{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:SPN2ASOSVUQIBIAL7L4GDOTZXS","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":"f042ac5061a6c0194855fe8eaf7d29830202ecc22422374125354a1a2c04bf33","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-02-02T01:43:57Z","title_canon_sha256":"d433ae25ab479e47a98af1e39f9fc313c677babfa60468c8b909f514a3379d5f"},"schema_version":"1.0","source":{"id":"1802.00531","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.00531","created_at":"2026-05-18T00:24:32Z"},{"alias_kind":"arxiv_version","alias_value":"1802.00531v1","created_at":"2026-05-18T00:24:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.00531","created_at":"2026-05-18T00:24:32Z"},{"alias_kind":"pith_short_12","alias_value":"SPN2ASOSVUQI","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_16","alias_value":"SPN2ASOSVUQIBIAL","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_8","alias_value":"SPN2ASOS","created_at":"2026-05-18T12:32:53Z"}],"graph_snapshots":[{"event_id":"sha256:3da3420fa0be33188213308ecf751609e9783fd6aaee1fee9fe4e4692d0f86cd","target":"graph","created_at":"2026-05-18T00:24:32Z","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":"The classical Menon's identity [7] states that\n  \\begin{equation*}\\label{oldbegin1} \\sum_{\\substack{a\\in\\Bbb Z_n^\\ast }}\\gcd(a -1,n)=\\varphi(n) \\sigma_{0} (n), \\end{equation*} where for a positive integer $n$, $\\Bbb Z_n^\\ast$ is the group of units of the ring $\\Bbb Z_n=\\Bbb Z/n\\Bbb Z$, $\\gcd(\\ ,\\ )$ represents the greatest common divisor, $\\varphi(n)$ is the Euler's totient function and $\\sigma_{k} (n) =\\sum_{d|n } d^{k}$ is the divisor function.\n  In this paper, we generalize Menon's identity with Dirichlet characters in the following way: \\begin{equation*}\n  \\sum_{\\substack{a\\in\\Bbb Z_n^\\ast","authors_text":"Daeyeoul Kim, Xiaoyu Hu, Yan Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-02-02T01:43:57Z","title":"A generalization of Menon's identity with Dirichlet characters"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.00531","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:d075994e0ffa2103c65c104388a1166a9a7e300ea3ac57bd0f3e4b394057496f","target":"record","created_at":"2026-05-18T00:24:32Z","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":"f042ac5061a6c0194855fe8eaf7d29830202ecc22422374125354a1a2c04bf33","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-02-02T01:43:57Z","title_canon_sha256":"d433ae25ab479e47a98af1e39f9fc313c677babfa60468c8b909f514a3379d5f"},"schema_version":"1.0","source":{"id":"1802.00531","kind":"arxiv","version":1}},"canonical_sha256":"93dba049d2ad2080a00bfaf861ba79bc8b7b316c533b0fbacf9ca32071efd760","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"93dba049d2ad2080a00bfaf861ba79bc8b7b316c533b0fbacf9ca32071efd760","first_computed_at":"2026-05-18T00:24:32.945870Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:24:32.945870Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FLzAuY0k1n6BeP67HZIfK3gDB3rGsBso0nKa9vjpfOmf2XxQCMeRzNe8zf4IXcr1BX3XIufrxs1lQBJF5rSJBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:24:32.946280Z","signed_message":"canonical_sha256_bytes"},"source_id":"1802.00531","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d075994e0ffa2103c65c104388a1166a9a7e300ea3ac57bd0f3e4b394057496f","sha256:3da3420fa0be33188213308ecf751609e9783fd6aaee1fee9fe4e4692d0f86cd"],"state_sha256":"b1312344e956025e5104a1368bfee3eeaeda312bf47bf71dba5767694a668472"}