{"paper":{"title":"Subword Complexity and (non)-automaticity of certain completely multiplicative functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Yining Hu","submitted_at":"2016-05-30T20:17:27Z","abstract_excerpt":"In this article, we prove that for a completely multiplicative function $f$ from $\\mathbb{N}^*$ to a field $K$ such that the set $$\\{p \\;|\\; f(p)\\neq 1_K \\;\\mbox{and }p \\mbox{ is prime}\\}$$ is finite, the asymptotic subword complexity of $f$ is $\\Theta(n^t)$, where $t$ is the number of primes $p$ that $f(p)\\neq 0_K, 1_K$. This proves in particular that sequences like $((-1)^{v_2(n)+v_3(n)})_n$ are not $k$-automatic for $k\\geq 2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.09403","kind":"arxiv","version":1},"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"}