Phi, primorials, and Poisson

Abstract

The primorial $p\#$ of a prime $p$ is the product of all primes $q\le p$. Let $\mathrm{pr}\left(n\right)$ denote the largest prime $p$ with $p\#\mid \varphi \left(n\right)$, where $\varphi$ is Euler’s totient function. We show that the normal order of $\mathrm{pr}\left(n\right)$ is $loglogn/logloglogn$; that is, $\mathrm{pr}\left(n\right)\sim loglogn/logloglogn$ 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 $k!\mid \varphi \left(n\right)$.