{"paper":{"title":"Probabilistic Proofs of Some Generalized Mertens' Formulas Via Generalized Dickman Distributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Ross G. Pinsky","submitted_at":"2018-09-13T11:13:09Z","abstract_excerpt":"The classical Mertens' formula states that $\n\\prod_{p\\le N}\\big(1-\\frac1p)^{-1}\\sim e^\\gamma\\log N, $ where the product is over all primes $p$ less than or equal to $N$, and $\\gamma$ is the Euler-Mascheroni constant. By the Euler product formula, this is equivalent to either of the following statements: $$ \\begin{aligned} &i. \\lim_{N\\to\\infty}\\frac{\\sum_{n:p|n\\Rightarrow p\\le N}\\thinspace\\frac1n}{\\sum_{n\\le N}\\frac1n}=e^\\gamma\\ \\ &ii. \\sum_{n:p|n\\Rightarrow p\\le N}\\thinspace\\frac1n\\sim e^\\gamma\\log N. \\end{aligned} $$ Via some random integer constructions and a criterion for weak convergence o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.04888","kind":"arxiv","version":3},"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"}