{"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"}