Number theory · Analysis MSC2020 11M26

conjecture

Riemann hypothesis

OpenGeneral case unresolved

Special cases & boundsRH verified to height 3,000,175,332,800 →More than 67.25% simple zeros on the line, asymptotically →Explore all 6 scoped results →

Still unresolved: RH requires every nontrivial zero at every height to lie on Re(s) = 1/2. Finite checks, positive proportions, sparse possible exceptions and positive Newman upper bounds do not settle that assertion; newer proposed results do not change the canonical status.

Source report · observed Oct 10, 2026

Source-linked discoveryspecialist backgroundChecked Oct 10, 2026Publication v3

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Question & scope

Do all nontrivial zeros of the Riemann zeta function have real part 1/2?

Classical Riemann ζ(s), meromorphically continued from ∑ n^(−s). Nontrivial zeros are those in 0 < Re(s) < 1; the conjecture requires Re(s) = 1/2 for every such zero, at every height. Trivial negative-even zeros and the pole at s = 1 are outside this question. GRH and finite-field analogues have distinct domains.

Question paraphrased by OpenMath. Use the cited source and locator for authoritative wording.

Research state

OpenGeneral case unresolved

Finite-height verification, asymptotic critical-line proportions, zero-density estimates, a zero-free region and heat-flow bounds establish different scopes. New AI-related claims have separate review labels.

Special cases & boundsRH verified to height 3,000,175,332,800 →More than 67.25% simple zeros on the line, asymptotically →Fewer possible zeros to the right of a fixed line →An explicit zero-free region near Re(s) = 1 →The Newman constant satisfies Λ ≥ 0 →A cited upper bound Λ ≤ 0.2 →

Source report · observed Oct 10, 2026 · Clay · Millennium Problems

This curated record attributes the general status and scoped results to their cited sources. Independent mathematical review is pending.

Clay · Millennium Problems

Superseded report

Source label: Unsolved

Source status date unavailable · Observed Oct 1, 2026

The cited source item and version only; this is not an independent OpenMath assessment.

Clay · Millennium Problems

Open

Source label: Unsolved

Source status date unavailable · Observed Oct 10, 2026

The cited source item and version only; this is not an independent OpenMath assessment.

What is established — and under which assumptions

Each result has its own scope and assumptions. Proved implications do not establish their premises; proof barriers and results in related models do not resolve the general question. Counts describe indexed statements, not a percentage solved.

Finite-height verification

Special case

RH verified to height 3,000,175,332,800 →

Every zero in the stated height range is on the line; rigorous counting excludes missed off-line zeros there.

Classical Riemann zeta function.
Positive ordinate 0 < γ ≤ 3,000,175,332,800; conjugation gives the negative counterpart.

Limit: RH still quantifies over every height. A complete check below this bound cannot exclude even one higher off-line zero.

Exact statement, overlap & source locators

Every nontrivial zero ρ = β + iγ of ζ with 0 < γ ≤ 3,000,175,332,800 has β = 1/2.

A distinct scope, not an additive fraction of the conjecture solved.

Platt–Trudgian · rigorous finite verification ↗ · Theorem 1, p. 2; §2 interval/ball arithmetic and Turing counting

Source: Published

Asymptotic critical-line proportion

Bound

More than 67.25% simple zeros on the line, asymptotically →

An unconditional lower bound above 67.25% for simple zeros on the line; two cited proofs support one Claim.

Classical ζ; no RH assumption.
Asymptotic proportions as T → ∞; the denominator counts multiplicity.

Limit: This is a lower bound as height tends to infinity, not a finite-window promise or a percentage of RH solved.

Exact statement, overlap & source locators

Let N(T) count nontrivial zeros with 0 < Im ρ ≤ T, with multiplicity, and N₀ˢ(T) count simple zeros on Re ρ = 1/2. Then liminf as T → ∞ of N₀ˢ(T)/N(T) ≥ k₀ = 3/2 − cot(1/√2)/√2 = 0.6725007037… .

Finite verification and asymptotic proportion address different regimes; their percentages and heights cannot be added.

Overlaps with: RH verified to height 3,000,175,332,800

Alpöge–Furman · simple zeros on the line ↗ · Theorem A, pp. 1–2; Montgomery–Taylor window; Remark 6.1

Lamzouri · another proof and further proportions ↗ · Theorem 1.1; definitions of N, N₀ˢ in §1

Source: Preprint

Zero-density estimate

Bound

Fewer possible zeros to the right of a fixed line →

The Guth–Maynard estimate improves quantitative bounds on possible zeros away from the critical line.

