Two dynamical perspectives on the randomness of the Mobius function

Abstract

Sarnak's approach to the Mobius randomness heuristic from the standpoint of dynamical

systems is studied in two complementary settings. First, the Mobius function is

realized in the context of symbolic dynamics, and we prove that its associated squarefree

factor has a unique measure of maximal entropy. We then show that the Mobius

function is linearly disjoint from all zero-entropy translations on homogeneous spaces

of connected Lie groups, generalizing work of Vinogradov, Green-Tao, and Bourgain-

Sarnak-Ziegler.