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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.

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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.

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