Möbius Randomness Law from the viewpoint of Chowla’s and Sarnak’s conjectures

The Liouville function assigns the value $+1$ to $n$ if the number of prime factors of $n$ counted with multiplicities is even and $-1$ if not. The Möbius function coincides with the Liouville function on the set of square-free numbers and assigns the value zero otherwise. Chowla’s conjecture predicts that the Liouville function is normal and Sarnak's conjecture asserts that all dynamical systems with zero topological entropy satisfy the Möbius randomness law. In my talk, I will survey some recent work on Sarnak’s conjecture and Chowla’s conjecture. I will also present my recent work on the connections between these two conjectures. In fact, based on Veech’s proof of Sarnak’s theorem on the dynamical Möbius flow and the Logarithmic theorem on Chowla and Sarnak’s conjectures due to Tao, I will present the ingredients of the proof of the fact that these two conjectures are equivalent.