Fix σ in [3/4,1); let T tend to infinity.
Unconditional large-value and analytic estimates; no RH assumption.

Limit: An upper bound on the number of zeros is not proof that there are none; the selected estimate leaves the strip just right of 1/2 untreated.

Exact statement, overlap & source locators

For fixed 3/4 ≤ σ < 1, let N(σ,T) count zeta zeros with Re ρ ≥ σ and |Im ρ| ≤ T, with multiplicity. Then N(σ,T) ≤ T^(15(1−σ)/(3+5σ)+o(1)) as T → ∞; combining classical estimates also gives T^(30(1−σ)/13+o(1)).

This counts zeros in a horizontal-position range, whereas the proportion bound counts simple zeros on the line.

Overlaps with: More than 67.25% simple zeros on the line, asymptotically

Guth–Maynard · zero-density estimates ↗ · Theorem 1.2 and equation (1.4), p. 2

Source: Published

Explicit zero-free region

Bound

An explicit zero-free region near Re(s) = 1 →

The March 2026 preprint improves a logarithmic exclusion region near the edge of the strip.

Real t ≥ 3.
Strict inequality σ > 1 − 1/(4.896 log t); classical ζ.

Limit: The boundary approaches 1 as t increases. RH needs the much stronger exclusion of every zero with Re ρ > 1/2.

Exact statement, overlap & source locators

ζ(σ + it) ≠ 0 whenever t ≥ 3 and σ > 1 − 1/(4.896 log t), with natural logarithm.

Excluding a region and bounding counts in a larger region are different conclusions; neither is an additive progress score.

Overlaps with: Fewer possible zeros to the right of a fixed line

Bellotti–Trudgian–Yang · explicit zero-free region ↗ · Theorem 1, p. 2 (strict inequality in theorem statement)

Source: Preprint

Heat-flow lower bound

Bound

The Newman constant satisfies Λ ≥ 0 →

Rodgers and Tao proved Newman’s nonnegativity conjecture: if RH is true, Λ must be exactly zero.

Classical Riemann ξ heat flow and the cited normalization.
No RH assumption.

Limit: Λ ≥ 0 alone allows positive Λ, which would make RH false. The required upper bound Λ ≤ 0 remains open.

Exact statement, overlap & source locators

For the classical de Bruijn–Newman heat-flow family, Λ ≥ 0. Λ is defined by H_t(z) = ∫₀∞ exp(tu²) Φ(u) cos(zu) du, using the classical Riemann kernel Φ and normalization in the cited paper; all H_t zeros are real exactly when t ≥ Λ.

A distinct scope, not an additive fraction of the conjecture solved.

Rodgers–Tao · Newman constant is nonnegative ↗ · Theorem 1.1; §1 normalization and RH equivalence

Source: Published

Heat-flow upper bound

Bound

A cited upper bound Λ ≤ 0.2 →

The finite-height theorem and Polymath15 criterion yield Λ ≤ 0.2; newer proposed bounds remain separate.

Use the same classical heat-flow normalization as the lower bound.
The rigorous finite-height verification satisfies the published Polymath15 criterion.

Limit: A positive upper bound leaves Λ = 0 unresolved. This is a cited baseline, not a claim that no later improvement exists.

Exact statement, overlap & source locators

In the classical heat-flow normalization, Λ ≤ 0.2. This is Corollary 2 of the finite-verification paper, applying Polymath15’s criterion with the verified height.

The upper bound depends on the same finite verification as the first result; the two heat-flow bounds constrain one constant.

Overlaps with: RH verified to height 3,000,175,332,800 The Newman constant satisfies Λ ≥ 0

Platt–Trudgian · rigorous finite verification ↗ · Corollary 2, §3.4, p. 5; Polymath15 Table 1 applied to Theorem 1

Source: Published

Formulations & supporting Claims

A Problem is the shared question. Its Claims record precise statements, their assumptions and how they relate to that question. Equivalent formulations share one answer.

Formulations

Equivalent formulation · equivalence

Equivalent formulation: the prime-counting error →

RH holds if and only if π(x) = Li(x) + O(√x log x) as x → ∞, where π counts primes and Li denotes the logarithmic integral.

The equivalence is established; the displayed asymptotic error assertion itself is unproved. The prime number theorem alone gives a weaker error guarantee.

unproved · Independent relationship review pending

Equivalent formulation · equivalence

Equivalent formulation: Λ = 0 →

For the classical de Bruijn–Newman constant, RH is equivalent to Λ = 0. Λ is defined by H_t(z) = ∫₀∞ exp(tu²) Φ(u) cos(zu) du, using the classical Riemann kernel Φ and normalization in the cited paper; all H_t zeros are real exactly when t ≥ Λ.

