{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:4HSMUC2D4O7IK2ANZZH6RZJITF","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":"d54067c1853067197cebbc7a7389ca2d5c786d356adae8b8d2afd2b971072b2a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-13T13:48:18Z","title_canon_sha256":"785acbc9575f7cb0b50ed3baafffabe23951bb8df06d0f352e8f644333f1d848"},"schema_version":"1.0","source":{"id":"1309.3455","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.3455","created_at":"2026-05-18T03:13:22Z"},{"alias_kind":"arxiv_version","alias_value":"1309.3455v1","created_at":"2026-05-18T03:13:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.3455","created_at":"2026-05-18T03:13:22Z"},{"alias_kind":"pith_short_12","alias_value":"4HSMUC2D4O7I","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"4HSMUC2D4O7IK2AN","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"4HSMUC2D","created_at":"2026-05-18T12:27:34Z"}],"graph_snapshots":[{"event_id":"sha256:1b058bbc2d1c794739470bc8254786f24ded44ee6e7b671cf87181d9bb644ed8","target":"graph","created_at":"2026-05-18T03:13:22Z","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"},"paper":{"abstract_excerpt":"Convergent infinite products, indexed by all natural numbers, in which each factor is a rational function of the index, can always be evaluated in terms of finite products of gamma functions. This goes back to Euler. A purpose of this note is to demonstrate the usefulness of this fact through a number of diverse applications involving multiplicative partitions, entries in Ramanujan's notebooks, the Chowla--Selberg formula, and the Thue--Morse sequence. In addition, we propose a numerical method for efficiently evaluating more general infinite series such as the slowly convergent Kepler--Bouwka","authors_text":"Armin Straub, Marc Chamberland","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-13T13:48:18Z","title":"On gamma quotients and infinite products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.3455","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":"sha256:2440a77325a8cfae5d7bc943e3230fe68745a26637a32644c5e0db6ec0f5647a","target":"record","created_at":"2026-05-18T03:13:22Z","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":"d54067c1853067197cebbc7a7389ca2d5c786d356adae8b8d2afd2b971072b2a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-13T13:48:18Z","title_canon_sha256":"785acbc9575f7cb0b50ed3baafffabe23951bb8df06d0f352e8f644333f1d848"},"schema_version":"1.0","source":{"id":"1309.3455","kind":"arxiv","version":1}},"canonical_sha256":"e1e4ca0b43e3be85680dce4fe8e52899625f2b154fea072a80bb78df2be5a633","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e1e4ca0b43e3be85680dce4fe8e52899625f2b154fea072a80bb78df2be5a633","first_computed_at":"2026-05-18T03:13:22.547095Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:13:22.547095Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xSOlQUDe1Q4vcgwiQOSUgPnAzCbSvDwdVzeKliEZ9HiQJOkX2ZocrlbfwY4jhw8J931J7HUHSimjGTab4wdDDg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:13:22.547743Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.3455","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2440a77325a8cfae5d7bc943e3230fe68745a26637a32644c5e0db6ec0f5647a","sha256:1b058bbc2d1c794739470bc8254786f24ded44ee6e7b671cf87181d9bb644ed8"],"state_sha256":"4919b0df79a20fbb4b79a8f3a2efebb745827f986b4b0eac8f5944c295d50e72"}