Ergodic theory · analytic number theory · zero-entropy dynamics

Sarnak’s Möbius Disjointness Conjecture

Collaboration beta

The conjecture says that the Möbius function has asymptotically zero correlation with every observable generated by a topological dynamical system of zero entropy.

h_{top} (T) = 0 \Rightarrow \frac{1}{N} \sum_{n \leq N} μ (n) f (T^{n} x) ⟶ 0 (f \in C (X), x \in X)

Known results and sources

Sarnak’s conjecture asks whether Möbius signs always cancel against every observable generated by zero-entropy dynamics.

Research problem

Exact mathematical statement

Let $(X,T)$ be a compact topological dynamical system with $h_{\mathrm{top}}(T)=0$ . For every $f\in C(X)$ and every $x\in X$ ,

\frac{1}{N}\sum_{n\le N}\mu(n)f(T^n x)\longrightarrow 0.

the source also records the equivalent formulation with the Liouville function $\lambda(n)=(-1)^{\Omega(n)}$ . Its current status is explicitly source-reported as open: no complete proof is claimed.

Problem infographic

Problem at a glance

A problem-first infographic states Sarnak's conjecture, then shows Target E as an open prerequisite before conditional finite-stage rough-boundary transport, followed by open nested-cloud and compactification-or-cancellation endpoints. — The exact Möbius-disjointness question remains open; the source reports deterministic stripping, while same-plateau realization, nested-cloud coherence, and the final rough-boundary endpoints remain unproved.

Current mathematical picture

Where work on Sarnak’s Möbius Disjointness Conjecture stands

Open conjecture

Selected route highlights from the current work, with the finite-cloud contradiction polarity corrected against the governing source. This is not yet a complete mathematical inventory.

Useful failureFinite-cloud-only contradiction

Finite-cloud entropy and agreement alone do not contradict a counterexample: the regular Toeplitz model satisfies those formal properties and defeats that inference. Cross-cloud coherence and the finite empirical-realization bridge remain necessary before the open rough-core compactification or normalized sifted-tail annihilation endpoints.

Route status · Refuted route

Main reductionNormalized rough-core transport

The source reports a quantitative comparison between the stripped base correlation and density-normalized correlation on the prime-rough set, with overlap controlled separately from stripping loss.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeRealize one positive-correlation code and all prescribed prime-edge defects on the same long plateau.Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected supporting details in the research record. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Sarnak’s Möbius Disjointness Conjecture in numbers

2.7kretained lines of mathematical investigation2,701 in the current working snapshot

Argument development: 2,295 · 85%
Explored or eliminated routes: 118 · 4%
Computational analysis: 17 · 1%
Open obligations: 69 · 3%
Definitions and setup: 202 · 7%

9selected mapped statements1routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byPeter SarnakProposed2010Status checkedAug 7, 2026

Source recordedAug 7, 2026

Recommended next task

Realize one positive-correlation code and all prescribed prime-edge defects on the same long plateau.

Suggested move: Fix one finite prime cloud and its tolerances, declare the finite scale and observable family, and prove the source’s inequalities (63.2)–(63.4) simultaneously for one remote code and one logarithmic plateau.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Refuted routeFinite-cloud-only contradiction

Route status · Refuted route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Realize one positive-correlation code and all prescribed prime-edge defects on the same long plateau.Suggested move: Fix one finite prime cloud and its tolerances, declare the finite scale and observable family, and prove the source’s inequalities (63.2)–(63.4) simultaneously for one remote code and one logarithmic plateau.

Ready to work on

Build a valid nested-cloud diagonal without reversing the current work’s finite-stage quantifier order.Suggested move: After each fixed-cloud realization and rough-core lower bound, choose the next cloud only after the endpoint limit, keeping valuation cutoffs, observables, code, and plateau parameters explicit at every stage.

Ready to work on

Eliminate every coherent prime-rough boundary that survives the finite-stage transport.Suggested move: Develop the compact branch toward a Besicovitch-almost-periodic boundary and, in parallel as mathematical alternatives, the normalized sifted-tail estimate with uniform complementary-prime nonpretentious control.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026

Current statusOpen conjecture

Open in the full ordinary Cesàro-average form for all zero-topological-entropy compact dynamical systems. Current peer-reviewed and 2026 preprint results prove the conjecture for specified classes or prove logarithmically averaged variants under additional hypotheses; those results do not establish the universal ordinary statement.

[2][8][10]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026-03-11
PreprintLiu, Ma, and Wang proved the conjecture for a specified distal transformation of the infinite-dimensional torus. The result is a preprint and its scope is one structured family.[11]
2025
Peer reviewedWei proved ordinary Möbius disjointness for products of zero-entropy affine linear flows with a class of rigid systems, plus a logarithmic theorem for a broader product class. These hypotheses describe special systems, not the universal conjecture.[10]
2021
Peer reviewedKanigowski, Lemańczyk, and Radziwiłł proved ordinary Möbius disjointness under specified quantitative rigidity hypotheses, yielding new cases including almost every interval-exchange map and selected smooth skew products and flows.[9]
2018
Peer reviewedFrantzikinakis and Host verified logarithmically averaged Möbius disjointness for a large class including all uniquely ergodic zero-entropy systems. The averaging and ergodic-weight hypotheses prevent promotion to the ordinary all-systems statement.[8]

Mathematical neighborhood

Related results and reusable starting points

Current focusSarnak's Möbius disjointness conjecture

Solved special caseprime number theorem as the trivial-flow case

For the one-point flow, Möbius disjointness reduces to cancellation of the Möbius summatory function, equivalent to the prime number theorem. This does not control arbitrary zero-entropy orbit sequences.

