Maximal order of elements of $\textrm{SL}(n, \mathbb{Z})$ In the case of $\textrm{SL}(2, \mathbb{Z})$, I know that the order of any finite order matrix in this group is at most $6$. This follows from the fact that $\textrm{SL}(2, \mathbb{Z}) \cong \textrm{Mod}(S_{1})$, the mapping class group of the torus. Furthermore, finite order elements of the mapping class group of a surface have order at most $4g + 2$ by a theorem of Wang. This bound is also attained, for example in this case by the matrix {{1, -1}, {1, 0}}. I would like to know if similar bounds hold when $n > 2$, that is:

What is the maximal order of a finite order matrix in $\textrm{SL}(n, \mathbb{Z})$ when $n > 2$?

 A: There is the following result (Landau's estimate):
Proposition Let $G(n)$ be the maximal order of a torsion element in $GL(n,\mathbb{Z})$. Then 
$$
\log G(n)\sim (n\log(n))^{1/2}.
$$
For a proof see here.
For example $G(2)=G(3)=6$, $G(6)=30$, or $G(12)=210$.
A: EDIT the $\phi$'s in the lcm were out of place, and there is a slight caveat in that I'm not sure the companion matrices for the cyclotomic polynomials have positive determinant, it depends on the constant coefficient of $\Phi_m$, so this formula works for $\mathrm{GL}$. It's also exactly the same formula and reasoning as in the article Dietrich Burde links to.

Take a finite order matrix $g$ in $\mathrm{SL}(n,\Bbb Z)$. It is diagonalisable over $\Bbb C$, and its complex eigenvalues are roots of unity. Its minimal polynomial $\mu$ is a unitary polynomial with rational coefficients and divides the characteristic polynomial, which is a polynomial with integer coefficients, hence $\mu$ actually has integer coefficients and is a product of distinct cyclotomic polynomials (because of its roots and having integer coefficients).
Each cyclotomic polynomial $\Phi_m$ is the minimal polynomial of its companion matrix $C_m\in\mathrm{SL}(\phi(m),\Bbb Z)$, which is integer valued and has finite order (because of diagonalisable with roots of unity as roots) equal to $\phi(m)$. You can create a matrix $\in\mathrm{SL}(n,\Bbb Z)$ as a bloc matrix using companion matrices, so the answer to the question ought to be
$$\max_{k_1,\dots,k_r\text{ s.t. }\phi(k_1)+\dots+\phi(k_r)\leq n}\,\mathrm{lcm}(k_1,\dots,k_r)$$
One could carry out the calculations for small $n$.
