Math Conjectures

Sarnak Möbius disjointness conjecture

OPENLandmarkConjectureProposed 2010 · Full conjecture

Canonical statement

Define the Möbius function by μ(1)=1\mu(1)=1, μ(n)=0\mu(n)=0 if a prime square divides nn, and μ(n)=(−1)k\mu(n)=(-1)^k if nn is a product of kk distinct primes. For every compact metric space XX, every continuous map T:X→XT:X\to X with zero topological entropy, every f∈C(X;C)f\in C(X;\mathbb C), and every x∈Xx\in X,
1N∑n=1Nμ(n)f(Tnx)⟶0as N→∞. \frac1N\sum_{n=1}^N\mu(n)f(T^n x)\longrightarrow0\qquad\text{as }N\to\infty.
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Define the Möbius function by \(\mu(1)=1\), \(\mu(n)=0\) if a prime square divides \(n\), and \(\mu(n)=(-1)^k\) if \(n\) is a product of \(k\) distinct primes. For every compact metric space \(X\), every continuous map \(T:X\to X\) with zero topological entropy, every \(f\in C(X;\mathbb C)\), and every \(x\in X\), \[ \frac1N\sum_{n=1}^N\mu(n)f(T^n x)\longrightarrow0\qquad\text{as }N\to\infty. \]
DYN-014:[Sarnak2010ThreeLecturesMobius][BourgainSarnakZiegler2013Mobius][FerencziKulagaPrzymusLemanczyk2018Sarnak][FrantzikinakisHost2018LogarithmicSarnak]
Dynamical systemsNumber theory

Notes

The limit is known for many algebraic, distal, substitution, horocycle, and other structured zero-entropy systems, and broad logarithmically averaged variants are proved. No argument covers every zero-topological-entropy system in the ordinary Cesàro average displayed above.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.