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I failed to find a way how to prove the limit and how to get the approximation below. Actually the formula and the comment were publised at MO with no clear explanations.

In other words I'd like to represent the infinite products of necklaces in a way like below. Any ideas on that?

"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 $n$: $\frac {a^n} {n!} \prod_{p=1}^n \frac {1-a^p} {1-a}-\prod_{p=1}^n N(p,a) \le \frac {a^n} {n!} O(1/(n^{1/2-\epsilon})$

$\frac {(a-1)^{n+1}} {(a-3) \cdot n!} \prod_{p=1}^n \frac {1-a^p} {1-a}$ gives a much better approximation to $\prod_{p=1}^n N(p,a)$ with $n \to \infty$ for $a > 3$ "

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  • $\begingroup$ The limit you talk about doesn't exist; the product diverges. From the second half of the post, it appears that what you're actually interested in is the asymptotic behaviour of the product. $\endgroup$
    – joriki
    Jun 9 '18 at 7:14

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