Analytic number theory · Riemann zeta zeros · pair statistics · prime variance

Montgomery Pair-Correlation Conjecture

Collaboration beta

Do the normalized spacings between high Riemann-zeta zeros follow the sine-kernel pair law predicted by random matrix theory?

\frac{1}{N (T)} \sum_{0 < γ, γ' \leq T γ \neq γ'} f (\frac{\log T}{2 π} (γ - γ')) \to \int_{ℝ} f (u) [1 - (\frac{\sin π u}{π u})^{2}] du

Known results and sources

Montgomery's conjecture asks whether normalized pairs of high zeta zeros approach the universal sine-kernel law, including strong repulsion at very small gaps.

Research problem

Exact mathematical statement

Assume the Riemann hypothesis and write each nontrivial zero of the Riemann zeta function as $\rho=\tfrac12+i\gamma$ . If $N(T)$ counts the ordinates $0<\gamma\le T$ , Montgomery's pair-correlation conjecture predicts that for every even Schwartz function $f$ ,

\frac{1}{N(T)}\sum_{\substack{0<\gamma,\gamma'\le T\\\gamma\ne\gamma'}} f\left(\frac{\log T}{2\pi}(\gamma-\gamma')\right) \longrightarrow \int_{\mathbb R} f(u)\left[1-\left(\frac{\sin(\pi u)}{\pi u}\right)^2\right]du.

Equivalently in the weighted Fourier normalization used by the source, the missing assertion is $F(\alpha,T)=1+o(1)$ for every fixed $|\alpha|>1$ . This remains open; the source's analytic program is conditional on RH and does not establish the conjecture.

Problem infographic

Problem at a glance

A scientific explainer places zeta zeros on the critical line, shows how two ordinates produce a normalized gap, and graphs the conjectured pair density one minus the squared sine kernel, with the dip near zero and approach to one at large separation marked as an open limiting law. — After scaling a difference of zero ordinates by log T over 2 pi, the conjecture predicts the sine-kernel pair density; the source retains substantial conditional reductions, but this limiting law remains unproved.

Current mathematical picture

Where work on Montgomery Pair-Correlation Conjecture stands

Recent proof claim under review

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

Main reductionAll-numerator top-shell map

Every top-shell numerator pair is mapped to an affine chart with active conductor cutoffs and a diagonal-completed two-dimensional dual decomposition.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeControl the active conductor-completion boundary between the actual finite packet and the completed zero-zero model at the required little-o scale.Task status · Work already reported in progress

Work mapped so far

Montgomery Pair-Correlation Conjecture in numbers

2kretained lines of mathematical investigation1,984 in the current working snapshot

Argument development: 1,759 · 89%
Explored or eliminated routes: 26 · 1%
Computational analysis: 4 · 0%
Open obligations: 94 · 5%
Definitions and setup: 101 · 5%

9selected mapped statements3open questions2contribution-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 byHugh L. MontgomeryProposed1973Status checkedAug 7, 2026

Source recordedAug 7, 2026

Recommended next task

Prove the logarithmically sharp averaged sifted reciprocal-Mertens bound for the completed model while preserving cancellation between diagonal and off-diagonal Möbius variables.

Suggested move: Start from the two-variable reciprocal-zeta factorization, shift both smoothed Mellin variables with sharp mean control, and restore the two hard cutoff boundaries by localized smoothing as prescribed in Work Order A1.

What would count as progress

Retain an exact proof or counterexample for the stated subproblem.

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

12 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.

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove the logarithmically sharp averaged sifted reciprocal-Mertens bound for the completed model while preserving cancellation between diagonal and off-diagonal Möbius variables.Suggested move: Start from the two-variable reciprocal-zeta factorization, shift both smoothed Mellin variables with sharp mean control, and restore the two hard cutoff boundaries by localized smoothing as prescribed in Work Order A1.

Ready to work on

Close every remaining active top-shell frequency sector and then the signed interior covariance and global transfer needed for the full conjecture.Suggested move: Prove the top-conductor edge estimate, then the active opposite-sign determinant and nonzero-resonance bounds, before returning to the interior finite-eta frame and the all-scale zero-side transfer.

Ready to work on

Control the active conductor-completion boundary between the actual finite packet and the completed zero-zero model at the required little-o scale.Suggested move: Use the exact truncated divisor kernel and its conductor floor, average over the common gcd and reduced slope before taking absolute values, and prove Work Order A0 without substituting the completed coefficient prematurely.

Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026

Current statusRecent proof claim under review

Recent proof claim under review: Jerby's November 2025 preprint asserts an RH-conditional proof of Montgomery pair correlation for zeros of Hardy's Z-function, but no peer-reviewed acceptance or authoritative independent validation was located. A February 2026 peer-reviewed article continues to present the classical zeta-zero formula as a conjecture, while accepted results still cover restricted Fourier support, analogues, conditional consequences, or averaged bounds. The canonical normalized sine-kernel pair-correlation statement therefore remains open in accepted literature.

