Sarnak's Möbius Disjointness Conjecture

Let \((X, T)\) be a topological dynamical system of zero entropy, \(f\) a continuous observable, and \(\mu\) the Möbius function. Then \(\frac{1}{N}\sum_{n \leq N} \mu(n)\, f(T^n x) \to 0\) as \(N \to \infty\) for every \(x \in X\): the Möbius function is disjoint from all deterministic zero-entropy dynamics.

Formulated by Peter Sarnak in the early 2010s as a dynamical counterpart to the Chowla conjecture: randomness of the Möbius function should defeat every deterministic system of zero entropy. It follows from Chowla's conjecture on self-correlations of \(\mu\), and it implies, for instance, the prime number theorem in arithmetic progressions along deterministic subsequences. The landmark progress is the Green–Tao theorem's orbit-counting machinery and later work of Matomäki, Radziwiłł and Tao on short-interval averages — all consistent with, but far short of, full disjointness. No zero-entropy counterexample is known, and no general case is proved.