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