{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:DGZ6NTAGIVCJSLM7KZZJTTULQC","short_pith_number":"pith:DGZ6NTAG","canonical_record":{"source":{"id":"2108.01822","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2021-08-04T02:47:44Z","cross_cats_sorted":[],"title_canon_sha256":"3858d3b0fba791e78845e593b3d6dec9dfb90be687d35cb6f168c390786608da","abstract_canon_sha256":"e30d5e1694899a32ee11fad0fa4dc7d4eb4cd0b21ebe312784b27e9b3bc8c01a"},"schema_version":"1.0"},"canonical_sha256":"19b3e6cc064544992d9f567299ce8b80b9802a648106af17cf16c635df919825","source":{"kind":"arxiv","id":"2108.01822","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2108.01822","created_at":"2026-07-05T03:03:11Z"},{"alias_kind":"arxiv_version","alias_value":"2108.01822v1","created_at":"2026-07-05T03:03:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.01822","created_at":"2026-07-05T03:03:11Z"},{"alias_kind":"pith_short_12","alias_value":"DGZ6NTAGIVCJ","created_at":"2026-07-05T03:03:11Z"},{"alias_kind":"pith_short_16","alias_value":"DGZ6NTAGIVCJSLM7","created_at":"2026-07-05T03:03:11Z"},{"alias_kind":"pith_short_8","alias_value":"DGZ6NTAG","created_at":"2026-07-05T03:03:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:DGZ6NTAGIVCJSLM7KZZJTTULQC","target":"record","payload":{"canonical_record":{"source":{"id":"2108.01822","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2021-08-04T02:47:44Z","cross_cats_sorted":[],"title_canon_sha256":"3858d3b0fba791e78845e593b3d6dec9dfb90be687d35cb6f168c390786608da","abstract_canon_sha256":"e30d5e1694899a32ee11fad0fa4dc7d4eb4cd0b21ebe312784b27e9b3bc8c01a"},"schema_version":"1.0"},"canonical_sha256":"19b3e6cc064544992d9f567299ce8b80b9802a648106af17cf16c635df919825","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:03:11.219612Z","signature_b64":"DTwq05DhvaG3GsM4xCmtR6scM4u31k11PDAPHTgrzCSy1WHeKjvg0IIGoMbjaahQQdHzBWsCES5oQheFJqybCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"19b3e6cc064544992d9f567299ce8b80b9802a648106af17cf16c635df919825","last_reissued_at":"2026-07-05T03:03:11.219167Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:03:11.219167Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2108.01822","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":"2026-07-05T03:03:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cCDMIAzrABp/GWOEswe7+UEbQrVdJku7K6mzHk7hyAoZENY/B/pkcJ66cVKiOnUKZ776bZXTowBh5L7zOdDqAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T16:08:18.045331Z"},"content_sha256":"90b7611f978f3bb353e82c1a9bcaf65d658f33569fb02a066cd4cb394c017ead","schema_version":"1.0","event_id":"sha256:90b7611f978f3bb353e82c1a9bcaf65d658f33569fb02a066cd4cb394c017ead"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:DGZ6NTAGIVCJSLM7KZZJTTULQC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"M\\\"obius functions of higher rank and Dirichlet series","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Masato Kobayashi","submitted_at":"2021-08-04T02:47:44Z","abstract_excerpt":"We introduce M\\\"obius functions of higher rank, a new class of arithmetic functions so that the classical M\\\"obius function is of rank 2. With this idea, we evaluate Dirichlet series on the sum of the reciprocal square of all $r$-free numbers. For the proof, the Riemann zeta function and cyclotomic polynomials play a key role."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.01822","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2108.01822/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":"2026-07-05T03:03:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QJPTmufJHzHq16MkZIeNtEkvDFrZGNoi9D0dlkOwKV0tKRYJ0rMyL9oO0R0D9eKbdfJf0zXSAv202tqXTZnRDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T16:08:18.046009Z"},"content_sha256":"3122aff0e29b0155fb79bf6911b8f792435b1e01b05930b018d3193e946b9684","schema_version":"1.0","event_id":"sha256:3122aff0e29b0155fb79bf6911b8f792435b1e01b05930b018d3193e946b9684"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DGZ6NTAGIVCJSLM7KZZJTTULQC/bundle.json","state_url":"https://pith.science