  Assuming Sarnak conjecture is true for any singular dynamical process, we
prove that the spectral measure of the M\"{o}bius function is equivalent to
Lebesgue measure. Conversely, under Elliott conjecture, we establish that the
M\"{o}bius function is orthogonal to any uniquely ergodic dynamical system with
singular spectrum. Furthermore, using Mirsky Theorem, we find a new simple
proof of Cellarosi-Sinai Theorem on the orthogonality of the square of the
M\"{o}bius function with respect to any weakly mixing dynamical system.
Finally, we establish Sarnak conjecture for a particular random model.
