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Analytic number theory · multiplicative functions · fixed-shift correlations

Chowla’s Conjecture for Liouville Correlations

Collaboration beta

For any distinct fixed shifts, does the average product of the Liouville signs at those shifted integers tend to zero?

Ch(X)=1X∑X<n≤2X∏i=1kλ(n+hi)⟶0
Known results and sources
Shifted rows of ivory and amber sign cells enter an averaging window whose signal fades without closing.
Chowla's conjecture asks whether every fixed-shift product of Liouville signs has average tending to zero.

Research problem

Exact mathematical statement

Let the completely multiplicative Liouville function be

λ(n)=(-1)Ω(n).\lambda(n)=(-1)^{\Omega(n)}.

Fix distinct integer shifts h=(h1,…,hk)\mathbf h=(h_1,…,h_k), and define the dyadic Cesàro correlation

Ch(X)=1X∑X<n≤2X∏i=1kλ(n+hi).C_{\mathbf h}(X)=\frac1X\sum_{X<n\le 2X}\prod_{i=1}^k\lambda(n+h_i).

The fixed-shift Chowla conjecture asserts that

Ch(X)⟶0(X→∞).C_{\mathbf h}(X)\longrightarrow 0\qquad(X\to\infty).

The source normalizes h1=0h_1=0 after translation and studies k≥2k\ge2; the full fixed-shift statement remains open in the source.

Problem infographic

Problem at a glance

Several shifted Liouville-sign strips combine columnwise into averages fluctuating around an unresolved neutral line.
Distinct fixed shifts align several Liouville sign sequences; the conjecture says their product should have vanishing long average.

Current mathematical picture

Where work on Chowla’s Conjecture for Liouville Correlations stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureGeneric-only activity tilting and Ramanujan substitution

At one active prime the correct kernel has value s−2 on the zero residue and −1 elsewhere; substituting cs changes the zero residue and destroys the exact unit-affine centering used downstream. The corrected principal-Ramanujan/active-centered-divisor endpoint and a parity-sensitive Type I/II dispersion analysis remain viable if the finite differences survive Cauchy–Schwarz and recombination.

Route status · Narrowed route
Main reductionCorrected two-point endpoint

The two-point scalar obstruction expands into fixed-determinant sums with coefficient aφ(b)μ(qP/a)μ(qA/b), replacing the erroneous v5 active coefficient.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeClose the corrected two-point Type I ranges with the active centered divisor retained.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Chowla’s Conjecture for Liouville Correlations in numbers

2.1kretained lines of mathematical investigation2,106 in the current working snapshot
Argument development
1,810 · 86%
Explored or eliminated routes
95 · 5%
Open obligations
55 · 3%
Definitions and setup
146 · 7%
7selected mapped statements1routes investigated5open questions5contribution-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 bySarvadaman ChowlaProposed1965Status checkedAug 14, 2026
Source recordedAug 14, 2026

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.

13 selected steps

Scroll horizontally to explore the route

Working route overview for Chowla’s Conjecture for Liouville CorrelationsA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Every fixed distinct-shift Liouville correlation should average to zero — Depends on missing premiseEvery fixed distinct-shiftLiouville correlation shouldaverage…Corrected active kernel — Depends on missing premiseCorrected active kernelCorrected two-point endpoint — Depends on missing premiseCorrected two-point endpointCurrent reduction — Depends on missing premiseCurrent reductionClosing target — Depends on missing premiseClosing targetExact unit-affine centering — Depends on missing premiseExact unit-affine centeringUncoupled divisor cancellation — Depends on missing premiseUncoupled divisorcancellationGeneric-only activity tilting and Ramanujan substitution — stoppedGeneric-only activitytilting and RamanujansubstitutionClose the corrected two-point Type I ranges with the active centered divisor retained. — OpenClose the correctedtwo-point Type I ranges withthe…Prove balanced Type II dispersion for the fixed-determinant system. — OpenProve balanced Type IIdispersion for thefixed-determinant…Extend the corrected endpoint and saving to the general-k colored tensor and recombine all scalar states. — OpenExtend the correctedendpoint and saving to thegeneral-k…Balanced centered-divisor Type II — OpenBalanced centered-divisorType IIGeneral-k colored endpoint — OpenGeneral-k colored endpoint
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
Narrowed routeGeneric-only activity tilting and Ramanujan substitution

At one active prime the correct kernel has value s−2 on the zero residue and −1 elsewhere; substituting cs changes the zero residue and destroys the exact unit-affine centering used downstream. The corrected principal-Ramanujan/active-centered-divisor endpoint and a parity-sensitive Type I/II dispersion analysis remain viable if the finite differences survive Cauchy–Schwarz and recombination.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks
01
Close the corrected two-point Type I ranges with the active centered divisor retained.Suggested move: Freeze the finite factor and small moduli, insert Λ using the stated prime-power lemma, and audit each unbalanced Vaughan range without replacing the φ(b) coefficient or paying the b-sum absolutely.
Ready to work on
02
Prove balanced Type II dispersion for the fixed-determinant system.Suggested move: Square only after preserving the active finite difference, compare one-copy and two-copy formulations, and locate a bilinear congruence or large-sieve saving that survives quotient Liouville weights.
Ready to work on
03
Balanced centered-divisor Type II