[9][10][7]

External progress

What the literature has established

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

2026-02-24
Peer reviewedKandhil, Languasco, and Moree developed a pair-correlation conjecture for Dirichlet L-functions and conditional consequences for primes in arithmetic progressions; their peer-reviewed article continued to present the classical Montgomery formula as a conjecture.[10]
2025-11-23
PreprintJerby posted a manuscript claiming, under RH, a proof of Montgomery pair correlation through variations of Hardy's Z-function. No authoritative independent acceptance was located in this collection.[9]
2025-03-21
PreprintGoldston, Lee, Schettler, and Suriajaya showed that the pair-correlation conjecture itself would imply that asymptotically 100 percent of nontrivial zeta zeros are both simple and on the critical line, without separately assuming RH.[8]
2025
Peer reviewedCarneiro, Milinovich, and Ramos bounded an appropriate long-interval average of the pair-correlation function between 0.9303 and 1.3208 under RH, around the conjectured value 1, without proving convergence.[7]

Mathematical neighborhood

Related results and reusable starting points

Current focusMontgomery pair-correlation conjecture

Equivalent formulationStrong weighted F(alpha,T) formulation

With the relevant RH, uniformity, weighting, and test-function conventions, the strong assertion that the weighted Fourier statistic F(alpha,T) tends to 1 for the missing range |alpha| greater than 1 yields the sine-kernel pair-correlation density and is commonly called the strong pair-correlation conjecture.

[1][2]

Equivalent formulationVariance of primes in short intervals

Under RH and appropriate uniformity ranges, strong pair correlation is equivalent to a precise asymptotic for the variance of primes in short intervals, translating the zero statistic into a prime-distribution problem.

[2]

Stronger or generalized formGUE hypothesis and n-level correlations

The GUE hypothesis predicts the full hierarchy of local n-level statistics; its two-point projection gives Montgomery pair correlation, so pair correlation alone does not establish the complete GUE law.

[4][5]

Stronger or generalized formPair correlation for other L-functions

Principal-L and Dirichlet-L pair-correlation statements generalize the statistical question to other L-functions and families. Proven restricted-support or family analogues do not settle the classical single-zeta conjecture.

[4][10]

Weaker or relaxed formRestricted-support Montgomery theorem

Restricted Fourier-support theorems establish pair-correlation identities only in the transform range accessible to prime-sum methods. The full conjecture requires the missing range and cannot be inferred from the restricted result alone.

[1][6]

Logical consequenceRiemann hypothesis and critical simple zeros

The canonical statement is traditionally formulated under RH, but recent work studies a formulation strong enough to imply that asymptotically all zeros are critical and simple. RH remains a distinct statement and official Clay problem.

[8][13]

Formal and computational footholds

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

formal library support · partial resource linkedLean mathlib Riemann-zeta infrastructure
Mathlib provides the Riemann zeta function, completed-zeta infrastructure, analytic facts about isolated zeros, and a definition of the Riemann hypothesis. The bounded documentation search found no exact Montgomery pair-correlation statement or proof.
[12]
computation · not independently reproducedOdlyzko zeta-zero spacing computation
The peer-reviewed computation compares normalized zeta-zero spacings with GUE predictions at several heights and reports increasingly close agreement. ProofAtlas did not rerun it, and finite numerical agreement does not prove the limiting conjecture.
[3]
dataset · source linked; not reproduced by ProofAtlasOdlyzko tables of Riemann-zeta zeros
The author's first-party page makes tables available for the first 2,001,052 zeros and blocks near very high-index zeros, supporting reproducible numerical exploration but not serving as a proof certificate.
[11]

Formalization opportunities

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

Formalization targetA reviewed formal definition of nontrivial Riemann-zeta zeros counted with multiplicity, their ordinates, the height-dependent logarithmic normalization, and the diagonal-exclusion convention.
Formalization targetFinite zero-counting and asymptotic-density infrastructure strong enough to define pair sums and their limiting measures.
Formalization targetA precise formal bridge among the sine-kernel test-function statement, the weighted Fourier statistic F(alpha,T), and each claimed uniformity range, including every RH assumption.
Formalization targetFormal analytic-number-theory infrastructure for the explicit formula, prime sums, Fourier transforms, and the restricted-support Montgomery theorem.
Formalization targetA checked proof of the unrestricted canonical statement or a reviewed verification of the recent proof claim; neither was located in the bounded public formalization search.

Detailed research inventory

Claims, milestones, and routes in the current map

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

7 standing statements2 proposed statements3 open questions

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction3 of 93
lemma3 of 93
negative result2 of 92

Selected mathematical clusters2 mathematical clusters

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

retained route statementDo normalized pairs of high zeta zeros converge to the sine-kernel law?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementSine-kernel pair lawintermediate
retained route statementFinite rational-frequency frameworkintermediate
retained route statementRational-frequency diagonalintermediate
retained route statementAll-numerator top-shell mapintermediate
retained route statementCompleted model is not the current workintermediate
retained route statementFull conjecture remains openintermediate
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 targetControl the active conductor-completion boundary between the actual finite packet and the completed zero-zero model at the required little-o scale.in progress reported
Research targetProve the logarithmically sharp averaged sifted reciprocal-Mertens bound for the completed model while preserving cancellation between diagonal and off-diagonal Möbius variables.open
Research targetClose every remaining active top-shell frequency sector and then the signed interior covariance and global transfer needed for the full conjecture.open

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 bridgeControl the active conductor-completion boundary between the actual finite packet and the completed zero-zero model at the required little-o scale.

