Bounds on the Minkowski constants and a function involving $\varphi$
Giulia Pelizzari, James Punch
TL;DR
This paper provides explicit, computable bounds for the Minkowski constant $M(n)$ and related arithmetic-geometric functions that arise in abelian-variety contexts. It uses elementary, prime-sum estimates, Legendre's formula, and standard bounds on primes to obtain two-sided bounds for the error term $E(n)=\log M(n)-\log n!-K n$ and to translate these into explicit bounds for $M(n)$, $G(n)$ and $H(n)$; it also analyzes the function $\Phi(n)=\max\{m: \varphi(m) \mid 2n\}$ and furnishes explicit applications to prior work. The main contributions include explicit bounds for $E(n)$ and the asymptotic form $\log M(n)= n\log n + n(K-1) + O(\sqrt{n}\log n)$, best-possible constants for the $M(n)$, $G(n)$, and $H(n)$ bounds, and tight estimates for $\Phi(n)$, including the odd-case bound $\Phi(n)\le 6n$ and the large-$n$ bound $\Phi(n) \sim 2e^{\gamma} n \log\log n$ for even $n$. These results refine prior work of Katznelson, Silverberg and Ozeki and provide explicit, computable constants for arithmetic-geometry applications.
Abstract
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 $Φ(n)$, 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 $Φ(n)$.