Bounds on the Minkowski constants and a function involving $\varphi$

In 1887, Minkowski determined the least common multiple of the orders of all finite subgroups of $GL_n(\mathbb{Q})$; we refer to this number as $M(n)$. In (Katznelson, 1994), Katznelson provides the asymptotic behaviour of $M(n)$, with a small error term. In this paper, we use elementary techniques to find explicit upper and lower bounds on $M(n)$ that improve on Katznelson's results; we also recover his asymptotic result. Our results immediately imply explicit bounds on functions closely related to $M(n)$, which appear in the study of abelian varieties (see, for example, (Silverberg, 1992), (Guralnick and Kedlaya, 2017) and (Ozeki, 2024)). Finally, we examine the function $\Phi(n)$, which also appears in (Ozeki, 2024), defined as the greatest positive integer $m$ for which $\varphi(m)$ divides $2n$. We provide explicit upper bounds on $\Phi(n)$.