Spectral properties of the Möbius function and a random Möbius model

Abstract

Assuming Sarnak’s conjecture is true for any singular dynamical process, we prove that the spectral measure of the Möbius function is equivalent to Lebesgue measure. Conversely, under Elliott’s conjecture, we establish that the Möbius function is orthogonal to any uniquely ergodic dynamical system with singular spectrum. Furthermore, using Mirsky’s theorem, we find a new simple proof of Cellarosi–Sinai’s theorem on the orthogonality of the square of the Möbius function with respect to any weakly mixing dynamical system. Finally, we establish Sarnak’s conjecture for a particular random model.