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