The current research map records this as an open mathematical step.

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 pointProve the logarithmically sharp averaged sifted reciprocal-Mertens bound for the completed model while preserving cancellation between diagonal and off-diagonal Möbius variables.

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Research contextPrepared context for any AI agent

Do the normalized spacings between high Riemann-zeta zeros follow the sine-kernel pair law predicted by random matrix theory?

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Sources and references13 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
The pair correlation of zeros of the zeta functionoriginal source · Hugh L. Montgomery · Proceedings of Symposia in Pure Mathematics · 1973 · DOI 10.1090/pspum/024/9944 · accessed Aug 7, 2026
2
Pair Correlation of Zeros and Primes in Short Intervalspeer reviewed result · Daniel A. Goldston, Hugh L. Montgomery · Progress in Mathematics 70, Birkhäuser · 1987 · accessed Aug 7, 2026
3
On the distribution of spacings between zeros of the zeta functionpeer reviewed result · Andrew M. Odlyzko · Mathematics of Computation · 1987 · DOI 10.1090/S0025-5718-1987-0866115-0 · accessed Aug 7, 2026
4
Zeros of principal L-functions and random matrix theorypeer reviewed result · Zeév Rudnick, Peter Sarnak · Duke Mathematical Journal · 1996 · DOI 10.1215/S0012-7094-96-08115-6 · accessed Aug 7, 2026
5
Random Matrices, Frobenius Eigenvalues, and Monodromysurvey or monograph · Nicholas M. Katz, Peter Sarnak · American Mathematical Society · 1999 · accessed Aug 7, 2026
6
An unconditional Montgomery theorem for pair correlation of zeros of the Riemann zeta-functionpeer reviewed result · Siegfred Alan C. Baluyot, Daniel Alan Goldston, Ade Irma Suriajaya, Caroline L. Turnage-Butterbaugh · Acta Arithmetica · 2024 · DOI 10.4064/aa230612-20-3 · accessed Aug 7, 2026
7
Fourier optimization and Montgomery's pair correlation conjecturepeer reviewed result · Emanuel Carneiro, Micah B. Milinovich, Antonio Pedro Ramos · Mathematics of Computation · 2025 · DOI 10.1090/mcom/3990 · accessed Aug 7, 2026
8
Pair Correlation Conjecture for the zeros of the Riemann zeta-function I: simple and critical zerospreprint · Daniel A. Goldston, Junghun Lee, Jordan Schettler, Ade Irma Suriajaya · arXiv · 2025-03-21 · ARXIV 2503.15449 · accessed Aug 7, 2026
9
Variations of the Hardy Z-Function and the Montgomery Pair Correlation Conjecturepreprint · Yochay Jerby · arXiv · 2025-11-23 · ARXIV 2511.18275 · accessed Aug 7, 2026
10
Pair correlation of zeros of Dirichlet L-functions: a possible path towards the conjectures of Chowla, Elliott-Halberstam and Montgomerypeer reviewed result · Neelam Kandhil, Alessandro Languasco, Pieter Moree · Mathematische Annalen · 2026-02-24 · DOI 10.1007/s00208-026-03383-y · accessed Aug 7, 2026
11
Tables of zeros of the Riemann zeta functionsoftware or dataset · Andrew M. Odlyzko · University of Minnesota · accessed Aug 7, 2026
12
Mathlib.NumberTheory.LSeries.RiemannZetaformalization · Lean mathlib contributors · Lean mathlib · accessed Aug 7, 2026
13
Riemann Hypothesisauthoritative webpage · Clay Mathematics Institute · accessed Aug 7, 2026

Important qualifications

The canonical target is the normalized pair correlation of nontrivial Riemann-zeta zeros, conventionally under the Riemann hypothesis; weighted Fourier, test-function, and spacing formulations require their precise hypotheses and normalizations.
Jerby's November 2025 preprint claims an RH-conditional proof using zeros of Hardy's Z-function, but no peer-reviewed acceptance or authoritative independent validation was located, so it is recorded as an unverified proof claim rather than a resolution.
Montgomery's restricted Fourier-support theorem, unconditional analogues of that theorem, averaged bounds, and results conditional on the full pair-correlation conjecture do not prove the missing unrestricted range.
The GUE hypothesis, n-level correlation, Dirichlet-L and general principal-L analogues, and the Riemann hypothesis are neighboring statements and are not silently identified with the canonical conjecture.
Function-field and family results establish powerful analogues but are not proofs for the single classical Riemann zeta function.
Odlyzko's computations and zero tables are numerical evidence for an asymptotic law, not theorem proofs; ProofAtlas did not independently reproduce them.
The bounded formalization search located Riemann-zeta and Riemann-hypothesis infrastructure in mathlib but no exact reviewed pair-correlation statement or proof; this does not establish global nonexistence.
No unreviewed source material or packet-derived source was read during this external collection.

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