Neither the lower bound Λ ≥ 0 nor a positive upper bound supplies Λ = 0. These are formulations of one Problem, not separate RH problems.

unproved · Independent relationship review pending

Supporting results

Supporting result · bound

At least 83.625% distinct zeros, asymptotically →

With N_d(T) counting distinct nontrivial zeros and N(T) counting total multiplicity in (0,T], liminf N_d(T)/N(T) ≥ (1+k₀)/2 = 0.8362503518…; k₀ = 3/2 − cot(1/√2)/√2.

RH permits multiple zeros. This bound is additional knowledge about multiplicity, not an 83.625% on-line guarantee.

Proved · cited result · Independent relationship review pending

Supporting result · bound

At least 88.762% simple OR on-line zeros, asymptotically →

Let N_{s∪0}(T) count zeros that are simple or on the critical line, with multiplicity in the union. Then liminf N_{s∪0}(T)/N(T) ≥ C₂ = (1+2√2+2k₀)/(3+2√2) = 0.8876200082… .

This must not be labeled “88.762% on the critical line.” A simple off-line zero belongs to this union.

Proved · cited result · Independent relationship review pending

Supporting result · bound

An average bound for simplicity and line membership →

Let N_s(T) count simple zeros anywhere in the strip and N₀(T) count on-line zeros with multiplicity. Then liminf (N_s(T)+N₀(T))/(2N(T)) ≥ (1+k₀)/2 = 0.8362503518… .

The average does not say that each separate proportion is at least 83.625%. It is a different conclusion from the distinct-zero bound with the same constant.

Proved · cited result · Independent relationship review pending

Related variants

Different statement · related variant · conjecture

Generalized RH for Dirichlet L-functions →

For every primitive Dirichlet character χ, each nontrivial zero of L(s,χ) in 0 < Re(s) < 1 has real part 1/2.

The conductor-one case recovers classical RH. Classical RH alone does not prove GRH for all other characters; more general automorphic GRH requires still broader domains.

unproved · Independent relationship review pending

Different statement · related variant · theorem

Partial bounds for a fixed Dirichlet character →

For each fixed primitive χ, Theorem A’s lower bounds k₀ for simple on-line zeros and (1+k₀)/2 for distinct zeros hold for L(s,χ), relative to its nontrivial-zero multiplicity count as T → ∞.

This extends the method to a GRH variant. It neither proves all zeros lie on the line nor asserts uniform bounds while the conductor grows with T.

Proved · cited result · Independent relationship review pending

Different statement · related variant · theorem

Proved analogue: curves over finite fields →

For a smooth projective geometrically connected curve C/F_q of genus g, Z_C(T) = P_C(T)/((1−T)(1−qT)), with P_C(T)=∏_{j=1}^{2g}(1−α_j T) and |α_j|=√q under every complex embedding.

This is proved in another mathematical domain. Its geometric proof gives insight but does not establish classical RH; it is not counted as a proved special case of ζ.

Proved · cited result · Independent relationship review pending

Different statement · related variant · conjecture

Separate conjecture: all nontrivial zeros are simple →

Every nontrivial zero of ζ(s) has multiplicity one.

RH by itself allows repeated on-line zeros. Simple off-line zeros would satisfy this conjecture but violate RH.

unproved · Independent relationship review pending

Proposed results awaiting assessment

Bound · bound

Proposed result: a fixed zero-free half-plane Re(s) > 7/8 →

The source claims ζ(s) ≠ 0 for Re(s) > 7/8, and the same region for every Dirichlet L-function, with principal-character poles at s = 1 excluded. It also claims this for finite-order Hecke L-functions over Q(√−3).

If validated, this would exclude zeros to the right of 7/8 at every height; it would still leave 1/2 < Re(s) ≤ 7/8 unresolved. This proposed Claim is not included in established progress.

Proposed · awaiting assessment · Independent relationship review pending

Bound · bound

Proposed result: Λ ≤ 0.1787854 →

The candidate claims Λ ≤ 893927/5000000 = 0.1787854 using X = 6000000185827, t₀ = 129/800 and y₀² = 87677/2500000 in the Polymath15 criterion.

Even a valid positive upper bound would leave Λ = 0 open. This candidate is not counted as established progress or used to replace the published baseline.

