{"paper":{"title":"Primitive sets with large counting functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Carl Pomerance, Greg Martin","submitted_at":"2010-09-06T10:34:26Z","abstract_excerpt":"A set of positive integers is said to be primitive if no element of the set is a multiple of another. If $S$ is a primitive set and $S(x)$ is the number of elements of $S$ not exceeding $x$, then a result of Erd\\H os implies that $\\int_2^\\infty (S(t)/t^2\\log t) dt$ converges. We establish an approximate converse to this theorem, showing that if $F$ satisfies some mild conditions and $\\int_2^\\infty (F(t)/t^2\\log t) dt$ converges, then there exists a primitive set $S$ with $S(x) \\gg F(x)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.1014","kind":"arxiv","version":2},"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"}