论文标题
PHI，原始人和泊松
Phi, Primorials, and Poisson
论文作者
论文摘要
Prime $ P $的原始$ P \＃$是所有Primes $ Q \ le P $的产品。令Pr $（n）$表示最大的Prime $ p $，其中$ p \＃\ mid ϕ（n）$，其中$ ϕ $是Euler的主机函数。我们表明，pr $（n）$的正常顺序是$ \ log \ log n/\ log \ log \ log \ log n $。也就是说，pr $（n）\ sim \ log \ log n/\ log \ log \ log \ log n $ as $ n \ to to \ to \ to \ infty $在一组渐近密度的整数上1。我们还显示了最大的整数$ k $，带有$ k！\ mid ϕ（n）$的结果。
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 $p\# \mid ϕ(n)$, 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$(n) \sim \log\log n/\log\log\log n$ 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 $k!\mid ϕ(n)$.
