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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1706.03120","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-06-09T20:47:32Z","cross_cats_sorted":[],"title_canon_sha256":"d297b17afd3612143577c16b53cfdcee2785c1e0b3182f93905b981695194e96","abstract_canon_sha256":"2a29993ec9d7085c3e29d16b57ecd9d66eb4b86841965f55fe10911c262888c4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:18:53.717803Z","signature_b64":"UnJhGk13QTGU7G4jDDNDGKkrGblEMoxe50WKomfR+XyEHo7GsbyeoBUVnEt2+8HERECPvL6AeBQHW45Ox7xYCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28f7e2e3e7f090a25c0e64342d19a98d7074ff2eab211244cffbe31569d6c4cd","last_reissued_at":"2026-05-18T00:18:53.717294Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:18:53.717294Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $x\\to\\infty$. 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