pi)\\leq 7.6063...$.\n  This result easily generalizes to series of the form $\\sum_{n=1}^{\\infty} \\frac{1}{n^u\\cdot |\\sin(n)|^v}$ where $u,v>0$. We use the currently known bound for $\\mu(\\pi)$ to derive conditions on $u$ and $v$ that"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.5100","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"}