{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:5SQE4WJNGSQIDFW7Y4RI77MEMN","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":"24657b83609babf3f43862d6e1eaf4a5706233c0aac8bfed28e6f664f5ea90f7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2020-05-13T15:50:21Z","title_canon_sha256":"691e32dfceab3f369e072b47b431c32425bbcf29174ab49863210caf3a09c4f5"},"schema_version":"1.0","source":{"id":"2005.06384","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2005.06384","created_at":"2026-07-05T01:02:40Z"},{"alias_kind":"arxiv_version","alias_value":"2005.06384v1","created_at":"2026-07-05T01:02:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2005.06384","created_at":"2026-07-05T01:02:40Z"},{"alias_kind":"pith_short_12","alias_value":"5SQE4WJNGSQI","created_at":"2026-07-05T01:02:40Z"},{"alias_kind":"pith_short_16","alias_value":"5SQE4WJNGSQIDFW7","created_at":"2026-07-05T01:02:40Z"},{"alias_kind":"pith_short_8","alias_value":"5SQE4WJN","created_at":"2026-07-05T01:02:40Z"}],"graph_snapshots":[{"event_id":"sha256:d36aaf696d1bb68cfa5aff8d98e3a7218add853ba5ab1df3d4c112f8f3aac7fe","target":"graph","created_at":"2026-07-05T01:02:40Z","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/2005.06384/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Lower and upper bounds $B_a(x)$ on the incomplete gamma function $\\Gamma(a,x)$ are given for all real $a$ and all real $x>0$. These bounds $B_a(x)$ are exact in the sense that $B_a(x)\\underset{x\\downarrow0}\\sim\\Gamma(a,x)$ and $B_a(x)\\underset{x\\to\\infty}\\sim\\Gamma(a,x)$. Moreover, the relative errors of these bounds are rather small for other values of $x$, away from $0$ and $\\infty$.","authors_text":"Iosif Pinelis","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2020-05-13T15:50:21Z","title":"Exact lower and upper bounds on the incomplete gamma function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.06384","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:ddc7e78cb78fb1732a7217ec35798dc30b66144fde2956ba3794ad9902f8d21b","target":"record","created_at":"2026-07-05T01:02:40Z","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":"24657b83609babf3f43862d6e1eaf4a5706233c0aac8bfed28e6f664f5ea90f7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2020-05-13T15:50:21Z","title_canon_sha256":"691e32dfceab3f369e072b47b431c32425bbcf29174ab49863210caf3a09c4f5"},"schema_version":"1.0","source":{"id":"2005.06384","kind":"arxiv","version":1}},"canonical_sha256":"eca04e592d34a08196dfc7228ffd846377aa4e7a1a158deddc946a706defcd26","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eca04e592d34a08196dfc7228ffd846377aa4e7a1a158deddc946a706defcd26","first_computed_at":"2026-07-05T01:02:40.850143Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:02:40.850143Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ovP5OrzQSroLhqiFF9IDzjJSLDExzRo48o4D8WiY+FqPxUigfu1ejxIEW10d7Lawe5bULW065bspADcAc7JCDw==","signature_status":"signed_v1","signed_at":"2026-07-05T01:02:40.850520Z","signed_message":"canonical_sha256_bytes"},"source_id":"2005.06384","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ddc7e78cb78fb1732a7217ec35798dc30b66144fde2956ba3794ad9902f8d21b","sha256:d36aaf696d1bb68cfa5aff8d98e3a7218add853ba5ab1df3d4c112f8f3aac7fe"],"state_sha256":"566ed9c020e0084866aeb7f9804f49627ef093854578719aa5021f390151b472"}