Proposed · awaiting assessment · Independent relationship review pending

Research activity & artifacts

Recent source updates, computational work and formalization projects provide context. These entries are not counted as additional proved progress. Artifact links carry their own scope and checking status.

formalization

Individual AI-assisted submission: reported 67.35478% bound

R. Arun Chandru’s submission reports κ = 66812491/99194876 ≈ 0.6735478050. The independent challenge lists kernel verification and an attestation of GPT-6 Astra use. The displayed bound is asymptotic and counts distinct on-line zeros.

Scope & limitation: The score is not automatically a simple-zero bound. Operator kernel checks, statement faithfulness and expert review are separate; this activity creates no accepted Claim, OpenMath Run or reproduction.

Independent Riemann challenge · October 2026 submission ↗

R. Arun Chandru (submitter) · Independent riemannzeta.fun project (operator) · Project documentation

Operator-reported current record observed 10 October 2026. Kernel checks and submitter attestations are external reports; independent OpenMath execution and expert mathematical review are pending.

Community submitted Lean proof ↗Source-reported evidence

candidate_critical_line_bound and its cumulative version count distinct critical-line zeros N0star, against all zeros Ncount with multiplicity. This does not state that the improved proportion counts simple zeros. OpenMath has not rebuilt it.

Exact submission and model attestation ↗Source-reported evidence

Revision & attribution

Source revision: git-d95a6d7d11d7836306ec7ec1eb5be3478663718e. Observed Oct 10, 2026.

Submission metadata names GPT-6 Astra. This is an individual’s model-use attestation, not an OpenAI company result or proof of RH. The challenge is independently operated and states it is not affiliated with Anthropic.

Community submitted Lean proof: d95a6d7d11d7836306ec7ec1eb5be3478663718e.

Exact submission and model attestation: d95a6d7d11d7836306ec7ec1eb5be3478663718e. The exact rational score and GPT-6 Astra name are submitter metadata; operator receipts and later revisions require separate review.

formalization

Claude discovery and Anthropic’s reported Lean verification

The paper’s human authors take responsibility for a Claude-discovered proof. The pinned Lean repository reports proofs for actual zeta, with standard axioms and comparator verification; dependencies and declarations are linked for inspection.

Scope & limitation: A model-written manuscript, an author-reviewed theorem and a kernel-check report are different evidence. OpenMath has not run the proof; no independent reproduction is asserted.

Alpöge–Furman · simple zeros on the line ↗

Levent Alpöge · Ralph Furman · Preprint

August 2026 preprint. Named mathematical authors explicitly state that they verified the proof and take responsibility. Journal acceptance and independent OpenMath expert review are not established.

Proof PDF · v2 ↗Manuscript / exposition · external

Anthropic Lean solution · pinned commit ↗Source-reported evidence

Repository reports a Mathlib riemannZeta proof with standard Lean axioms, comparator checks and no RH premise. Challenge.lean contains statement placeholders; Solution.lean supplies the proof chain. OpenMath has not built or audited the dependencies.

Anthropic statement and dependency audit ↗Source-reported evidence

Revision & attribution

Source revision: arxiv-v2. Observed Oct 10, 2026.

The paper credits Claude (Anthropic) with discovery and writing; Alpöge and Furman with verification, Jarred Sumner with guiding the investigation, and Eric Easley with orchestrating Lean formalization. Company attribution is source-reported, not independently reproduced here.

Source updated Aug 19, 2026.

Proof PDF · v2: arXiv:2608.13637v2. External source evidence; no OpenMath execution or independent reproduction is attached.

Anthropic Lean solution · pinned commit: 2bafb8c88f177284a2123b5fefa2ff84e2365eb6.

Anthropic statement and dependency audit: 2bafb8c88f177284a2123b5fefa2ff84e2365eb6. Author/operator audit and toolchain are documented at this commit. This is an external checking report, not an OpenMath Run.

formalization

Axiom Math formalization: paper certificate and later repository

The September paper describes a finite-set certificate and a transfer assuming known analytic inputs. The later pinned repository exports all four actual-zeta proportion targets and reports comparator verification; its dependencies include PrimeNumberTheoremAnd.

Scope & limitation: Do not conflate the early conditional transfer certificate with the later full target types. OpenMath has not audited that expanded proof chain or checked manuscript-to-Lean faithfulness.

Lamzouri · another proof and further proportions ↗

Youness Lamzouri · Preprint

Author preprint at the cited revision; journal acceptance has not been established by this curation.

