Spectral properties of the M\"{o}bius function and a random M\"{o}bius model

TL;DR

This paper explores the spectral properties of the M"{o}bius function, establishing connections with conjectures in dynamical systems and proving the Sarnak conjecture for a specific random model.

Contribution

It links the M"{o}bius function's spectral measure to Lebesgue measure under Sarnak conjecture assumptions and proves orthogonality results assuming Elliott conjecture, also confirming Sarnak conjecture for a random model.

Findings

Spectral measure of M"{o}bius function is equivalent to Lebesgue measure.

M"{o}bius function is orthogonal to certain dynamical systems under Elliott conjecture.

Sarnak conjecture holds for a particular random model.

Abstract

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.