gr.group theory - Maximum Order of elements in $GL(n,Z)$

From the fact you're citing, it looks like

$$ f(n) = max { m : Phi(m) le n }. $$

For example, $Phi(30) = 7$ and $Phi(m) ge 11$ for all $m ge 31$ (note that since $phi(n) ge sqrt{n}$ we only need to check finitely many values!) -- so $f(7) = f(8) = f(9) = f(10) = 30$.

Now, consider the fact that
$$ lim inf phi(n) {log log n over n} = e^{-gamma} $$
which is equation (20) in this Mathworld article. Of course this holds if we replace $phi$ by $Phi$.

So $f(n)$ should grow like the inverse of the function
$$ n to {e^{-gamma} n over log log n} $$. It appears, then, that $f(n) sim e^gamma n log log n$ as $n to infty$.

Unfortunately this disagrees with your estimate. One of us is wrong somewhere.

EDIT: I believe my argument is basically right, but the original fact was stated incorrectly. From the paper of Levitt that Stanley pointed to, we should actually have

$$ Phi( p_1^{alpha_1} cdots p_k^{alpha_k}) = phi(p_1^{alpha_1}) + cdots + phi(p_k^{alpha_k}) - [k equiv 2 mod 4] $$

and so $Phi(x)$ is usually much smaller than $phi(x)$ -- therefore $f$ grows much faster than I said it did.