An elemental Erdős-Kac theorem for algebraic number fields

2016/03/17 by Paul Pollack, Pollack, Paul
Mathematics · #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #Rings, Modules, and Algebras #math.NT

paper · pdf · doi:10.48550/arxiv.1603.05352

15 pages

arxiv created 2016/03/17 · openalex publication_date 2016/03/17 · arxiv updated 2016/03/18 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28

Abstract

Fix a number field K. For each nonzero α∈ ℤ_K, let ν(α) denote the number of distinct, nonassociate irreducible divisors of α. We show that ν(α) is normally distributed with mean proportional to (loglog |N(α)|)^D and standard deviation proportional to (loglog|N(α)|)^D-1/2. Here D, as well as the constants of proportionality, depend only on the class group of K. For example, for each fixed real λ, the proportion of α∈ ℤ[√(-5)] with ν(α) ≤ (1)/(8)(loglogN(α))² + \fracλ2√(2) (loglogN(α))^3/2 is given by (1)/(√(2π)) ∫_-∞^λ e^-t2/2 dt. As further evidence that "irreducibles play a game of chance", we show that the values ν(α) are equidistributed modulo m for every fixed m.

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