{"paper":{"title":"Divisor-sum fibers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Carl Pomerance, Lola Thompson, Paul Pollack","submitted_at":"2017-06-09T20:47:32Z","abstract_excerpt":"Let $s(\\cdot)$ denote the sum-of-proper-divisors function, that is, $s(n) = \\sum_{d\\mid n,~d<n}d$. Erd\\H{o}s-Granville-Pomerance-Spiro conjectured that for any set $\\mathcal{A}$ of asymptotic density zero, the preimage set $s^{-1}(\\mathcal{A})$ also has density zero. We prove a weak form of this conjecture: If $\\epsilon(x)$ is any function tending to $0$ as $x\\to\\infty$, and $\\mathcal{A}$ is a set of integers of cardinality at most $x^{\\frac12+\\epsilon(x)}$, then the number of integers $n\\le x$ with $s(n) \\in \\mathcal{A}$ is $o(x)$, as $x\\to\\infty$. In particular, the EGPS conjecture holds for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.03120","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"}