{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:NRUQUNSNSZMVGQCV4F7XEEEDL7","short_pith_number":"pith:NRUQUNSN","canonical_record":{"source":{"id":"1812.00379","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-12-02T12:15:22Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"bae9189a74aa41430ab57982a3429b6272d84167d84658d232e7099283b3a749","abstract_canon_sha256":"4d0b5a3fdd3727c5d944bec0d42f0c2ea86b06c511b0fb347cb46bed806b7d5f"},"schema_version":"1.0"},"canonical_sha256":"6c690a364d9659534055e17f7210835fe6d9344016fc25a40f97bb3db5e2009b","source":{"kind":"arxiv","id":"1812.00379","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.00379","created_at":"2026-07-05T00:55:55Z"},{"alias_kind":"arxiv_version","alias_value":"1812.00379v2","created_at":"2026-07-05T00:55:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.00379","created_at":"2026-07-05T00:55:55Z"},{"alias_kind":"pith_short_12","alias_value":"NRUQUNSNSZMV","created_at":"2026-07-05T00:55:55Z"},{"alias_kind":"pith_short_16","alias_value":"NRUQUNSNSZMVGQCV","created_at":"2026-07-05T00:55:55Z"},{"alias_kind":"pith_short_8","alias_value":"NRUQUNSN","created_at":"2026-07-05T00:55:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:NRUQUNSNSZMVGQCV4F7XEEEDL7","target":"record","payload":{"canonical_record":{"source":{"id":"1812.00379","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-12-02T12:15:22Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"bae9189a74aa41430ab57982a3429b6272d84167d84658d232e7099283b3a749","abstract_canon_sha256":"4d0b5a3fdd3727c5d944bec0d42f0c2ea86b06c511b0fb347cb46bed806b7d5f"},"schema_version":"1.0"},"canonical_sha256":"6c690a364d9659534055e17f7210835fe6d9344016fc25a40f97bb3db5e2009b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:55:55.793301Z","signature_b64":"/uVs2LH03KaHXbjeo8LR6WHlRZE9GHTYQbQRIksV0KTs/YOww8Rl8I7uCYwfEMrf3CS0swsWhz7dMDp257hoCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6c690a364d9659534055e17f7210835fe6d9344016fc25a40f97bb3db5e2009b","last_reissued_at":"2026-07-05T00:55:55.792751Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:55:55.792751Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1812.00379","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:55:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SalcJikCzrHjarIswG/L3OGOG7vXqy1HtnoU4bkqnytG/P5F+OyDPE5fY2m3dM8BFaxaJueAV0lTtdaHETpAAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T15:53:18.325853Z"},"content_sha256":"f80b2bd2df34576980b0a2285682bdfe7178e8bfd1b60bacef0edf12ec248178","schema_version":"1.0","event_id":"sha256:f80b2bd2df34576980b0a2285682bdfe7178e8bfd1b60bacef0edf12ec248178"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:NRUQUNSNSZMVGQCV4F7XEEEDL7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The smallest parts function associated with $\\omega(q)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Liuquan Wang, Yifan Yang","submitted_at":"2018-12-02T12:15:22Z","abstract_excerpt":"We establish two families of congruences modulo powers of 5 for the Fourier coefficients of $(2E_2(2\\tau)-E_2(\\tau))\\eta(2\\tau)^{-1}$, where $E_2(\\tau)$ is the weight 2 Eisenstein series and $\\eta(\\tau)$ is the Dedekind eta function. This allows us to prove similar congruences for two smallest parts functions. The first function is $\\mathrm{spt}_{\\omega}(n)$, which was introduced by Andrews, Dixit and Yee and associated with Ramanujan/Watson's third order mock theta function $\\omega(q)$. The second one is $\\mathrm{spt}_{C5}(n)$, which appeared in the work of Garvan and Jennings-Shaffer. Moreov"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.00379","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1812.00379/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T00:55:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uFVtO1KSbKzwrbuAWZJlduLlnvm4G5Nd0pTGwL4Ujz/L0+nfyrxo/uln613yCGHk6xYJg8iCOd3Wqw1Bdun4CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T15:53:18.338272Z"},"content_sha256":"5428dd2a059a7e8432ff2fa56d593f7a2c37c094d4b64d64e2fdf8a152ffe3f7","schema_version":"1.0","event_id":"sha256:5428dd2a059a7e8432ff2fa56d593f7a2c37c094d4b64d64e2fdf8a152ffe3f7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NRUQUNSNSZMVGQCV4F7XEEEDL7/bundle.json","state_url":"https://pith.science/pith/NRUQUNSNSZMVGQCV