Sarnak has recently initiated the study of the Möbius function and its square, the characteristic function of square-free integers, from a dynamical point of view, introducing the Möbius flow and the square-free flow as the action of the shift map on the respective subshfits generated by these functions. In this paper, we extend the study of the square-free flow to the more general context of B-free integers, that is to say integers with no factor in a given family B of pairwise relatively prime integers, the sum of whose reciprocals is finite. Relying on dynamical arguments, we prove in particular that the distribution of patterns in the characteristic function of the B-free integers follows a shift-invariant probability measure, and gives rise to a measurable dynamical system isomorphic to a specific minimal rotation on a compact group. As a by-product, we get the abundance of twin B-free numbers. Moreover, we show that the distribution of patterns in small intervals of the form [N, N + √ N) also conforms to the same measure. When elements of B are squares, we introduce a generalization of the Möbius function, and discuss a conjecture of Chowla in this broader context.

We show that Sarnak's conjecture on Möbius disjointness holds in every uniquely ergodic model of a quasi-discrete spectrum automorphism. A consequence of this result is that, for each non constant polynomial P ∈ R[x] with irrational leading coefficient and for each multiplicative function ν : N → C, |ν| ≤ 1, we have 1 M M ≤m<2M 1 H m≤n