Let us consider the $\lim\_{n\to \infty}\prod\_{p=1}^n N(p,a)$, where the number of fixed necklaces of length n composed of a types of beads $N(n,a)$ can be calculated via totient function: <http://mathworld.wolfram.com/Necklace.html>


It is possible to show that for large $n$:
$\frac {a^n} {n!} \prod\_{p=1}^n \frac {1-a^p} {1-a}-\prod\_{p=1}^n N(p,a) \le \frac {a^n} {n!} O(1/(n^{1/2-\epsilon})$


Can we improve the error of the approximation?
Thank you for any hints.



Error term in formula for products of necklaces
Just a minor note: from what I have found $\frac {(a-1)^{n+1}} {(a-3) \cdot n!} \prod\_{p=1}^n \frac {1-a^p} {1-a}$ gives a much better approximation to $\prod\_{p=1}^n N(p,a)$ with $n \to \infty$ for $a > 3$, though the exact form of the error estimation is not clear.


