{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:HFC5DPRTUKKMBWR2OQB52RJSJ4","short_pith_number":"pith:HFC5DPRT","canonical_record":{"source":{"id":"1801.04643","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-15T02:32:36Z","cross_cats_sorted":[],"title_canon_sha256":"995cbb24d144b96c8974f2812c44f188db3931eb12f3a2d510552d55514d73c8","abstract_canon_sha256":"08f82cb9b3624ce066b6adcc9dcb9dab2379859ee303aed1711acd99116aef70"},"schema_version":"1.0"},"canonical_sha256":"3945d1be33a294c0da3a7403dd45324f3a08c8744c822622b3c2bdc0c877f552","source":{"kind":"arxiv","id":"1801.04643","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.04643","created_at":"2026-05-18T00:25:50Z"},{"alias_kind":"arxiv_version","alias_value":"1801.04643v2","created_at":"2026-05-18T00:25:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.04643","created_at":"2026-05-18T00:25:50Z"},{"alias_kind":"pith_short_12","alias_value":"HFC5DPRTUKKM","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_16","alias_value":"HFC5DPRTUKKMBWR2","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_8","alias_value":"HFC5DPRT","created_at":"2026-05-18T12:32:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:HFC5DPRTUKKMBWR2OQB52RJSJ4","target":"record","payload":{"canonical_record":{"source":{"id":"1801.04643","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-15T02:32:36Z","cross_cats_sorted":[],"title_canon_sha256":"995cbb24d144b96c8974f2812c44f188db3931eb12f3a2d510552d55514d73c8","abstract_canon_sha256":"08f82cb9b3624ce066b6adcc9dcb9dab2379859ee303aed1711acd99116aef70"},"schema_version":"1.0"},"canonical_sha256":"3945d1be33a294c0da3a7403dd45324f3a08c8744c822622b3c2bdc0c877f552","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:25:50.628843Z","signature_b64":"t25AKu4baOE00zTySod0Z1jKIPJSc+VEcpQ2zJ9NYFnRah74ZWFWkRZVc64X4oI4ZYqukdWqEgvoqrY8jGjTAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3945d1be33a294c0da3a7403dd45324f3a08c8744c822622b3c2bdc0c877f552","last_reissued_at":"2026-05-18T00:25:50.628165Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:25:50.628165Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1801.04643","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":"2026-05-18T00:25:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MMV6tTKisd+mVEjdGzEtUD1p1ATgLkvCNBBkk2MK8lVWwYe0O9aap15J8xKUpu7rb1ARBjaVE2rJZnxqCG/uAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T21:29:24.113236Z"},"content_sha256":"ae9042a3d48b6e63e6733e6945b84b6a2402269c5d00ff077ff9d1228d8d8fe0","schema_version":"1.0","event_id":"sha256:ae9042a3d48b6e63e6733e6945b84b6a2402269c5d00ff077ff9d1228d8d8fe0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:HFC5DPRTUKKMBWR2OQB52RJSJ4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalized Lambert Series Identities and Applications in Rank Differences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bin Wei, Helen W.J. Zhang","submitted_at":"2018-01-15T02:32:36Z","abstract_excerpt":"In this article, we prove two identities of generalized Lambert series. By introducing what we call $\\mathcal{S}$-series, we establish relationships between multiple generalized Lambert series and multiple infinite products. Compared with Chan's work, these new identities are useful in generating various formulas for generalized Lambert series with the same poles. Using these formulas, we study the 3-dissection properties of ranks for overpartitions modulo 6. In this case, $-1$ appears as a unit root, so that double poles occur. We also relate these ranks to the third order mock theta function"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.04643","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":""},"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":"2026-05-18T00:25:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"talb4g65ROM/M9bKXJjYmW+HO5phJ7msOSii7hF8pnpize6FmLBltu3QNHhXHPV/KkLlYytjMQ6Iaiw5VpUuCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T21:29:24.113601Z"},"content_sha256":"3bfcd0b74e5fd835266c7056dadcfdf7253bc96639424169abe047ff42b1319a","schema_version":"1.0","event_id":"sha256:3bfcd0b74e5fd835266c7056dadcfdf7253bc96639424169abe047ff42b1319a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HFC5DPRTUKKMBWR2OQB52RJSJ4/bundle.json