TITLE: Maximum Order of elements in $GL(n,Z)$
QUESTION [8 upvotes]: Hi,
I know that $\mathrm{GL}(n,\mathbb{Z})$ has an element of order $m$ iff $\Phi(m)\leq n$, where $\Phi(m) = \varphi(m)$ if $p_1^{\alpha_1}\neq 2$ or $m=2$, $\Phi(m) = \varphi(m)-1$ if $p_1^{\alpha_1}= 2$ or $m\not=2$, and $\varphi$ is Euler totient.
From there I can show that the maximum order of a element of finite order in $\mathrm{GL}(n,\mathbb{Z})$, $f(n)$ statisfies $\displaystyle \lim_{n\to \infty} \frac{\ln f(n)}{\sqrt{n\ln n}} =1$.
Here is my question:
Can we find the asymptotic behavior of $f(n)$ (and not $\ln \bigl(f(n)\bigr)$)?

REPLY [5 votes]: See MR1655470 (99m:20111) on MathSciNet.