[1]

Stronger or generalized formordinary Chowla conjecture

The ordinary Chowla correlation conjecture implies ordinary Sarnak disjointness. It asserts fixed-shift correlations of Liouville or Möbius and remains stronger than the dynamical conclusion; an averaged-shift theorem is not the fixed-shift conjecture.

[6][5]

Weaker or relaxed formlogarithmically averaged Sarnak conjecture

Replacing uniform Cesàro weights by logarithmic weights gives a weaker conjecture. Important large-class logarithmic theorems and equivalences do not automatically recover the ordinary average.

[6][8]

Solved special casehorocycle flows

Discrete horocycle flows satisfy the ordinary disjointness conclusion through bilinear estimates and joining rigidity. This supplies both a solved class and a criterion reused elsewhere.

[4]

Solved special caseautomatic sequences

Automatic sequences satisfy Sarnak's conjecture, giving a complete result for sequences produced by finite automata but not for all zero-entropy symbolic dynamics.

[7]

Dependency or reductionrigidity criteria for Möbius disjointness

Quantitative rigidity, bounded-prime-volume rigidity, and affine/rigid product decompositions provide sufficient mechanisms for disjointness in broad classes. A universal route must cover zero-entropy systems without these verified hypotheses.

[9][10]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

formal library support · source linked; not reproduced by ProofAtlasMathlib arithmetic Möbius function
Mathlib defines the arithmetic Möbius function, proves its squarefree formula and multiplicativity, and identifies it as the Dirichlet-convolution inverse of zeta. This is foundational support rather than the disjointness statement.
[12]
formal library support · partial resource linkedMathlib topological entropy foundations
Mathlib has current cover-based topological entropy definitions and structural lemmas for dynamical systems and subsets. The documentation does not claim a formal connection to Möbius-weighted orbit averages or Sarnak's conjecture.
[13]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA reviewed formal statement combining compact metric dynamics, continuous observables, zero topological entropy, the arithmetic Möbius function, and convergence of complex Cesàro orbit averages for every point.
Formalization targetFormal analytic number theory for exponential and bilinear sums, short intervals, multiplicative-function estimates, and quantitative error terms used by current special-case proofs.
Formalization targetFormal ergodic-theory infrastructure for invariant measures, joinings, rigidity, nilsystems, horocycle flows, and the correspondence between topological and measure-theoretic hypotheses.
Formalization targetStatement-alignment lemmas separating ordinary, logarithmic, averaged-shift, Möbius, and Liouville variants and verifying every claimed implication in the selected route.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected supporting details in the research record. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected the cited passages. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Cited passages corrected

Research-record correctionWe corrected the cited passages. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Cited passages corrected

Research-record correctionWe corrected supporting details in the research record. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

6 standing statements3 proposed statements3 open questions

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction3 of 93
lemma4 of 94
equivalence1 of 91

Selected mathematical clusters3 mathematical clusters

Statements and reductionsClaims, implications, and derivations in the current map.17 displayed rows

retained route statementIf h_top(T)=0, then (1/N) Σ_{n≤N} μ(n)f(Tⁿx) → 0 for every continuous f and every x.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementScaled-window stripping telescopeintermediate
retained route statementFinite valuation truncationintermediate
retained route statementLong windows control scaled windowsintermediate
retained route statementExact moving-window rough identityintermediate
retained route statementNormalized rough-core transportintermediate
retained route statementEmpirical realization is the next bridgeintermediate
Recorded relationshipThe source material reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe current work reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed

Open questionsSpecific obligations that remain open in the current routes.3 displayed rows

Research targetRealize one positive-correlation code and all prescribed prime-edge defects on the same long plateau.open
Research targetBuild a valid nested-cloud diagonal without reversing the current work’s finite-stage quantifier order.open
Research targetEliminate every coherent prime-rough boundary that survives the finite-stage transport.open

Explored routes and evidenceChallenges, computations, and approaches that have already narrowed the search.3 displayed rows · 1 route included

Useful failureFinite-cloud contradiction from entropy and agreement alonereported failure
ComputationSource-reported symbolic and measure-theoretic derivations for finite-cloud stripping, scaled-window transport, and stress modelsProofAtlas did not execute attachments or independently recheck the mathematics. The source reports that the deterministic logarithmic stripping gap is closed, while Target E and both final rough-boundary endpoints remain open. · reported unreproduced
Refuted routeFinite-cloud-only contradictionFinite-cloud entropy and agreement alone do not contradict a counterexample: the regular Toeplitz model satisfies those formal properties and defeats that inference. Cross-cloud coherence and the finite empirical-realization bridge remain necessary before the open rough-core compactification or normalized sifted-tail annihilation endpoints.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeRealize one positive-correlation code and all prescribed prime-edge defects on the same long plateau.

