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 $p\#\mid \ddot{I}•(n)$, where $\ddot{I}•$ 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 $k!\mid \ddot{I}•(n)$.

10 pages