4F7XEEEDL7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NRUQUNSNSZMVGQCV4F7XEEEDL7/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":"2026-08-16T15:53:18Z","links":{"resolver":"https://pith.science/pith/NRUQUNSNSZMVGQCV4F7XEEEDL7","bundle":"https://pith.science/pith/NRUQUNSNSZMVGQCV4F7XEEEDL7/bundle.json","state":"https://pith.science/pith/NRUQUNSNSZMVGQCV4F7XEEEDL7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NRUQUNSNSZMVGQCV4F7XEEEDL7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:NRUQUNSNSZMVGQCV4F7XEEEDL7","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":"4d0b5a3fdd3727c5d944bec0d42f0c2ea86b06c511b0fb347cb46bed806b7d5f","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-12-02T12:15:22Z","title_canon_sha256":"bae9189a74aa41430ab57982a3429b6272d84167d84658d232e7099283b3a749"},"schema_version":"1.0","source":{"id":"1812.00379","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.00379","created_at":"2026-07-05T00:55:55Z"},{"alias_kind":"arxiv_version","alias_value":"1812.00379v2","created_at":"2026-07-05T00:55:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.00379","created_at":"2026-07-05T00:55:55Z"},{"alias_kind":"pith_short_12","alias_value":"NRUQUNSNSZMV","created_at":"2026-07-05T00:55:55Z"},{"alias_kind":"pith_short_16","alias_value":"NRUQUNSNSZMVGQCV","created_at":"2026-07-05T00:55:55Z"},{"alias_kind":"pith_short_8","alias_value":"NRUQUNSN","created_at":"2026-07-05T00:55:55Z"}],"graph_snapshots":[{"event_id":"sha256:5428dd2a059a7e8432ff2fa56d593f7a2c37c094d4b64d64e2fdf8a152ffe3f7","target":"graph","created_at":"2026-07-05T00:55:55Z","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/1812.00379/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We establish two families of congruences modulo powers of 5 for the Fourier coefficients of $(2E_2(2\\tau)-E_2(\\tau))\\eta(2\\tau)^{-1}$, where $E_2(\\tau)$ is the weight 2 Eisenstein series and $\\eta(\\tau)$ is the Dedekind eta function. This allows us to prove similar congruences for two smallest parts functions. The first function is $\\mathrm{spt}_{\\omega}(n)$, which was introduced by Andrews, Dixit and Yee and associated with Ramanujan/Watson's third order mock theta function $\\omega(q)$. The second one is $\\mathrm{spt}_{C5}(n)$, which appeared in the work of Garvan and Jennings-Shaffer. Moreov","authors_text":"Liuquan Wang, Yifan Yang","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-12-02T12:15:22Z","title":"The smallest parts function associated with $\\omega(q)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.00379","kind":"arxiv","version":2},"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:f80b2bd2df34576980b0a2285682bdfe7178e8bfd1b60bacef0edf12ec248178","target":"record","created_at":"2026-07-05T00:55:55Z","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":"4d0b5a3fdd3727c5d944bec0d42f0c2ea86b06c511b0fb347cb46bed806b7d5f","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-12-02T12:15:22Z","title_canon_sha256":"bae9189a74aa41430ab57982a3429b6272d84167d84658d232e7099283b3a749"},"schema_version":"1.0","source":{"id":"1812.00379","kind":"arxiv","version":2}},"canonical_sha256":"6c690a364d9659534055e17f7210835fe6d9344016fc25a40f97bb3db5e2009b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6c690a364d9659534055e17f7210835fe6d9344016fc25a40f97bb3db5e2009b","first_computed_at":"2026-07-05T00:55:55.792751Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:55:55.792751Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/uVs2LH03KaHXbjeo8LR6WHlRZE9GHTYQbQRIksV0KTs/YOww8Rl8I7uCYwfEMrf3CS0swsWhz7dMDp257hoCw==","signature_status":"signed_v1","signed_at":"2026-07-05T00:55:55.793301Z","signed_message":"canonical_sha256_bytes"},"source_id":"1812.00379","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f80b2bd2df34576980b0a2285682bdfe7178e8bfd1b60bacef0edf12ec248178","sha256:5428dd2a059a7e8432ff2fa56d593f7a2c37c094d4b64d64e2fdf8a152ffe3f7"],"state_sha256":"554553f7561c4bd98324315680894bc6640458ce00a77caefd27da679302ffab"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lqm+zouT7DRIr+LFPw2sFNbk/TS5kBzUVVLa7i2Wv41JqeUujAhX+xAGgQHtcwdqzP+2HHEa15TORHgI1b8yDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T15:53:18.405345Z","bundle_sha256":"2156f3aa6684fed67df119c4f8f2ef914801345e93bbeff21b08935dd4889d80"}}