On the distribution of the density of maximal order elements in general linear groups

In this paper we consider the density of maximal order elements in $\mathrm{GL}_n(q)$. Fixing any of the rank $n$ of the group, the characteristic $p$ or the degree $r$ of the extension of the underlying field $\mathbb{F}_q$ of size $q=p^r$, we compute the expected value of the said density and establish that it follows a distribution law.