An elemental Erdős-Kac theorem for algebraic number fields [article]

Paul Pollack
2016 arXiv   pre-print
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 ( |N(α)|)^D and standard deviation proportional to (|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(N(α))^2 + λ/2√(2) (N(α))^3/2 is given by 1/√(2π)∫_-∞^λ e^-t^2/2 dt.
more » ... s further evidence that "irreducibles play a game of chance", we show that the values ν(α) are equidistributed modulo m for every fixed m.
arXiv:1603.05352v1 fatcat:dy3wscucpnealnf74xse4jppiu