{"paper":{"title":"A Natural Probabilistic Model on the Integers and its Relation to Dickman-Type Distributions and Buchstab's Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ross G. Pinsky","submitted_at":"2016-06-09T13:59:27Z","abstract_excerpt":"Let $\\{p_j\\}_{j=1}^\\infty$ denote the set of prime numbers in increasing order, let $\\Omega_N\\subset \\mathbb{N}$ denote the set of positive integers with no prime factor larger than $p_N$ and let $P_N$ denote the probability measure on $\\Omega_N$ which gives to each $n\\in\\Omega_N$ a probability proportional to $\\frac1n$. This measure is in fact the distribution of the random integer $I_N\\in\\Omega_N$ defined by $I_N=\\prod_{j=1}^Np_j^{X_{p_j}}$, where $\\{X_{p_j}\\}_{j=1}^\\infty$ are independent random variables and $X_{p_j}$ is distributed as Geom$(1-\\frac1{p_j})$. We show that $\\frac{\\log n}{\\lo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.02965","kind":"arxiv","version":6},"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"}