Target E is the immediate next bridge. A nested-cloud diagonal and the compactification or normalized sifted-tail endpoint remain open afterward, together with the stated uniformity obligations.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

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Prepared starting pointRealize one positive-correlation code and all prescribed prime-edge defects on the same long plateau.

Sarnak’s Möbius Disjointness Conjecture · ready to start

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Research contextPrepared context for any AI agent

The conjecture says that the Möbius function has asymptotically zero correlation with every observable generated by a topological dynamical system of zero entropy.

Exact question and boundaries
Current routes and known obstacles
What a useful result should report

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Sources and references14 cited works · next context review by Nov 7, 2026

The mathematical context was checked on Aug 7, 2026. Status can be refreshed sooner after a material result or claim.

1
Three Lectures on the Möbius Function, Randomness and Dynamicsoriginal source · Peter Sarnak · Institute for Advanced Study · 2011 · accessed Aug 7, 2026
2
Sarnak Conjecture Learning Seminar — Aboutauthoritative webpage · Ohio State University Department of Mathematics · 2022 · accessed Aug 7, 2026
3
Sarnak's Conjecture: What's Newsurvey or monograph · Sébastien Ferenczi, Joanna Kułaga-Przymus, Mariusz Lemańczyk · Springer · 2018 · ARXIV 1710.04039 · DOI 10.1007/978-3-319-74908-2_11 · accessed Aug 7, 2026
4
Disjointness of Möbius from horocycle flowspeer reviewed result · Jean Bourgain, Peter Sarnak, Tamar Ziegler · Springer · 2013 · ARXIV 1110.0992 · accessed Aug 7, 2026
5
An averaged form of Chowla's conjecturepeer reviewed result · Kaisa Matomäki, Maksym Radziwiłł, Terence Tao · Algebra & Number Theory · 2015 · ARXIV 1503.05121 · DOI 10.2140/ant.2015.9.2167 · accessed Aug 7, 2026
6
Equivalence of the logarithmically averaged Chowla and Sarnak conjecturespreprint · Terence Tao · arXiv · 2016 · ARXIV 1605.04628 · accessed Aug 7, 2026
7
Automatic sequences fulfill the Sarnak conjecturepeer reviewed result · Clemens Müllner · Duke Mathematical Journal · 2017 · ARXIV 1602.03042 · DOI 10.1215/00127094-2017-0024 · accessed Aug 7, 2026
8
The logarithmic Sarnak conjecture for ergodic weightspeer reviewed result · Nikos Frantzikinakis, Bernard Host · Annals of Mathematics · 2018-03-27 · DOI 10.4007/annals.2018.187.3.6 · MR 3779960 · accessed Aug 7, 2026
9
Rigidity in dynamics and Möbius disjointnesspeer reviewed result · Adam Kanigowski, Mariusz Lemańczyk, Maksym Radziwiłł · Fundamenta Mathematicae · 2021 · ARXIV 1905.13256 · DOI 10.4064/fm931-11-2020 · accessed Aug 7, 2026
10
Möbius Disjointness for product flows of rigid dynamical systems and affine linear flowspeer reviewed result · Fei Wei · Discrete and Continuous Dynamical Systems · 2025 · ARXIV 2101.10134 · DOI 10.3934/dcds.2024120 · accessed Aug 7, 2026
11
The Möbius Disjointness Conjecture on infinite-dimensional toruspreprint · Qingyang Liu, Jing Ma, Hongbo Wang · arXiv · 2026-03-11 · ARXIV 2603.11087 · accessed Aug 7, 2026
12
Mathlib.NumberTheory.ArithmeticFunction.Moebiusformalization · Mathlib · accessed Aug 7, 2026
13
Mathlib.Dynamics.TopologicalEntropy.Subsetformalization · Mathlib · accessed Aug 7, 2026
14
List of conjecturesencyclopedia · Wikimedia Foundation · accessed Aug 7, 2026

Important qualifications

This record centers Sarnak's full ordinary Cesàro-average conjecture for every zero-topological-entropy compact dynamical system. Logarithmic averages, averaged shifts, individual dynamical classes, Liouville variants, and stronger Chowla correlations remain explicitly distinguished.
The literature contains hundreds of proved system classes. Milestones are representative and recorded rather than an exhaustive bibliography.
The 2026 infinite-dimensional-torus result retains preprint posture and establishes one specified distal flow, not all zero-entropy systems.
No aligned public FrontierMath: Open Problems entry was verified. Absence from this record is a scoped-search result, not a claim that no future or private listing exists.
The scoped Lean, Rocq/Coq, Isabelle, and public-web search verified Mathlib support for the Möbius function and topological entropy but no formal statement or proof of the full conjecture. This does not establish nonexistence.
No packet source or attachment was read, executed, or used as evidence in this administrative collection.

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