{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:MVRF4ZD37QQRAFW6AANYS2VQ3Z","short_pith_number":"pith:MVRF4ZD3","canonical_record":{"source":{"id":"1011.0019","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-10-29T21:14:11Z","cross_cats_sorted":[],"title_canon_sha256":"41bcab9ee9a1c297ec10683ccb4162f45a2ab6891ee9582871bde0b0bebfa4ad","abstract_canon_sha256":"c3a1399a3a1c82f7c040012b18c9d1fe7bf20560733d45e29dab9c69e721dfbf"},"schema_version":"1.0"},"canonical_sha256":"65625e647bfc211016de001b896ab0de42e395374bd662a28b01d4f93375564d","source":{"kind":"arxiv","id":"1011.0019","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.0019","created_at":"2026-05-18T02:58:01Z"},{"alias_kind":"arxiv_version","alias_value":"1011.0019v4","created_at":"2026-05-18T02:58:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.0019","created_at":"2026-05-18T02:58:01Z"},{"alias_kind":"pith_short_12","alias_value":"MVRF4ZD37QQR","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"MVRF4ZD37QQRAFW6","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"MVRF4ZD3","created_at":"2026-05-18T12:26:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:MVRF4ZD37QQRAFW6AANYS2VQ3Z","target":"record","payload":{"canonical_record":{"source":{"id":"1011.0019","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-10-29T21:14:11Z","cross_cats_sorted":[],"title_canon_sha256":"41bcab9ee9a1c297ec10683ccb4162f45a2ab6891ee9582871bde0b0bebfa4ad","abstract_canon_sha256":"c3a1399a3a1c82f7c040012b18c9d1fe7bf20560733d45e29dab9c69e721dfbf"},"schema_version":"1.0"},"canonical_sha256":"65625e647bfc211016de001b896ab0de42e395374bd662a28b01d4f93375564d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:01.602015Z","signature_b64":"vaJ0IkJJEmpwD0dvj4nfkHornsZIQpuj35sAnal5gxQwpA+uaXHcf2Vr7d8mh+aLDlaRtQ9y6e8Nq+208Qv7DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"65625e647bfc211016de001b896ab0de42e395374bd662a28b01d4f93375564d","last_reissued_at":"2026-05-18T02:58:01.601376Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:01.601376Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1011.0019","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:58:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M0sUaTk4uLq0gcD9uvztli5gP/aKMvVdZG0Lk5pb0A1SEsAM1+IakeqgKzYxPXRP/IeNzixgyO85XhjvTZdyBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T09:41:54.615139Z"},"content_sha256":"10ffaf3f1dad4504909f550856489bdc075d85aeaef26e0b1e7791b6cd22fc9e","schema_version":"1.0","event_id":"sha256:10ffaf3f1dad4504909f550856489bdc075d85aeaef26e0b1e7791b6cd22fc9e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:MVRF4ZD37QQRAFW6AANYS2VQ3Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Correlations of the divisor function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Lilian Matthiesen","submitted_at":"2010-10-29T21:14:11Z","abstract_excerpt":"In this paper we study linear correlations of the divisor function tau(n) = sum_{d|n} 1 using methods developed by Green and Tao. For example, we obtain an asymptotic for sum_{n,d} tau(n) tau(n+d) ... tau(n+ (k-1)d)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.0019","kind":"arxiv","version":4},"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":"2026-05-18T02:58:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OG4NHqUcNg1ROyj1Rtuz/tpQVD+PlWhY5AtEu3xnDhDm0FlLGk47MPYfjZdrDFUVfj6g/nThE7/0rZIxyRzMAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T09:41:54.615765Z"},"content_sha256":"f9e035db796157bfa494d94082c9b9b4fc54a1b5847c519b680a1672c3572829","schema_version":"1.0","event_id":"sha256:f9e035db796157bfa494d94082c9b9b4fc54a1b5847c519b680a1672c3572829"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MVRF4ZD37QQRAFW6AANYS2VQ3Z/bundle.json","state_url":"https://pith.science