{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:5UA6SOGT3A7Q24ZSSDXFSWQJQN","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":"19c13c57eab327edffb35bac1eeefd20eaee7e628e66cb347e977db0480ed782","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-06-02T03:11:12Z","title_canon_sha256":"4b3a71125e596684697d3d5669571f3bd6266ae1e5799eaaa2cb4b45b2fb0f41"},"schema_version":"1.0","source":{"id":"1206.0340","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.0340","created_at":"2026-05-18T03:54:25Z"},{"alias_kind":"arxiv_version","alias_value":"1206.0340v1","created_at":"2026-05-18T03:54:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.0340","created_at":"2026-05-18T03:54:25Z"},{"alias_kind":"pith_short_12","alias_value":"5UA6SOGT3A7Q","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_16","alias_value":"5UA6SOGT3A7Q24ZS","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_8","alias_value":"5UA6SOGT","created_at":"2026-05-18T12:26:56Z"}],"graph_snapshots":[{"event_id":"sha256:e52b41d30817f4c9015622c923029ccfdcb554a81583a1505f78fb89d10f04b6","target":"graph","created_at":"2026-05-18T03:54:25Z","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":"We show that the Lambert series $f(x)=\\sum d(n) x^n$ is irrational at $x=1/b$ for negative integers $b < -1$ using an elementary proof that finishes an incomplete proof of Erdos.","authors_text":"Joseph Vandehey","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-06-02T03:11:12Z","title":"On an incomplete argument of Erdos on the irrationality of Lambert series"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.0340","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":"sha256:4e9e14a7a3c939f4956a8d112d68f7f9964cd45c3dcc4d8cd25ee62f4e36b60f","target":"record","created_at":"2026-05-18T03:54:25Z","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":"19c13c57eab327edffb35bac1eeefd20eaee7e628e66cb347e977db0480ed782","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-06-02T03:11:12Z","title_canon_sha256":"4b3a71125e596684697d3d5669571f3bd6266ae1e5799eaaa2cb4b45b2fb0f41"},"schema_version":"1.0","source":{"id":"1206.0340","kind":"arxiv","version":1}},"canonical_sha256":"ed01e938d3d83f0d733290ee595a098342aac0fb98fff7b6c58cee9e850e5d0c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ed01e938d3d83f0d733290ee595a098342aac0fb98fff7b6c58cee9e850e5d0c","first_computed_at":"2026-05-18T03:54:25.203181Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:54:25.203181Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YzLmAeI57BKOONOguyoGs+vSYXMwt3agcUg/gUnVhTvAjwCAbyP4u5LahncJz3DlWoXED5xjuWzoQFwWagz5AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:54:25.203630Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.0340","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4e9e14a7a3c939f4956a8d112d68f7f9964cd45c3dcc4d8cd25ee62f4e36b60f","sha256:e52b41d30817f4c9015622c923029ccfdcb554a81583a1505f78fb89d10f04b6"],"state_sha256":"b7778c8e218cb93233fa3bf492071b59f22dc8ff170cf8be360862e6c6dd79b3"}