Alternative proof PDF · v2 ↗Manuscript / exposition · external

Axiom Math Lean solution · pinned commit ↗Source-reported evidence

Paper Appendix A describes an early certificate assuming known analytic inputs. The later October repository exports the four proportions for actual Mathlib zeta without such premises in the target types, depends on PrimeNumberTheoremAnd, and reports comparator verification. The dependency proof chain has not been audited or executed by OpenMath.

Axiom formal challenge and count definitions ↗Source-reported evidence

Revision & attribution

Source revision: arxiv-v2. Observed Oct 10, 2026.

Lamzouri gives a human-authored proof inspired by the Claude result. Appendix A reports AxiomProver (Axiom Math) autoformalization; this is a different role from discovering the original bound.

Source updated Sep 8, 2026.

Alternative proof PDF · v2: arXiv:2609.02882v2. External source evidence; no OpenMath execution or independent reproduction is attached.

Axiom Math Lean solution · pinned commit: 1d7edeb4e67096f9da1b37cd4b763e8b73886655.

Axiom formal challenge and count definitions: 1d7edeb4e67096f9da1b37cd4b763e8b73886655. Challenge targets intentionally use sorry; Solution/Basic.lean supplies proofs. Counts distinguish simple, distinct, on-line multiplicity and their union. A target file alone is not a proof.

research direction

Recent AI-related proposals awaiting assessment

OpenAI reports a seven-eighths zero-free half-plane with Lean artifacts; Gomila reports an improved positive Newman upper bound with interval certificates. Each proposed Claim has its own scope and pinned source evidence.

Scope & limitation: These are proposals in this publication, excluded from established progress. No cited source establishes the full classical RH here.

OpenAI · reported seven-eighths half-plane ↗

OpenAI · Preprint

Preprint released in the 6 October 2026 collection. The repository’s formalization metadata marks review status unchecked. Recorded here as a proposed result awaiting assessment; formal checking and mathematical faithfulness are separate.

Proposed seven-eighths proof PDF ↗Source-reported evidence

Alternate eleven-twelfths proof PDF ↗Source-reported evidence

OpenAI Lean implementation · pinned commit ↗Source-reported evidence

The formalization catalogue maps actual Mathlib riemannZeta and Dirichlet L-function targets to this file. Comparator target files contain intentional sorry placeholders. No local compilation, axiom audit or statement-faithfulness review is asserted here.

Formalization scope · family 003 ↗Source-reported evidence

Release and review metadata ↗Source-reported evidence

Revision & attribution

Source revision: git-fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. Observed Oct 10, 2026.

OpenAI attributes this collection to an unreleased internal model. The zero-free-region work is an exception to the usual fixed evaluation procedure; the alternate 11/12 writeup was human-edited for readability. Do not infer a public model version or attribute individual GPT-6 submissions to this company release.

Source updated Sep 30, 2026.

Proposed seven-eighths proof PDF: fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. Manuscript claims a fixed Re(s) > 7/8 region. Its proof and faithfulness to the formal target have not been independently assessed by OpenMath.

Alternate eleven-twelfths proof PDF: fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. Weaker fixed region Re(s) > 11/12; an alternate argument within the same source family, not a duplicate new RH Problem.

OpenAI Lean implementation · pinned commit: fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb.

Formalization scope · family 003: fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. Scope includes zeta, all positive Dirichlet moduli and finite-order Hecke L-functions over Q(√−3); excludes principal poles and later arithmetic applications.

Release and review metadata: fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. Metadata declares review.status unchecked for the collection. Exact pin preserves later correction history.

Gomila · proposed improved Newman bound ↗

Jude Gomila · Project documentation

Public candidate audit package, not yet peer reviewed. The author reports interval certificates; AI referee checks do not establish independent human mathematical acceptance.

Candidate Newman-bound paper ↗Source-reported evidence

Interval certificates and replay package ↗No OpenMath execution recorded

Candidate review questions ↗Source-reported evidence

Revision & attribution

Source revision: git-a74738deb6d5e0f76887cb36901da08b68dca705. Observed Oct 10, 2026.

The author discloses human–AI collaboration using Claude and ChatGPT/Codex. Model versions are not established by this curation; the attributed author and operator are separate from AI companies.

Candidate Newman-bound paper: a74738deb6d5e0f76887cb36901da08b68dca705. Claimed bound Λ ≤ 893927/5000000; certificate inputs and theorem-to-code alignment remain unassessed by OpenMath.

Interval certificates and replay package: a74738deb6d5e0f76887cb36901da08b68dca705. Read-only artifact links; OpenMath has not run the verifier or replayed the interval certificates.

