{"paper":{"title":"The distribution functions of $\\sigma(n)/n$ and $n/\\phi(n)$, II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andreas Weingartner","submitted_at":"2010-11-18T19:35:34Z","abstract_excerpt":"Let $\\sigma(n)$ be the sum of the positive divisors of $n$, and let $A(t)$ be the natural density of the set of positive integers $n$ satisfying $\\sigma(n)/n \\ge t$. We give an improved asymptotic result for $\\log A(t)$ as $t$ grows unbounded. The same result holds if $\\sigma(n)/n$ is replaced by $n/\\phi(n)$, where $\\phi(n)$ is Euler's totient function."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.4262","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"}