Phi, Primorials, and Poisson

Abstract

The primorial $p\#$ of a prime $p$ is the product of all primes $q\le p$. Let pr$(n)$ denote the largest prime $p$ with $p\# \mid \phi(n)$, where $\phi$ 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$(n) \sim \log\log n/\log\log\log n$ as $n\to\infty$ 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 $k!\mid \phi(n)$.