Candidate review questions: a74738deb6d5e0f76887cb36901da08b68dca705. Author lists outstanding review questions. An AI review is not independent expert certification.

research direction

From infinitely many on-line zeros to proportion bounds

Hardy proved infinitely many zeros lie on the line; later work established positive proportions. The August 2026 paper identifies the previous simple-on-line benchmark as 5/12 and improves it to k₀ ≈ 0.6725007. Its proportion metric differs from finite-height verification.

Scope & limitation: A historical source’s old numerical records are not today’s best bounds. The selected papers are milestones, not an exhaustive priority history; new submissions stay separately attributed.

Bombieri · official RH description ↗

Enrico Bombieri · Published

Official Clay exposition. Historical results and approaches must not be read as current record bounds.

Official description PDF ↗Manuscript / exposition · external

Revision & attribution

Source revision: official-description. Observed Oct 10, 2026.

The cited work does not report AI involvement; absence of attribution is not evidence of absence.

Official description PDF: official-description. External source evidence; no OpenMath execution or independent reproduction is attached.

Alpöge–Furman · simple zeros on the line ↗

Levent Alpöge · Ralph Furman · Preprint

August 2026 preprint. Named mathematical authors explicitly state that they verified the proof and take responsibility. Journal acceptance and independent OpenMath expert review are not established.

Proof PDF · v2 ↗Manuscript / exposition · external

Anthropic Lean solution · pinned commit ↗Source-reported evidence

Anthropic statement and dependency audit ↗Source-reported evidence

Revision & attribution

Source revision: arxiv-v2. Observed Oct 10, 2026.

Source updated Aug 19, 2026.

Proof PDF · v2: arXiv:2608.13637v2. External source evidence; no OpenMath execution or independent reproduction is attached.

Anthropic Lean solution · pinned commit: 2bafb8c88f177284a2123b5fefa2ff84e2365eb6.

research direction

Geometry, positivity and spectral research directions

The official exposition connects finite-field geometry, positivity criteria and spectral ideas to the classical problem. These explain possible routes and why the finite-field analogue is informative.

Scope & limitation: An analogy, positivity reformulation or proposed spectral operator does not establish the missing classical theorem. No solution is asserted by this research-direction entry.

Bombieri · official RH description ↗

Enrico Bombieri · Published

Official Clay exposition. Historical results and approaches must not be read as current record bounds.

Official description PDF ↗Manuscript / exposition · external

Revision & attribution

Source revision: official-description. Observed Oct 10, 2026.

The cited work does not report AI involvement; absence of attribution is not evidence of absence.

Official description PDF: official-description. External source evidence; no OpenMath execution or independent reproduction is attached.

Literature & sources

Clay current RH status ↗

Clay reports the full classical conjecture unsolved; observed 10 October 2026. This observation is separate from older theorem dates and recent proposed bounds.

Bombieri · official RH description ↗

Enrico Bombieri · Published

Official Clay exposition. Historical results and approaches must not be read as current record bounds.

Official description PDF ↗Manuscript / exposition · external

Revision & attribution

Source revision: official-description. Observed Oct 10, 2026.

The cited work does not report AI involvement; absence of attribution is not evidence of absence.

Official description PDF: official-description. External source evidence; no OpenMath execution or independent reproduction is attached.

Platt–Trudgian · rigorous finite verification ↗

Dave Platt · Tim Trudgian · Published

Published in Bulletin of the London Mathematical Society, DOI 10.1112/blms.12460. Statements here use the pinned author manuscript, Theorem 1 and Corollary 2.

Verification proof PDF · v1 ↗Manuscript / exposition · external

Revision & attribution

Source revision: arxiv-v1. Observed Oct 10, 2026.

The cited work does not report AI involvement; absence of attribution is not evidence of absence.

Source updated Apr 21, 2020.

Verification proof PDF · v1: arXiv:2004.09765v1. External source evidence; no OpenMath execution or independent reproduction is attached.

Alpöge–Furman · simple zeros on the line ↗

Levent Alpöge · Ralph Furman · Preprint

August 2026 preprint. Named mathematical authors explicitly state that they verified the proof and take responsibility. Journal acceptance and independent OpenMath expert review are not established.

Proof PDF · v2 ↗Manuscript / exposition · external

Anthropic Lean solution · pinned commit ↗Source-reported evidence

Anthropic statement and dependency audit ↗Source-reported evidence

Revision & attribution

Source revision: arxiv-v2. Observed Oct 10, 2026.

Source updated Aug 19, 2026.

Proof PDF · v2: arXiv:2608.13637v2. External source evidence; no OpenMath execution or independent reproduction is attached.

