{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:FNSB4E6TZVP7M2M63RUARL4R3A","short_pith_number":"pith:FNSB4E6T","canonical_record":{"source":{"id":"1507.07977","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-28T22:40:51Z","cross_cats_sorted":[],"title_canon_sha256":"103d9f97640f32b5ab49ffbb1c06ebc88ac6fc9837fdf26764b9847ba756903c","abstract_canon_sha256":"633a9957b933156859edeb7fc6753fc0bd85e1d55757f0856f2ab54f8fec61e1"},"schema_version":"1.0"},"canonical_sha256":"2b641e13d3cd5ff6699edc6808af91d80e1c203601d2a82b244f48aedf48b408","source":{"kind":"arxiv","id":"1507.07977","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.07977","created_at":"2026-05-18T01:36:08Z"},{"alias_kind":"arxiv_version","alias_value":"1507.07977v1","created_at":"2026-05-18T01:36:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.07977","created_at":"2026-05-18T01:36:08Z"},{"alias_kind":"pith_short_12","alias_value":"FNSB4E6TZVP7","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"FNSB4E6TZVP7M2M6","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"FNSB4E6T","created_at":"2026-05-18T12:29:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:FNSB4E6TZVP7M2M63RUARL4R3A","target":"record","payload":{"canonical_record":{"source":{"id":"1507.07977","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-28T22:40:51Z","cross_cats_sorted":[],"title_canon_sha256":"103d9f97640f32b5ab49ffbb1c06ebc88ac6fc9837fdf26764b9847ba756903c","abstract_canon_sha256":"633a9957b933156859edeb7fc6753fc0bd85e1d55757f0856f2ab54f8fec61e1"},"schema_version":"1.0"},"canonical_sha256":"2b641e13d3cd5ff6699edc6808af91d80e1c203601d2a82b244f48aedf48b408","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:08.766781Z","signature_b64":"Aq6j9jrvACf/SCsYP60NJs20NpBOenTHOMB3fEywrNdYmacW0THt/nrPMWjqgG1YuKvmaqIBtiHd/SDt76k8AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b641e13d3cd5ff6699edc6808af91d80e1c203601d2a82b244f48aedf48b408","last_reissued_at":"2026-05-18T01:36:08.766202Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:08.766202Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.07977","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:36:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DQTDIaD5dViXTLDylpJkCVvMDDObhA7zv0gOwwB8JaEPnq+Z05rRGS/0NvIdnRUKImffUsgaWeqyLj4+iY9hCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T01:38:16.096124Z"},"content_sha256":"087ad0cbc68d4ff51dd847fcf1f70fcd510fc79894cbbe198dcb48333ffcde70","schema_version":"1.0","event_id":"sha256:087ad0cbc68d4ff51dd847fcf1f70fcd510fc79894cbbe198dcb48333ffcde70"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:FNSB4E6TZVP7M2M63RUARL4R3A","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Asymptotics for the partial fractions of the restricted partition generating function II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Cormac O'Sullivan","submitted_at":"2015-07-28T22:40:51Z","abstract_excerpt":"The generating function for $p_N(n)$, the number of partitions of $n$ into at most $N$ parts, may be written as a product of $N$ factors. In part I, we studied the behavior of coefficients in the partial fraction decomposition of this product as $N \\to \\infty$ by applying the saddle-point method to get the asymptotics of the main terms. In this second part we bound the error terms. This involves estimating products of sines and further saddle-point arguments. The saddle-points needed are associated to zeros of the analytically continued dilogarithm."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07977","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:36:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ywg4DxxVywTHBio0b6KkO9etF/YegwZ6xQyG4LUxBd4n+//F9pjzT3R74ZEiBprLzXqH1Ozb7JaTFOMGRNXfDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T01:38:16.096555Z"},"content_sha256":"e17c8e190c226bfb403e81a2d804ec9d165d914da87f20e7dfacf640b6950bc7","schema_version":"1.0","event_id":"sha256:e17c8e190c226bfb403e81a2d804ec9d165d914da87f20e7dfacf640b6950bc7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FNSB4E6TZVP7M2M63RUARL4R3A/bundle.json","state_url":"https://pith.science/pith