{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:GLVINKWWW3OAKYY4N7MHAG5QFH","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":"e2915e9ca1cf203b28307f37f9a4aa2aff832d051a9ff4f4633286b5cff8c935","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-01-04T16:06:47Z","title_canon_sha256":"04c63f2d40022f6f1cdee5f601e29fe4a66e1edd2cce29d0c63df145e1d60b75"},"schema_version":"1.0","source":{"id":"1201.0931","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.0931","created_at":"2026-05-18T03:32:57Z"},{"alias_kind":"arxiv_version","alias_value":"1201.0931v3","created_at":"2026-05-18T03:32:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.0931","created_at":"2026-05-18T03:32:57Z"},{"alias_kind":"pith_short_12","alias_value":"GLVINKWWW3OA","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"GLVINKWWW3OAKYY4","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"GLVINKWW","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:35530837c6e4d68a2a04006edf0e275515825d8f02defaa8933c5d6cc4717d19","target":"graph","created_at":"2026-05-18T03:32:57Z","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 $k\\ge 1,a\\ge 1,b\\ge 0$ and $ c\\ge 1$ be integers. Let $f$ be a multiplicative function with $f(n)\\ne 0$ for all positive integers $n$. We define the arithmetic function $g_{k,f}$ for any positive integer $n$ by $g_{k,f}(n):=\\frac{\\prod_{i=0}^k f(b+a(n+ic))} {f({\\rm lcm}_{0\\le i\\le k} \\{b+a(n+ic)\\})}$. We first show that $g_{k,f}$ is periodic and $c {\\rm lcm}(1,...,k)$ is its period. Consequently, we provide a detailed local analysis to the periodic function $g_{k,\\varphi}$, and determine the smallest period of $g_{k,\\varphi}$, where $\\varphi$ is the Euler phi function.","authors_text":"Guoyou Qian, Qianrong Tan, Shaofang Hong","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-01-04T16:06:47Z","title":"On the periodicity of a class of arithmetic functions associated with multiplicative functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.0931","kind":"arxiv","version":3},"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":"sha256:921534e2832bf14d617d985e06442be64fb9c8b9ccbffefed673c69777b7e800","target":"record","created_at":"2026-05-18T03:32:57Z","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":"e2915e9ca1cf203b28307f37f9a4aa2aff832d051a9ff4f4633286b5cff8c935","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-01-04T16:06:47Z","title_canon_sha256":"04c63f2d40022f6f1cdee5f601e29fe4a66e1edd2cce29d0c63df145e1d60b75"},"schema_version":"1.0","source":{"id":"1201.0931","kind":"arxiv","version":3}},"canonical_sha256":"32ea86aad6b6dc05631c6fd8701bb029e1c26aa388db4addfb34dd5461a0472e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"32ea86aad6b6dc05631c6fd8701bb029e1c26aa388db4addfb34dd5461a0472e","first_computed_at":"2026-05-18T03:32:57.196471Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:32:57.196471Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bIXF9OZIQ4TULPuBfP/fb2WPsbPm0bkemJ4iz7vD1nA5F2fvqUUbGBx3+2XPPpQnxxyde3yWZg2Wyqo+TP9kCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:32:57.197164Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.0931","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:921534e2832bf14d617d985e06442be64fb9c8b9ccbffefed673c69777b7e800","sha256:35530837c6e4d68a2a04006edf0e275515825d8f02defaa8933c5d6cc4717d19"],"state_sha256":"9c193c5a0003528f4a4bcf243f48d3d59ada14ef8703791e01bd2b1dcd5ea557"}