Anthropic Lean solution · pinned commit: 2bafb8c88f177284a2123b5fefa2ff84e2365eb6.

Lamzouri · another proof and further proportions ↗

Youness Lamzouri · Preprint

Author preprint at the cited revision; journal acceptance has not been established by this curation.

Alternative proof PDF · v2 ↗Manuscript / exposition · external

Axiom Math Lean solution · pinned commit ↗Source-reported evidence

Axiom formal challenge and count definitions ↗Source-reported evidence

Revision & attribution

Source revision: arxiv-v2. Observed Oct 10, 2026.

Lamzouri gives a human-authored proof inspired by the Claude result. Appendix A reports AxiomProver (Axiom Math) autoformalization; this is a different role from discovering the original bound.

Source updated Sep 8, 2026.

Alternative proof PDF · v2: arXiv:2609.02882v2. External source evidence; no OpenMath execution or independent reproduction is attached.

Axiom Math Lean solution · pinned commit: 1d7edeb4e67096f9da1b37cd4b763e8b73886655.

Guth–Maynard · zero-density estimates ↗

Larry Guth · James Maynard · Published

Published online 1 March 2026 in Annals of Mathematics 203, pp. 623–675, DOI 10.4007/annals.2026.203.2.6. The author manuscript was revised April 2026; the pinned theorem statement is cited here.

Zero-density proof PDF · v2 ↗Manuscript / exposition · external

Annals publication and acceptance record ↗Manuscript / exposition · external

Revision & attribution

Source revision: arxiv-v2. Observed Oct 10, 2026.

The cited work does not report AI involvement; absence of attribution is not evidence of absence.

Source updated Apr 7, 2026.

Zero-density proof PDF · v2: arXiv:2405.20552v2. External source evidence; no OpenMath execution or independent reproduction is attached.

Annals publication and acceptance record: Annals 203 (2026), issue 2. Publisher confirms publication and dates; this bibliographic record does not certify an independent OpenMath proof review.

Bellotti–Trudgian–Yang · explicit zero-free region ↗

Chiara Bellotti · Timothy S. Trudgian · Andrew Yang · Preprint

Author preprint at the cited revision; journal acceptance has not been established by this curation.

Zero-free-region proof PDF · v1 ↗Manuscript / exposition · external

Revision & attribution

Source revision: arxiv-v1. Observed Oct 10, 2026.

The cited work does not report AI involvement; absence of attribution is not evidence of absence.

Source updated Mar 23, 2026.

Zero-free-region proof PDF · v1: arXiv:2603.21490v1. External source evidence; no OpenMath execution or independent reproduction is attached.

Rodgers–Tao · Newman constant is nonnegative ↗

Brad Rodgers · Terence Tao · Published

Published in Forum of Mathematics, Pi (2020). The pinned 2021 author revision records a correction to Section 8; earlier manuscript history is not silently substituted.

Corrected Newman proof PDF · v5 ↗Manuscript / exposition · external

Revision & attribution

Source revision: arxiv-v5. Observed Oct 10, 2026.

The cited work does not report AI involvement; absence of attribution is not evidence of absence.

Source updated Jul 3, 2021.

Corrected Newman proof PDF · v5: arXiv:1801.05914v5. External source evidence; no OpenMath execution or independent reproduction is attached.

OpenAI · reported seven-eighths half-plane ↗

OpenAI · Preprint

Proposed seven-eighths proof PDF ↗Source-reported evidence

Alternate eleven-twelfths proof PDF ↗Source-reported evidence

OpenAI Lean implementation · pinned commit ↗Source-reported evidence

Formalization scope · family 003 ↗Source-reported evidence

Release and review metadata ↗Source-reported evidence

Revision & attribution

Source revision: git-fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. Observed Oct 10, 2026.

Source updated Sep 30, 2026.

OpenAI Lean implementation · pinned commit: fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb.

Release and review metadata: fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. Metadata declares review.status unchecked for the collection. Exact pin preserves later correction history.

Gomila · proposed improved Newman bound ↗

Jude Gomila · Project documentation

Public candidate audit package, not yet peer reviewed. The author reports interval certificates; AI referee checks do not establish independent human mathematical acceptance.

Candidate Newman-bound paper ↗Source-reported evidence

Interval certificates and replay package ↗No OpenMath execution recorded

Candidate review questions ↗Source-reported evidence

Revision & attribution

Source revision: git-a74738deb6d5e0f76887cb36901da08b68dca705. Observed Oct 10, 2026.

The author discloses human–AI collaboration using Claude and ChatGPT/Codex. Model versions are not established by this curation; the attributed author and operator are separate from AI companies.

