# Lemma for fixed necklaces at infinity

The number of fixed necklaces of length $n$ with $a$ types of beads is

$$N(n,a)=\frac1n\sum_{d|n}\phi(d)a^{n/d}\;.$$ http://mathworld.wolfram.com/Necklace.html

1) I am looking for how to prove a formula (Lemma) which represents the case for $n \to \infty$ like this: $\Pi_{p=1}^{n}N(n,a) \approx \frac {a^n} {n!} \Pi_{p=1}^{n} \frac {1-a^p} {1-a}$

2) What is the error of the approximation? Thank you in advance.

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