ProofAtlas

Analytic number theory · primes in arithmetic progressions · level of distribution

Elliott–Halberstam Conjecture

Collaboration beta

Are primes evenly distributed among reduced residue classes on average for moduli almost as large as x? The conjecture remains open; this source isolates a difficult upper-transition estimate rather than proving it.

∑q≤xθmax(a,q)=1|ψ(x;q,a)-xφ(q)|≪θ,Ax(logx)A
Known results and sources
A dark landscape of evenly spaced arithmetic-progression lanes carries scattered luminous primes toward an unresolved horizon near level one.
Elliott–Halberstam asks for nearly full average equidistribution of primes among reduced residue classes; the conjecture remains open.

Research problem

Exact mathematical statement

Let

ψ(x;q,a)=∑n≤x, n≡a(modq)Λ(n).\psi(x;q,a)=\sum_{n\le x,\ n\equiv a\pmod q}\Lambda(n).

For every fixed θ<1\theta<1 and A>0A>0, the Elliott–Halberstam conjecture asks whether

∑q≤xθmax(a,q)=1|ψ(x;q,a)-xφ(q)|≪θ,Ax(logx)A.\sum_{q\le x^\theta}\max_{(a,q)=1}\left|\psi(x;q,a)-\frac{x}{\phi(q)}\right|\ll_{\theta,A}\frac{x}{(\log x)^A}.

Problem infographic

Problem at a glance

A landscape diagram shows primes distributed across residue-class lanes for increasing modulus ranges, a classical midpoint boundary, and an open near-one horizon.
The exact question compares prime-counting error across every reduced residue class and averages its maximum over moduli up to x to a fixed power below one.

Current mathematical picture

Where work on Elliott–Halberstam Conjecture stands

Open conjecture

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

Useful failureExtending the asymmetric lower-transition cutoff through the endpoint range

At rho > 4/7, the hard-source threshold H is below the required p^(3/4+epsilon) cutoff, making the Type-I constraints incompatible. A family Burgess moment or prime-switching argument that preserves the structured quotient and prime averages remains a proposed route.

Route status · Narrowed route
Main reductionCurrent reduction

A Vaughan decomposition and exact rational-output analysis reduce critical blocks to coefficient-sensitive large-sieve and shifted-energy estimates. The upper range leaves a structured long quotient of Burgess scale and requires the target UOASH_2 estimate.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeProve the exact coefficient-sensitive UOASH_2 upper-transition estimate.Task status · Ready to work on
Research-record correctionResearch-record correction

We 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.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Elliott–Halberstam Conjecture in numbers

8.8kretained lines of mathematical investigation8,756 in the current working snapshot
Argument development
7,760 · 89%
Explored or eliminated routes
152 · 2%
Computational analysis
72 · 1%
Open obligations
336 · 4%
Definitions and setup
436 · 5%
7selected 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 D. T. A. Elliott and Heini HalberstamProposed1970Status 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

11 selected steps

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

11 selected steps

Scroll horizontally to explore the route

Working route overview for Elliott–Halberstam ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Do primes have level of distribution every fixed theta < 1? — Depends on missing premiseDo primes have level ofdistribution every fixedtheta…Conditional lower transition — Depends on missing premiseConditional lower transitionCurrent reduction — Depends on missing premiseCurrent reductionExact Elliott–Halberstam target — Depends on missing premiseExact Elliott–HalberstamtargetClosing target — Depends on missing premiseClosing targetExact phase boundary — Depends on missing premiseExact phase boundaryPrime slice is insufficient — Depends on missing premisePrime slice is insufficientExtending the asymmetric lower-transition cutoff through the endpoint range — stoppedExtending the asymmetriclower-transition cutoffthrough…Prove the exact coefficient-sensitive UOASH_2 upper-transition estimate. — OpenProve the exactcoefficient-sensitiveUOASH_2…Complete the prime fourth-moment positive-norm packet ledger. — OpenComplete the primefourth-moment positive-normpacket…Extend the estimate through descended/composite conductors and all fixed higher moments. — OpenExtend the estimate throughdescended/compositeconductors…
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 routeExtending the asymmetric lower-transition cutoff through the endpoint range

At rho > 4/7, the hard-source threshold H is below the required p^(3/4+epsilon) cutoff, making the Type-I constraints incompatible. A family Burgess moment or prime-switching argument that preserves the structured quotient and prime averages remains a proposed route.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Prove the exact coefficient-sensitive UOASH_2 upper-transition estimate.Suggested move: Normalize one family-Burgess candidate against the positive residual norm and verify every fixed source, residual, Mellin, numerator, and defect label.
Ready to work on
02
Complete the prime fourth-moment positive-norm packet ledger.Suggested move: Assemble the eleven stated diagonal, defect, boundary, mixed-factorization, zero-mode, and large-denominator components without upgrading conditional inputs.
Ready to work on
03
Extend the estimate through descended/composite conductors and all fixed higher moments.Suggested move: Prove coefficient stability under conductor descent and define a checked residual 2k-moment for each fixed k before inferring levels approaching one.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

No checked source establishes Elliott–Halberstam. Recent work proves distribution beyond one half only with restricted weights, residue classes, or conditional hypotheses; those results do not imply the full maximum-over-reduced-residue-classes conjecture for every theta below one.