The route needs cancellation for balanced bilinear pieces while retaining quotient rough-Liouville factors and the active finite difference; this closing estimate is not supplied.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
General-k colored endpoint

For fixed higher order, the source leaves a colored centered-divisor dispersion theorem uniform over all retained moduli, colorings, annuli, weights and scalar-state constants.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Extend the corrected endpoint and saving to the general-k colored tensor and recombine all scalar states.Suggested move: Control the q0,q2,…,qk colorings uniformly and audit constants, norms, annulus losses and cross-state recombination before returning to the original correlation.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The ordinary Cesàro fixed-distinct-shift conjecture remains open for correlations of order at least two. Logarithmically averaged two-point and odd-order cases and broad averaged-shift variants are proved but do not imply the exact target.

[2][3][4]
External progress

What the literature has established

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

  1. Peer reviewedHigher short-interval uniformity yielded a new averaged-shift version of Chowla's conjecture.[4]
  2. PreprintTao and Teräväinen proved logarithmically averaged odd-order cases.[3]
  3. PreprintTao proved the logarithmically averaged two-point case.[2]
  4. Historical sourceChowla formulated the conjectured cancellation of correlations of the Liouville function.[1]
4 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusChowla conjecture for Liouville correlations
Weaker or relaxed formLogarithmically averaged two-point Chowla conjecture

The logarithmically averaged two-point statement is proved, but logarithmic weights do not settle the ordinary Cesàro average.

[2]
Weaker or relaxed formLogarithmically averaged odd-order correlations

Odd-order logarithmically averaged cases are proved, with a different averaging regime from the exact target.

[3]
Weaker or relaxed formShift-averaged Chowla variants

An averaged version over shifts follows from higher uniformity in short intervals; it does not fix every shift tuple individually.

[4]

Formalization opportunities

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

  • Formalization targetFormalized analytic-number-theory infrastructure for multiplicative-function correlations and the cited entropy/uniformity methods was not identified.

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 the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. 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.

5 standing statements2 proposed statements5 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction2 of 72
  • lemma3 of 73
  • equivalence1 of 71
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
  • retained route statementEvery fixed distinct-shift Liouville correlation should average to zero
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementCorrected active kernelintermediate
  • retained route statementExact unit-affine centeringintermediate
  • retained route statementUncoupled divisor cancellationintermediate
  • retained route statementCorrected two-point endpointintermediate
  • Recorded relationshipThe source 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 source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureGeneric-only activity tilting and Ramanujan substitutionreported failure
  • Research targetClose the corrected two-point Type I ranges with the active centered divisor retained.open
  • Research targetProve balanced Type II dispersion for the fixed-determinant system.open
  • Research targetExtend the corrected endpoint and saving to the general-k colored tensor and recombine all scalar states.open
  • Research targetBalanced centered-divisor Type IIopen
  • Research targetGeneral-k colored endpointopen
  • Narrowed routeGeneric-only activity tilting and Ramanujan substitutionAt one active prime the correct kernel has value s−2 on the zero residue and −1 elsewhere; substituting cs changes the zero residue and destroys the exact unit-affine centering used downstream. The corrected principal-Ramanujan/active-centered-divisor endpoint and a parity-sensitive Type I/II dispersion analysis remain viable if the finite differences survive Cauchy–Schwarz and recombination.
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 bridgeClose the corrected two-point Type I ranges with the active centered divisor retained.

This is the first prioritized work order for the corrected two-point route. Balanced Type II dispersion remains the central unresolved obstacle, followed by general-k extension and recombination.

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.

Continue the mathematics

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Prepared starting pointClose the corrected two-point Type I ranges with the active centered divisor retained.

Chowla’s Conjecture for Liouville Correlations · ready to start

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Research contextPrepared context for any AI agent

For any distinct fixed shifts, does the average product of the Liouville signs at those shifted integers tend to zero?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references4 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    The Riemann Hypothesis and Hilbert's Tenth Problemoriginal source · Sarvadaman Chowla · Gordon and Breach · 1965 · accessed Aug 14, 2026
  2. 2
    The logarithmically averaged Chowla and Elliott conjectures for two-point correlationspreprint · Terence Tao · arXiv · 2015-09-17 · ARXIV 1509.05422 · accessed Aug 14, 2026
  3. 3
    Odd order cases of the logarithmically averaged Chowla conjecturepreprint · Terence Tao, Joni Teräväinen · arXiv · 2017-10-05 · ARXIV 1710.02112 · accessed Aug 14, 2026
  4. 4
    Higher uniformity of bounded multiplicative functions in short intervals on averagepeer reviewed result · Kaisa Matomäki, Maksym Radziwiłł, Terence Tao, Joni Teräväinen, Tamar Ziegler · Annals of Mathematics · 2023 · DOI 10.4007/annals.2023.197.2.3 · MR 4543441 · accessed Aug 14, 2026

Important qualifications

  • Scoped to the historical book record and primary papers for logarithmic and averaged-shift progress; the literature was not exhaustively surveyed.
  • The ordinary fixed-shift Cesàro statement is kept distinct from logarithmically averaged and shift-averaged variants.
  • No formalization or maintained computation was identified in the scoped search.

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