P. Sarnak has recently initiated the study of the topological and ergodic dynamics issues of the Möbius function and its square. Furthermore, he state a conjecture on the orthogonality of the Möbius function and the topological dynamical flow with zero topological entropy. He announced that almost surely the Möbius
function is orthogonal to any dynamical sequence, that is, the ergodic average with Möbius weight converge almost surely. He also announced that the well known Chowla conjecture on the correlation of the Möbius implies Sarnak conjecture.

In this talk, I will present a ergodic proof of Sarnak's results. I will further introduce the notion of Mirsky dynamical systems and its generalization.

My talk is based on my recent joint work with Mariusz Lemanzyk (Poland)-Thierry de la Rue (France) and Joanna Kulaga-Przymus (Poland) and my joint work with M. Disertori (Germany).