{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:5JVNF5NTK5RPMHAF2M5W33XCWM","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"1e76241d9b0e70add487a62d689ee0e37267da934d16510d072b9db013f333c8","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-08-29T03:59:18Z","title_canon_sha256":"570e32a0029ff7bb0b44c426b1674225608d9651498fea49f6ea04f194f80f72"},"schema_version":"1.0","source":{"id":"2208.13356","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2208.13356","created_at":"2026-07-05T04:52:07Z"},{"alias_kind":"arxiv_version","alias_value":"2208.13356v1","created_at":"2026-07-05T04:52:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2208.13356","created_at":"2026-07-05T04:52:07Z"},{"alias_kind":"pith_short_12","alias_value":"5JVNF5NTK5RP","created_at":"2026-07-05T04:52:07Z"},{"alias_kind":"pith_short_16","alias_value":"5JVNF5NTK5RPMHAF","created_at":"2026-07-05T04:52:07Z"},{"alias_kind":"pith_short_8","alias_value":"5JVNF5NT","created_at":"2026-07-05T04:52:07Z"}],"graph_snapshots":[{"event_id":"sha256:584f8448f1532096111072b86e6f03aaf7f6c709614a447e08e45def120c432a","target":"graph","created_at":"2026-07-05T04:52:07Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2208.13356/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"It is unknown whether the Flint-Hills series $\\sum_{n=1}^\\infty \\frac{1}{n^3\\sin^2(n)}$ converges. Alekseyev (2011) connected this question to the irrationality measure of $\\pi$, that $\\mu(\\pi) > \\frac{5}{2}$ would imply divergence of the Flint-Hills series. In this paper we established a near-complete converse, that $\\mu(\\pi) < \\frac{5}{2}$ would imply convergence. The associated results on the density of close rational approximations may be of independent interest. The remaining edge case of $\\mu(\\pi) = \\frac{5}{2}$ is briefly addressed, with evidence that it would be hard to resolve.","authors_text":"Alex Meiburg","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-08-29T03:59:18Z","title":"Bounds on Irrationality Measures and the Flint-Hills Series"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2208.13356","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:64a6d11cdd49717535bc99fb12ef987ab68b6c3d855c54d653a46f04cf85f46c","target":"record","created_at":"2026-07-05T04:52:07Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"1e76241d9b0e70add487a62d689ee0e37267da934d16510d072b9db013f333c8","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-08-29T03:59:18Z","title_canon_sha256":"570e32a0029ff7bb0b44c426b1674225608d9651498fea49f6ea04f194f80f72"},"schema_version":"1.0","source":{"id":"2208.13356","kind":"arxiv","version":1}},"canonical_sha256":"ea6ad2f5b35762f61c05d33b6deee2b32cd8338e4a6981a0fe6e08b489c5ed56","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ea6ad2f5b35762f61c05d33b6deee2b32cd8338e4a6981a0fe6e08b489c5ed56","first_computed_at":"2026-07-05T04:52:07.949060Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:52:07.949060Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UMSRbFqtArw7vRmLtNTfbRyt26zT70zHP4y+Z3YjdSSJnjzqM8Eq0Jvc4lfK0lEgprkhBl1zdNvP5L9lzdnzAg==","signature_status":"signed_v1","signed_at":"2026-07-05T04:52:07.949479Z","signed_message":"canonical_sha256_bytes"},"source_id":"2208.13356","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:64a6d11cdd49717535bc99fb12ef987ab68b6c3d855c54d653a46f04cf85f46c","sha256:584f8448f1532096111072b86e6f03aaf7f6c709614a447e08e45def120c432a"],"state_sha256":"d6d85c5aabb3c076dbbb13fcdddde4ec24e5167116ffb7719318c9452992c5d8"}