{"paper":{"title":"On the closure of the image of the generalized divisor function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Carlo Sanna","submitted_at":"2015-07-29T10:17:49Z","abstract_excerpt":"For any real number $s$, let $\\sigma_s$ be the generalized divisor function, i.e., the arithmetic function defined by $\\sigma_s(n) := \\sum_{d \\, \\mid \\, n} d^s$, for all positive integers $n$. We prove that for any $r > 1$ the topological closure of $\\sigma_{-r}(\\mathbf{N}^+)$ is the union of a finite number of pairwise disjoint closed intervals $I_1, \\ldots, I_\\ell$. Moreover, for $k=1,\\ldots,\\ell$, we show that the set of positive integers $n$ such that $\\sigma_{-r}(n) \\in I_k$ has a positive rational asymptotic density $d_k$. In fact, we provide a method to give exact closed form expression"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08091","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"}