Candidate Newman-bound paper: a74738deb6d5e0f76887cb36901da08b68dca705. Claimed bound Λ ≤ 893927/5000000; certificate inputs and theorem-to-code alignment remain unassessed by OpenMath.

Interval certificates and replay package: a74738deb6d5e0f76887cb36901da08b68dca705. Read-only artifact links; OpenMath has not run the verifier or replayed the interval certificates.

Candidate review questions: a74738deb6d5e0f76887cb36901da08b68dca705. Author lists outstanding review questions. An AI review is not independent expert certification.

Independent Riemann challenge · October 2026 submission ↗

R. Arun Chandru (submitter) · Independent riemannzeta.fun project (operator) · Project documentation

Operator-reported current record observed 10 October 2026. Kernel checks and submitter attestations are external reports; independent OpenMath execution and expert mathematical review are pending.

Community submitted Lean proof ↗Source-reported evidence

Exact submission and model attestation ↗Source-reported evidence

Revision & attribution

Source revision: git-d95a6d7d11d7836306ec7ec1eb5be3478663718e. Observed Oct 10, 2026.

Community submitted Lean proof: d95a6d7d11d7836306ec7ec1eb5be3478663718e.

Provenance & coverage

Coverage of this curation

Six cited scopes of established progress, two equivalent formulations, supporting multiplicity results and four related variants. Recent AI/company/individual proposals are separated from accepted progress. Counts and percentages use explicitly stated conventions.

Formal proofs: Anthropic and Axiom repositories report Lean proofs with exact zeta counts. OpenAI supplies targets and proof code for its proposed bound, with collection review unchecked. Comparator challenge files contain deliberate proof placeholders and must be paired with solutions. All code is linked at exact commits; OpenMath has not compiled, axiom-audited or independently reproduced it.

AI-assisted work: The original proportion theorem credits Claude; Lamzouri’s proof is human-authored with AxiomProver formalization. OpenAI’s recent company preprint is proposed here. Individual GPT-6 Astra use and Gomila’s Claude/ChatGPT/Codex collaboration are self-reported, not company endorsements or proof of RH.

Search scope & review

Checked Clay/Bombieri, pinned arXiv theorem statements and PDFs (Platt–Trudgian, Alpöge–Furman, Lamzouri, Guth–Maynard, Bellotti–Trudgian–Yang, Rodgers–Tao), official Anthropic and OpenAI releases, Axiom Math proof repository, individual challenge records and the Gomila candidate package. Targeted Google/DeepMind searches did not establish an RH result for inclusion. This is a selected search as of 10 October 2026, not an exhaustive absence claim.

Checked Oct 10, 2026 · independent review pending.

Published after staff preview and source checks. This page covers the cited items; it does not represent a complete list of problems or a literature-wide review.

Selected theorem-level milestones, not exhaustive literature coverage, a certified latest-record ranking or a percentage of RH solved.
Independent mathematical and relationship review is pending; bibliographic checking does not establish expert certification.
External Lean and numerical verification reports are distinct from OpenMath execution; no new Run or reproduction is attached.
Recent proof claims remain proposed until assessed. AI attribution is source-reported; search coverage cannot establish that no other AI work exists.

Clay · Millennium Problems ↗

Clay Mathematics Institute · metadata-link · Not established; links and OpenMath paraphrases only

Store metadata and original OpenMath paraphrases; follow the original source for authoritative wording. Rights / source terms

1 cited entry · exact source identity

Version: clay-millennium@observed-2026-10-01
Revision: observed-2026-10-01
Content SHA-256: No upstream byte snapshot retained; linked metadata only
Source update: Unavailable
Observed: 2026-10-01
Check cadence: 30 days

https://www.claymath.org/millennium/riemann-hypothesis/ ↗riemann-hypothesis:entry:0:v1

Clay · Millennium Problems ↗

Clay Mathematics Institute · metadata-link · Not established; links and OpenMath paraphrases only

Store metadata and original OpenMath paraphrases; follow the original source for authoritative wording. Rights / source terms

1 cited entry · exact source identity

Version: clay-millennium@observed-2026-10-10
Revision: observed-2026-10-10
Content SHA-256: No upstream byte snapshot retained; linked metadata only
Source update: Unavailable
Observed: 2026-10-10
Check cadence: 30 days

https://www.claymath.org/millennium/riemann-hypothesis/ ↗riemann-hypothesis:entry:0:v2

Bombieri · official RH description ↗

Enrico Bombieri · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.