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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1104.5100","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-04-27T09:50:10Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"25b5f7d59b232999a8a3a8cf109339a85ab109b1c39ce234b78379887860924c","abstract_canon_sha256":"22107f4480187e0b159153982d34241bf3d52045d376a9127f18a8ffad1f62ef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:23:22.650051Z","signature_b64":"ChSmkNukxKl1asn+yZNNNgG4ysSmfLfvZsJEbyCJfxeu/6QzUQkYjIC1SPA/XaQtwb2VRsXVXwptTzCvMcYBBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d944b69285ffdd80f861452cd48a781e5aff4f62ecb8777bbc64430a58219f30","last_reissued_at":"2026-05-18T04:23:22.649495Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:23:22.649495Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1104.5100","created_at":"2026-05-18T04:23:22.649588+00:00"},{"alias_kind":"arxiv_version","alias_value":"1104.5100v1","created_at":"2026-05-18T04:23:22.649588+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.5100","created_at":"2026-05-18T04:23:22.649588+00:00"},{"alias_kind":"pith_short_12","alias_value":"3FCLNEUF77OY","created_at":"2026-05-18T12:26:18.847500+00:00"},{"alias_kind":"pith_short_16","alias_value":"3FCLNEUF77OYB6DB","created_at":"2026-05-18T12:26:18.847500+00:00"},{"alias_kind":"pith_short_8","alias_value":"3FCLNEUF","created_at":"2026-05-18T12:26:18.847500+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":2,"sample":[{"citing_arxiv_id":"2109.00295","citing_title":"On the flint hills series","ref_index":2,"is_internal_anchor":true},{"citing_arxiv_id":"2603.09719","citing_title":"On the Critical Line Re(s) = 1/2, the Irrationality Measure of {\\pi}, and the Automorphic Structure of the Flint Hills Series","ref_index":2,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3FCLNEUF77OYB6DBIUWNJCTYDZ","json":"https://pith.science/pith/3FCLNEUF77OYB6DBIUWNJCTYDZ.json","graph_json":"https://pith.science/api/pith-number/3FCLNEUF77OYB6DBIUWNJCTYDZ/graph.json","events_json":"https://pith.science/api/pith-number/3FCLNEUF77OYB6DBIUWNJCTYDZ/events.json","paper":"https://pith.science/paper/3FCLNEUF"},"agent_actions":{"view_html":"https://pith.science/pith/3FCLNEUF77OYB6DBIUWNJCTYDZ","download_json":"https://pith.science/pith/3FCLNEUF77OYB6DBIUWNJCTYDZ.json","view_paper":"https://pith.science/paper/3FCLNEUF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1104.5100&json=true","fetch_graph":"https://pith.science/api/pith-number/3FCLNEUF77OYB6DBIUWNJCTYDZ/graph.json","fetch_events":"https://pith.science/api/pith-number/3FCLNEUF77OYB6DBIUWNJCTYDZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3FCLNEUF77OYB6DBIUWNJCTYDZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3FCLNEUF77OYB6DBIUWNJCTYDZ/action/storage_attestation","attest_author":"https://pith.science/pith/3FCLNEUF77OYB6DBIUWNJCTYDZ/action/author_attestation","sign_citation":"https://pith.science/pith/3FCLNEUF77OYB6DBIUWNJCTYDZ/action/citation_signature","submit_replication":"https://pith.science/pith/3FCLNEUF77OYB6DBIUWNJCTYDZ/action/replication_record"}},"created_at":"2026-05-18T04:23:22.649588+00:00","updated_at":"2026-05-18T04:23:22.649588+00:00"}