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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1201.0931","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-01-04T16:06:47Z","cross_cats_sorted":[],"title_canon_sha256":"04c63f2d40022f6f1cdee5f601e29fe4a66e1edd2cce29d0c63df145e1d60b75","abstract_canon_sha256":"e2915e9ca1cf203b28307f37f9a4aa2aff832d051a9ff4f4633286b5cff8c935"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:32:57.197164Z","signature_b64":"bIXF9OZIQ4TULPuBfP/fb2WPsbPm0bkemJ4iz7vD1nA5F2fvqUUbGBx3+2XPPpQnxxyde3yWZg2Wyqo+TP9kCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"32ea86aad6b6dc05631c6fd8701bb029e1c26aa388db4addfb34dd5461a0472e","last_reissued_at":"2026-05-18T03:32:57.196471Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:32:57.196471Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the periodicity of a class of arithmetic functions associated with multiplicative functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Guoyou Qian, Qianrong Tan, Shaofang Hong","submitted_at":"2012-01-04T16:06:47Z","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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.0931","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1201.0931","created_at":"2026-05-18T03:32:57.196580+00:00"},{"alias_kind":"arxiv_version","alias_value":"1201.0931v3","created_at":"2026-05-18T03:32:57.196580+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.0931","created_at":"2026-05-18T03:32:57.196580+00:00"},{"alias_kind":"pith_short_12","alias_value":"GLVINKWWW3OA","created_at":"2026-05-18T12:27:06.952714+00:00"},{"alias_kind":"pith_short_16","alias_value":"GLVINKWWW3OAKYY4","created_at":"2026-05-18T12:27:06.952714+00:00"},{"alias_kind":"pith_short_8","alias_value":"GLVINKWW","created_at":"2026-05-18T12:27:06.952714+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GLVINKWWW3OAKYY4N7MHAG5QFH","json":"https://pith.science/pith/GLVINKWWW3OAKYY4N7MHAG5QFH.json","graph_json":"https://pith.science/api/pith-number/GLVINKWWW3OAKYY4N7MHAG5QFH/graph.json","events_json":"https://pith.science/api/pith-number/GLVINKWWW3OAKYY4N7MHAG5QFH/events.json","paper":"https://pith.science/paper/GLVINKWW"},"agent_actions":{"view_html":"https://pith.science/pith/GLVINKWWW3OAKYY4N7MHAG5QFH","download_json":"https://pith.science/pith/GLVINKWWW3OAKYY4N7MHAG5QFH.json","view_paper":"https://pith.science/paper/GLVINKWW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1201.0931&json=true","fetch_graph":"https://pith.science/api/pith-number/GLVINKWWW3OAKYY4N7MHAG5QFH/graph.json","fetch_events":"https://pith.science/api/pith-number/GLVINKWWW3OAKYY4N7MHAG5QFH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GLVINKWWW3OAKYY4N7MHAG5QFH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GLVINKWWW3OAKYY4N7MHAG5QFH/action/storage_attestation","attest_author":"https://pith.science/pith/GLVINKWWW3OAKYY4N7MHAG5QFH/action/author_attestation","sign_citation":"https://pith.science/pith/GLVINKWWW3OAKYY4N7MHAG5QFH/action/citation_signature","submit_replication":"https://pith.science/pith/GLVINKWWW3OAKYY4N7MHAG5QFH/action/replication_record"}},"created_at":"2026-05-18T03:32:57.196580+00:00","updated_at":"2026-05-18T03:32:57.196580+00:00"}