For every dynamical system (X,T) on a compact metric space with zero topological entropy, every f∈ C(X), and every x∈ X, one has frac1N∑_n=1^N f(T^n x)mu(n)=o(1) as N→∞, where mu is the Möbius function.

Sarnak’s Conjecture

Sarnak's Conjecture asserts that every zero-topological-entropy dynamical system is asymptotically orthogonal to the Möbius function: for continuous $f$ and every $x$ , $\frac{1}{N}\sum_{n=1}^{N} f(T^n x)\mu(n)=o(1)$ . The universal statement remains unsettled.

Known results

Rigidity criteria prove the conjecture for systems with bounded-prime-volume or polynomial-rate rigidity, and for systems with countably many invariant ergodic measures (2019).
Fei Wei characterizes the rigid-system case via asymptotically periodic functions and proves further sufficient conditions (2022 version).
Arbitrarily slow convergence has been constructed for zero-entropy systems, but all examples still satisfy the conjecture (2022).
Results cover specified skew products, nilmanifolds, automatic sequences, and selected distal systems, not the universal claim (2026).

2026 examples and counterexample claim

A new preprint constructs continuum-many semi-irrational and totally irrational examples with uniformly sublinear but unbounded lifted displacement; it explicitly does not resolve the conjecture. Separately, an undated article claims that a construction called “Burns Law” gives a zero-entropy counterexample, but the claim has no independent verification and conflicts with sources reporting the universal conjecture as open.

Current status (as of August 2026): The conjecture is proved only for various specified classes and remains open in general; an asserted counterexample is unverified.

Sarnak’s Conjecture

Primary source

Additional references

Progress summary

Solutions 0