/pith/MVRF4ZD37QQRAFW6AANYS2VQ3Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MVRF4ZD37QQRAFW6AANYS2VQ3Z/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":"2026-07-01T09:41:54Z","links":{"resolver":"https://pith.science/pith/MVRF4ZD37QQRAFW6AANYS2VQ3Z","bundle":"https://pith.science/pith/MVRF4ZD37QQRAFW6AANYS2VQ3Z/bundle.json","state":"https://pith.science/pith/MVRF4ZD37QQRAFW6AANYS2VQ3Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MVRF4ZD37QQRAFW6AANYS2VQ3Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:MVRF4ZD37QQRAFW6AANYS2VQ3Z","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":"c3a1399a3a1c82f7c040012b18c9d1fe7bf20560733d45e29dab9c69e721dfbf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-10-29T21:14:11Z","title_canon_sha256":"41bcab9ee9a1c297ec10683ccb4162f45a2ab6891ee9582871bde0b0bebfa4ad"},"schema_version":"1.0","source":{"id":"1011.0019","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.0019","created_at":"2026-05-18T02:58:01Z"},{"alias_kind":"arxiv_version","alias_value":"1011.0019v4","created_at":"2026-05-18T02:58:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.0019","created_at":"2026-05-18T02:58:01Z"},{"alias_kind":"pith_short_12","alias_value":"MVRF4ZD37QQR","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"MVRF4ZD37QQRAFW6","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"MVRF4ZD3","created_at":"2026-05-18T12:26:10Z"}],"graph_snapshots":[{"event_id":"sha256:f9e035db796157bfa494d94082c9b9b4fc54a1b5847c519b680a1672c3572829","target":"graph","created_at":"2026-05-18T02:58:01Z","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 paper we study linear correlations of the divisor function tau(n) = sum_{d|n} 1 using methods developed by Green and Tao. For example, we obtain an asymptotic for sum_{n,d} tau(n) tau(n+d) ... tau(n+ (k-1)d).","authors_text":"Lilian Matthiesen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-10-29T21:14:11Z","title":"Correlations of the divisor function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.0019","kind":"arxiv","version":4},"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":"sha256:10ffaf3f1dad4504909f550856489bdc075d85aeaef26e0b1e7791b6cd22fc9e","target":"record","created_at":"2026-05-18T02:58:01Z","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":"c3a1399a3a1c82f7c040012b18c9d1fe7bf20560733d45e29dab9c69e721dfbf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-10-29T21:14:11Z","title_canon_sha256":"41bcab9ee9a1c297ec10683ccb4162f45a2ab6891ee9582871bde0b0bebfa4ad"},"schema_version":"1.0","source":{"id":"1011.0019","kind":"arxiv","version":4}},"canonical_sha256":"65625e647bfc211016de001b896ab0de42e395374bd662a28b01d4f93375564d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"65625e647bfc211016de001b896ab0de42e395374bd662a28b01d4f93375564d","first_computed_at":"2026-05-18T02:58:01.601376Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:01.601376Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vaJ0IkJJEmpwD0dvj4nfkHornsZIQpuj35sAnal5gxQwpA+uaXHcf2Vr7d8mh+aLDlaRtQ9y6e8Nq+208Qv7DA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:01.602015Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.0019","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:10ffaf3f1dad4504909f550856489bdc075d85aeaef26e0b1e7791b6cd22fc9e","sha256:f9e035db796157bfa494d94082c9b9b4fc54a1b5847c519b680a1672c3572829"],"state_sha256":"9ba81147337c5705fb1e008a659d4b0ca74105d44bda24ba9f4d534fe7c47e84"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I2rZ19wxYthxzynhfAPrhYw0CKG6Hru1QvVAAXiMuLgP/zSHHgrnhjrkkbSUm+LXioF/e3G3CXdP1Rf72u1DCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T09:41:54.620379Z","bundle_sha256":"14f75d60d1735da036467f3cff4b2e97f148cebc57ab1bfa317030dcebd5c369"}}