2015/11/25 by Alexander P. Mangerel, Mangerel, Alexander P.
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1511.08038
38 pages
openalex publication_date 2015/11/25 · arxiv created 2015/12/11 · arxiv updated 2015/12/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let E0,…,En be a partition of the set of prime numbers, and define Ej(x) := ∑p ∈ Ej \atop p ≤ x (1)/(p). Define π(x;E,k) to be the number of integers n ≤ x with kj prime factors in Ej for each j. Basic probabilistic heuristics suggest that x-1π(x;E,k), modelled as the distribution function of a random variable, should satisfy a joint Poisson law with parameter vector (E0(x),…,En(x)), as x → ∞. We prove an asymptotic formula for π(x;E,k) which contradicts these heuristics in the case that for each j, Ej(x)2 ≤ kj ≤ log(2)/(3)-ε x for each j under mild hypotheses. As a particular application, we prove an asymptotic formula regarding integers with prime factors from specific arithmetic progressions, which generalizes a result due to Delange.