","state_url":"https://pith.science/pith/HFC5DPRTUKKMBWR2OQB52RJSJ4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HFC5DPRTUKKMBWR2OQB52RJSJ4/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":"2026-06-20T21:29:24Z","links":{"resolver":"https://pith.science/pith/HFC5DPRTUKKMBWR2OQB52RJSJ4","bundle":"https://pith.science/pith/HFC5DPRTUKKMBWR2OQB52RJSJ4/bundle.json","state":"https://pith.science/pith/HFC5DPRTUKKMBWR2OQB52RJSJ4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HFC5DPRTUKKMBWR2OQB52RJSJ4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:HFC5DPRTUKKMBWR2OQB52RJSJ4","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":"08f82cb9b3624ce066b6adcc9dcb9dab2379859ee303aed1711acd99116aef70","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-15T02:32:36Z","title_canon_sha256":"995cbb24d144b96c8974f2812c44f188db3931eb12f3a2d510552d55514d73c8"},"schema_version":"1.0","source":{"id":"1801.04643","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.04643","created_at":"2026-05-18T00:25:50Z"},{"alias_kind":"arxiv_version","alias_value":"1801.04643v2","created_at":"2026-05-18T00:25:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.04643","created_at":"2026-05-18T00:25:50Z"},{"alias_kind":"pith_short_12","alias_value":"HFC5DPRTUKKM","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_16","alias_value":"HFC5DPRTUKKMBWR2","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_8","alias_value":"HFC5DPRT","created_at":"2026-05-18T12:32:28Z"}],"graph_snapshots":[{"event_id":"sha256:3bfcd0b74e5fd835266c7056dadcfdf7253bc96639424169abe047ff42b1319a","target":"graph","created_at":"2026-05-18T00:25:50Z","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":"In this article, we prove two identities of generalized Lambert series. By introducing what we call $\\mathcal{S}$-series, we establish relationships between multiple generalized Lambert series and multiple infinite products. Compared with Chan's work, these new identities are useful in generating various formulas for generalized Lambert series with the same poles. Using these formulas, we study the 3-dissection properties of ranks for overpartitions modulo 6. In this case, $-1$ appears as a unit root, so that double poles occur. We also relate these ranks to the third order mock theta function","authors_text":"Bin Wei, Helen W.J. Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-15T02:32:36Z","title":"Generalized Lambert Series Identities and Applications in Rank Differences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.04643","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:ae9042a3d48b6e63e6733e6945b84b6a2402269c5d00ff077ff9d1228d8d8fe0","target":"record","created_at":"2026-05-18T00:25:50Z","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":"08f82cb9b3624ce066b6adcc9dcb9dab2379859ee303aed1711acd99116aef70","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-15T02:32:36Z","title_canon_sha256":"995cbb24d144b96c8974f2812c44f188db3931eb12f3a2d510552d55514d73c8"},"schema_version":"1.0","source":{"id":"1801.04643","kind":"arxiv","version":2}},"canonical_sha256":"3945d1be33a294c0da3a7403dd45324f3a08c8744c822622b3c2bdc0c877f552","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3945d1be33a294c0da3a7403dd45324f3a08c8744c822622b3c2bdc0c877f552","first_computed_at":"2026-05-18T00:25:50.628165Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:25:50.628165Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"t25AKu4baOE00zTySod0Z1jKIPJSc+VEcpQ2zJ9NYFnRah74ZWFWkRZVc64X4oI4ZYqukdWqEgvoqrY8jGjTAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:25:50.628843Z","signed_message":"canonical_sha256_bytes"},"source_id":"1801.04643","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ae9042a3d48b6e63e6733e6945b84b6a2402269c5d00ff077ff9d1228d8d8fe0","sha256:3bfcd0b74e5fd835266c7056dadcfdf7253bc96639424169abe047ff42b1319a"],"state_sha256":"5e3654b1c4285fef62488ad69aa322420ba4e099d8be5f56441f73c7181f7564"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dh4FEV9C7YQlJ9twnF1asSTlIsCm6y5ovBtOQzHI9nsN/T0m8cBmT7IAmTNFRiEubLLuzyzyYz+abFOv9TP9Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T21:29:24.115451Z","bundle_sha256":"ad1c94fc271a1bbe076b4b5a605e5ad7cac851b047294576e1c38e529c37f244"}}