[2][3][4]
External progress

What the literature has established

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

  1. Peer reviewedMaynard obtained moduli up to x^(3/5-epsilon) with suitably well-factorable weights, extending a restricted large-modulus regime but not the full conjecture.[2]
  2. PreprintWright obtained conditional almost-all results up to x^(2/3-epsilon) under a strong exceptional-character hypothesis; this is not an unconditional Elliott–Halberstam theorem.[4]
  3. Peer reviewedPolymath8b analyzed Elliott–Halberstam and generalized variants as conditional inputs for bounded-prime-gap results, without proving them.[3]
  4. Historical sourceElliott and Halberstam formulated the conjectural near-level-one distribution of primes in arithmetic progressions.[1]
5 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusElliott–Halberstam Conjecture
Weaker or relaxed formBombieri–Vinogradov and weighted large-modulus estimates

Classical and weighted estimates establish smaller or restricted distribution ranges, not the exact nearly-full uniform maximum statement.

[2]
Logical consequenceConditional bounded prime-gap improvements

Elliott–Halberstam-type hypotheses yield stronger prime-gap conclusions; those consequences are not evidence that the conjecture is true.

[3]

Formal and computational footholds

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

  • formal library support · partial resource linkedLean mathlib primes in arithmetic progressions infrastructure

    Mathlib includes Dirichlet-theorem and L-series infrastructure for primes in arithmetic progressions, but no formal Elliott–Halberstam statement or proof was identified.

    [5]

Formalization opportunities

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

  • Formalization targetA formal von Mangoldt function in residue classes and the exact uniform asymptotic notation.
  • Formalization targetFormal analytic number theory for large-sieve, character-sum, and dispersion estimates.
  • Formalization targetA checked statement and proof of the missing level-below-one bound.

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 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
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.

4 standing statements3 proposed statements3 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction2 of 72
  • lemma1 of 71
  • equivalence1 of 71
  • negative result2 of 72
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 statementDo primes have level of distribution every fixed theta < 1?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact Elliott–Halberstam targetintermediate
  • retained route statementPrime slice is insufficientintermediate
  • retained route statementConditional lower transitionintermediate
  • retained route statementExact phase boundaryintermediate
  • 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 failureExtending the asymmetric lower-transition cutoff through the endpoint rangereported failure
  • Research targetProve the exact coefficient-sensitive UOASH_2 upper-transition estimate.open
  • Research targetComplete the prime fourth-moment positive-norm packet ledger.open
  • Research targetExtend the estimate through descended/composite conductors and all fixed higher moments.open
  • Research targetUpper OASH estimatesuperseded
  • Research targetFull assembly after UOASHsuperseded
  • Narrowed routeExtending the asymmetric lower-transition cutoff through the endpoint rangeAt rho > 4/7, the hard-source threshold H is below the required p^(3/4+epsilon) cutoff, making the Type-I constraints incompatible. A family Burgess moment or prime-switching argument that preserves the structured quotient and prime averages remains a proposed route.
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 bridgeProve the exact coefficient-sensitive UOASH_2 upper-transition estimate.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

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 pointProve the exact coefficient-sensitive UOASH_2 upper-transition estimate.

Elliott–Halberstam Conjecture · ready to start

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Research contextPrepared context for any AI agent

Are primes evenly distributed among reduced residue classes on average for moduli almost as large as x? The conjecture remains open; this source isolates a difficult upper-transition estimate rather than proving it.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references5 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
    A conjecture in prime number theoryoriginal source · Peter D. T. A. Elliott, Heini Halberstam · Symposia Mathematica IV · 1970 · accessed Aug 14, 2026
  2. 2
    Primes in arithmetic progressions to large moduli II: well-factorable estimatespeer reviewed result · James Maynard · Memoirs of the American Mathematical Society 306 · 2025 · ARXIV 2006.07088 · DOI 10.1090/memo/1543 · accessed Aug 14, 2026
  3. 3
    Variants of the Selberg sieve, and bounded intervals containing many primespeer reviewed result · D. H. J. Polymath · Research in the Mathematical Sciences · 2014 · ARXIV 1407.4897 · DOI 10.1186/s40687-014-0012-7 · accessed Aug 14, 2026
  4. 4
    Primes in Arithmetic Progressions to Large Moduli and Siegel Zeroespreprint · Thomas Wright · arXiv · 2025 · ARXIV 2507.10780 · accessed Aug 14, 2026
  5. 5
    Mathlib.NumberTheory.LSeries.PrimesInAPformalization · Lean mathlib · accessed Aug 14, 2026

Important qualifications

  • Original bibliographic identity; recent peer-reviewed weighted large-modulus milestone; a recent conditional preprint; visible Lean library support.
  • The private 2026 packet and its URLs were not used as external authority. No submitted URL or attachment was fetched.
  • This bounded search is not an exhaustive bibliography, theorem audit, or priority review. Restricted distribution estimates were not promoted to the full conjecture.

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