Phi, primorials, and Poisson

Source abstract

The primorial p # of a prime p is the product of all primes q ≤ p . Let pr ( n ) denote the largest prime p with p # ∣ ϕ ( n ) , where ϕ is Euler’s totient function. We show that the normal order of pr ( n ) is log log n / log log log n ; that is, pr ( n ) ∼ log log n / log log log n as n → ∞ on a set of integers of asymptotic density 1. In fact, we show there is an asymptotic secondary term and, on a tertiary level, there is an asymptotic Poisson distribution. We also show an analogous result for the largest integer k with k ! ∣ ϕ ( n ) .