The intermediate prime divisors of integers

Abstract

Let p 1 > p 2 > ⋯ > p ω {p_1} > {p_2} > \cdots > {p_\omega } be the distinct prime divisors of the integer n n . If ω = ω ( n ) → + ∞ \omega = \omega (n) \to + \infty with n n , then p j {p_j} is called an intermediate prime divisor of n n if both j j and ω − j \omega - j tend to infinity with n n . We show that log ⁡ log ⁡ p j \log \log {p_j} , as j j goes through the indices for which p j {p_j} is intermediate, forms a limiting Poisson process in the sense of natural density.