{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:KFYDJNV27S4REAIQ5S4BNLSBKK","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":"55ee471a1a92e4949ebbb9a34e3e7847343bf5897032c22cf01183ff01490120","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-05-26T11:34:47Z","title_canon_sha256":"e2496df2059e2f7c193629ad071fa6581f084867b844ba8f94b2f75701426fff"},"schema_version":"1.0","source":{"id":"1305.6015","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.6015","created_at":"2026-05-18T02:58:23Z"},{"alias_kind":"arxiv_version","alias_value":"1305.6015v2","created_at":"2026-05-18T02:58:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.6015","created_at":"2026-05-18T02:58:23Z"},{"alias_kind":"pith_short_12","alias_value":"KFYDJNV27S4R","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"KFYDJNV27S4REAIQ","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"KFYDJNV2","created_at":"2026-05-18T12:27:49Z"}],"graph_snapshots":[{"event_id":"sha256:305647445c4df4819a43b02c0e413a2870a4b169c2028c0f42487d1dc7175e5e","target":"graph","created_at":"2026-05-18T02:58:23Z","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":"Let $F$ be a number field, $k$ a positive integer. In this paper, we define the Mobius and Liouville functions of order $k$ in $F$. We give a formula about the partial sums of them by using elementary number theory and complex analysis. Moreover, we also consider the number of $k$-free ideals of the integer ring of $F$.","authors_text":"Yusuke Fujisawa","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-05-26T11:34:47Z","title":"On Mobius and Liouville functions of order $k$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.6015","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:f0b7bb77f2346532695606923a287b66f24433f27b6ae2ab19fe2577a401dc3b","target":"record","created_at":"2026-05-18T02:58:23Z","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":"55ee471a1a92e4949ebbb9a34e3e7847343bf5897032c22cf01183ff01490120","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-05-26T11:34:47Z","title_canon_sha256":"e2496df2059e2f7c193629ad071fa6581f084867b844ba8f94b2f75701426fff"},"schema_version":"1.0","source":{"id":"1305.6015","kind":"arxiv","version":2}},"canonical_sha256":"517034b6bafcb9120110ecb816ae4152a5bbabdee16154184bca58f660b45e46","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"517034b6bafcb9120110ecb816ae4152a5bbabdee16154184bca58f660b45e46","first_computed_at":"2026-05-18T02:58:23.558894Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:23.558894Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mGVe+pb6rHeM6kdCcSEw2UsCdVAqPElOknwkS180vUVMdkhSOgly0LoBjGPoW7UfEYkFIwXwdVpX70XXbYf6BA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:23.559459Z","signed_message":"canonical_sha256_bytes"},"source_id":"1305.6015","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f0b7bb77f2346532695606923a287b66f24433f27b6ae2ab19fe2577a401dc3b","sha256:305647445c4df4819a43b02c0e413a2870a4b169c2028c0f42487d1dc7175e5e"],"state_sha256":"95e5e8d5a218ad11cac432807e83d689d1b9bb47ee1ce9956bb6874a53ca207d"}