/pith/DGZ6NTAGIVCJSLM7KZZJTTULQC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DGZ6NTAGIVCJSLM7KZZJTTULQC/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":"2026-08-06T16:08:18Z","links":{"resolver":"https://pith.science/pith/DGZ6NTAGIVCJSLM7KZZJTTULQC","bundle":"https://pith.science/pith/DGZ6NTAGIVCJSLM7KZZJTTULQC/bundle.json","state":"https://pith.science/pith/DGZ6NTAGIVCJSLM7KZZJTTULQC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DGZ6NTAGIVCJSLM7KZZJTTULQC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:DGZ6NTAGIVCJSLM7KZZJTTULQC","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":"e30d5e1694899a32ee11fad0fa4dc7d4eb4cd0b21ebe312784b27e9b3bc8c01a","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2021-08-04T02:47:44Z","title_canon_sha256":"3858d3b0fba791e78845e593b3d6dec9dfb90be687d35cb6f168c390786608da"},"schema_version":"1.0","source":{"id":"2108.01822","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2108.01822","created_at":"2026-07-05T03:03:11Z"},{"alias_kind":"arxiv_version","alias_value":"2108.01822v1","created_at":"2026-07-05T03:03:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.01822","created_at":"2026-07-05T03:03:11Z"},{"alias_kind":"pith_short_12","alias_value":"DGZ6NTAGIVCJ","created_at":"2026-07-05T03:03:11Z"},{"alias_kind":"pith_short_16","alias_value":"DGZ6NTAGIVCJSLM7","created_at":"2026-07-05T03:03:11Z"},{"alias_kind":"pith_short_8","alias_value":"DGZ6NTAG","created_at":"2026-07-05T03:03:11Z"}],"graph_snapshots":[{"event_id":"sha256:3122aff0e29b0155fb79bf6911b8f792435b1e01b05930b018d3193e946b9684","target":"graph","created_at":"2026-07-05T03:03:11Z","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/2108.01822/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce M\\\"obius functions of higher rank, a new class of arithmetic functions so that the classical M\\\"obius function is of rank 2. With this idea, we evaluate Dirichlet series on the sum of the reciprocal square of all $r$-free numbers. For the proof, the Riemann zeta function and cyclotomic polynomials play a key role.","authors_text":"Masato Kobayashi","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2021-08-04T02:47:44Z","title":"M\\\"obius functions of higher rank and Dirichlet series"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.01822","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":"sha256:90b7611f978f3bb353e82c1a9bcaf65d658f33569fb02a066cd4cb394c017ead","target":"record","created_at":"2026-07-05T03:03:11Z","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":"e30d5e1694899a32ee11fad0fa4dc7d4eb4cd0b21ebe312784b27e9b3bc8c01a","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2021-08-04T02:47:44Z","title_canon_sha256":"3858d3b0fba791e78845e593b3d6dec9dfb90be687d35cb6f168c390786608da"},"schema_version":"1.0","source":{"id":"2108.01822","kind":"arxiv","version":1}},"canonical_sha256":"19b3e6cc064544992d9f567299ce8b80b9802a648106af17cf16c635df919825","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"19b3e6cc064544992d9f567299ce8b80b9802a648106af17cf16c635df919825","first_computed_at":"2026-07-05T03:03:11.219167Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:03:11.219167Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DTwq05DhvaG3GsM4xCmtR6scM4u31k11PDAPHTgrzCSy1WHeKjvg0IIGoMbjaahQQdHzBWsCES5oQheFJqybCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T03:03:11.219612Z","signed_message":"canonical_sha256_bytes"},"source_id":"2108.01822","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:90b7611f978f3bb353e82c1a9bcaf65d658f33569fb02a066cd4cb394c017ead","sha256:3122aff0e29b0155fb79bf6911b8f792435b1e01b05930b018d3193e946b9684"],"state_sha256":"bf2394c52e5a251ff3fb9e86e4cf5b5775abf81e60a39ef465ba77406515e7b4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mQalKoKAD24mQ+g8QoOIET2oMYuIFw6mkaD0ONj0itsPWiKFrzapagaKZE4LakPvhCVx7NuNgF0xmMtnfL8lAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T16:08:18.052372Z","bundle_sha256":"e6a587c8338cf71be97c294a6ae95f6fcf21d926518cd5ed4313800a98ec945d"}}