/FNSB4E6TZVP7M2M63RUARL4R3A/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FNSB4E6TZVP7M2M63RUARL4R3A/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-23T01:38:16Z","links":{"resolver":"https://pith.science/pith/FNSB4E6TZVP7M2M63RUARL4R3A","bundle":"https://pith.science/pith/FNSB4E6TZVP7M2M63RUARL4R3A/bundle.json","state":"https://pith.science/pith/FNSB4E6TZVP7M2M63RUARL4R3A/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FNSB4E6TZVP7M2M63RUARL4R3A/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:FNSB4E6TZVP7M2M63RUARL4R3A","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":"633a9957b933156859edeb7fc6753fc0bd85e1d55757f0856f2ab54f8fec61e1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-28T22:40:51Z","title_canon_sha256":"103d9f97640f32b5ab49ffbb1c06ebc88ac6fc9837fdf26764b9847ba756903c"},"schema_version":"1.0","source":{"id":"1507.07977","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.07977","created_at":"2026-05-18T01:36:08Z"},{"alias_kind":"arxiv_version","alias_value":"1507.07977v1","created_at":"2026-05-18T01:36:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.07977","created_at":"2026-05-18T01:36:08Z"},{"alias_kind":"pith_short_12","alias_value":"FNSB4E6TZVP7","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"FNSB4E6TZVP7M2M6","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"FNSB4E6T","created_at":"2026-05-18T12:29:19Z"}],"graph_snapshots":[{"event_id":"sha256:e17c8e190c226bfb403e81a2d804ec9d165d914da87f20e7dfacf640b6950bc7","target":"graph","created_at":"2026-05-18T01:36:08Z","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":"The generating function for $p_N(n)$, the number of partitions of $n$ into at most $N$ parts, may be written as a product of $N$ factors. In part I, we studied the behavior of coefficients in the partial fraction decomposition of this product as $N \\to \\infty$ by applying the saddle-point method to get the asymptotics of the main terms. In this second part we bound the error terms. This involves estimating products of sines and further saddle-point arguments. The saddle-points needed are associated to zeros of the analytically continued dilogarithm.","authors_text":"Cormac O'Sullivan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-28T22:40:51Z","title":"Asymptotics for the partial fractions of the restricted partition generating function II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07977","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":"sha256:087ad0cbc68d4ff51dd847fcf1f70fcd510fc79894cbbe198dcb48333ffcde70","target":"record","created_at":"2026-05-18T01:36:08Z","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":"633a9957b933156859edeb7fc6753fc0bd85e1d55757f0856f2ab54f8fec61e1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-28T22:40:51Z","title_canon_sha256":"103d9f97640f32b5ab49ffbb1c06ebc88ac6fc9837fdf26764b9847ba756903c"},"schema_version":"1.0","source":{"id":"1507.07977","kind":"arxiv","version":1}},"canonical_sha256":"2b641e13d3cd5ff6699edc6808af91d80e1c203601d2a82b244f48aedf48b408","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2b641e13d3cd5ff6699edc6808af91d80e1c203601d2a82b244f48aedf48b408","first_computed_at":"2026-05-18T01:36:08.766202Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:08.766202Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Aq6j9jrvACf/SCsYP60NJs20NpBOenTHOMB3fEywrNdYmacW0THt/nrPMWjqgG1YuKvmaqIBtiHd/SDt76k8AA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:08.766781Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.07977","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:087ad0cbc68d4ff51dd847fcf1f70fcd510fc79894cbbe198dcb48333ffcde70","sha256:e17c8e190c226bfb403e81a2d804ec9d165d914da87f20e7dfacf640b6950bc7"],"state_sha256":"10e9236be59920d8a8cb3d2b073c8c4a8c09add0615c53904e4b78c061c9236d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M/v8xBJVnwHsezY/orO/hxCZF4HhdUvd/wQ/AoEISUbVkxPXI80I4JFmqCx6RWLg5IXlEWuINJhI4nOXskdECA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T01:38:16.098911Z","bundle_sha256":"e7eaa1bd791227f7cedcfe69ba59d21645f8f95a588d99abff6261239d23e631"}}