To get the asymptotic, one can replace the factorial in the exact formula by the Stirling formula. Then the desired quantity can be estimated as follows: $$ c\cdot A_N\le (\dots)\le C\cdot A_N $$ with $$ A_N=N^{\phi(N)}\cdot \prod_{d|N}(\sqrt{2\pi d}\cdot e^{-d})^{\mu(N/d)}=(N/e)^{\phi(N)}. $$
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