Clay · Millennium Problems ↗
Unsolved · observed Oct 1, 2026 · superseded
Source status date unavailable · observed Oct 1, 2026
The cited source item and version only; this is not an independent OpenMath assessment.
Publication v5
Record details
Riemann hypothesis →Coverage & review
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. Includes selected source-dated history, exact Claim connections, interpretations, approaches and grounded conditional impacts.
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.
- 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.
- Timeline dates refer to cited research events at their supported precision. Overlapping results and conditional consequences do not define a percentage solved.
General status observations
Clay · Millennium Problems ↗
Unsolved · observed Oct 10, 2026
Source status date unavailable · observed Oct 10, 2026
The cited source item and version only; this is not an independent OpenMath assessment.
Source identities & rights
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.
No upstream full text or code is republished by this record. Rights / source terms
6 cited entries · exact source identity
- Version
- clay-bombieri-riemann@official-description
- Revision
- official-description
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- Unavailable
- Observed
- 2026-10-10
- Check cadence
- 90 days
- §II, p. 4, equation π(x) = Li(x) + O(√x log x) ↗
riemann-hypothesis:rh-prime-error:v1 - §IV, pp. 6–8: curve zeta functions and Weil’s theorem ↗
riemann-hypothesis:rh-finite-field-curves:v1 - §III: Hardy, Selberg, Levinson and Conrey; compare current AF Theorem A introduction ↗
riemann-hypothesis:rh-proportion-history:v1 - §IV finite-field geometry and §V approaches ↗
riemann-hypothesis:rh-research-directions:v1 - §I, PDF page 1 ↗
riemann-hypothesis:rh-classical-target:v1 - §I, PDF page 1: Riemann’s 1859 memoir ↗
riemann-hypothesis:rh-origin-history:v1
Platt–Trudgian · rigorous finite verification ↗
Dave Platt; Tim Trudgian · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
2 cited entries · exact source identity
- Version
- arxiv-2004.09765@arxiv-v1
- Revision
- arxiv-v1
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2020-04-21
- Observed
- 2026-10-10
- Check cadence
- 90 days
- Theorem 1, p. 2; §2 interval/ball arithmetic and Turing counting ↗
riemann-hypothesis:rh-finite-height:v1 - Corollary 2, §3.4, p. 5; Polymath15 Table 1 applied to Theorem 1 ↗
riemann-hypothesis:rh-newman-upper:v1
Alpöge–Furman · simple zeros on the line ↗
Levent Alpöge; Ralph Furman · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
5 cited entries · exact source identity
- Version
- arxiv-2608.13637@arxiv-v2
- Revision
- arxiv-v2
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2026-08-19
- Observed
- 2026-10-10
- Check cadence
- 30 days
- Theorem A, pp. 1–2; Montgomery–Taylor window; Remark 6.1 ↗
riemann-hypothesis:rh-simple-line-af:v1 - Theorem A(ii), Montgomery–Taylor refinement ↗
riemann-hypothesis:rh-distinct-proportion:v1 - Theorem B and §1: primitive Dirichlet L-functions; RH not assumed in the proportion bound ↗
riemann-hypothesis:rh-dirichlet-grh:v1 - Theorem B, p. 2: fixed primitive χ; constants may depend on conductor ↗
riemann-hypothesis:rh-dirichlet-fixed-bound:v1 - Pinned Anthropic README, Solution.lean and formalization.yaml; Appendix A ↗
riemann-hypothesis:rh-anthropic-formal-audit:v1
Lamzouri · another proof and further proportions ↗
Youness Lamzouri · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
5 cited entries · exact source identity
- Version
- arxiv-2609.02882@arxiv-v2
- Revision
- arxiv-v2
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2026-09-08
- Observed
- 2026-10-10
- Check cadence
- 30 days
- Theorem 1.1; definitions of N, N₀ˢ in §1 ↗
riemann-hypothesis:rh-simple-line-lamzouri:v1 - Theorem 1.1; §1 definition of N_{s∪0} ↗
riemann-hypothesis:rh-simple-or-line:v1 - Theorem 1.1: average of N_s and N₀ ↗
riemann-hypothesis:rh-average-proportion:v1 - §1 discussion of simplicity and critical-line conjectures ↗
riemann-hypothesis:rh-simple-zero-conjecture:v1 - Appendix A versus October 3 repository Solution/Basic.lean and Challenge/Basic.lean ↗
riemann-hypothesis:rh-axiom-formal-audit:v1
Guth–Maynard · zero-density estimates ↗
Larry Guth; James Maynard · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
1 cited entry · exact source identity
- Version
- arxiv-2405.20552@arxiv-v2
- Revision
- arxiv-v2
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2026-04-07
- Observed
- 2026-10-10
- Check cadence
- 90 days
- Theorem 1.2 and equation (1.4), p. 2 ↗
riemann-hypothesis:rh-zero-density:v1
Bellotti–Trudgian–Yang · explicit zero-free region ↗
Chiara Bellotti; Timothy S. Trudgian; Andrew Yang · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
1 cited entry · exact source identity
- Version
- arxiv-2603.21490@arxiv-v1
- Revision
- arxiv-v1
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2026-03-23
- Observed
- 2026-10-10
- Check cadence
- 90 days
- Theorem 1, p. 2 (strict inequality in theorem statement) ↗
riemann-hypothesis:rh-zero-free-region:v1
Rodgers–Tao · Newman constant is nonnegative ↗
Brad Rodgers; Terence Tao · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
2 cited entries · exact source identity
- Version
- arxiv-1801.05914@arxiv-v5
- Revision
- arxiv-v5
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2021-07-03
- Observed
- 2026-10-10
- Check cadence
- 90 days
- Theorem 1.1; §1 normalization and RH equivalence ↗
riemann-hypothesis:rh-newman-lower:v1 - §1: RH iff Λ ≤ 0; Theorem 1.1: Λ ≥ 0 ↗
riemann-hypothesis:rh-newman-equivalence:v1
OpenAI · reported seven-eighths half-plane ↗
OpenAI · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
1 cited entry · exact source identity
- Version
- openai-quasi-rh@git-fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb
- Revision
- git-fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2026-09-30
- Observed
- 2026-10-10
- Check cadence
- 30 days
- Theorem 1.1; formalization scope family 003; comparator OAI.riemannZeta_ne_zero_of_seven_eighths_lt_re ↗
riemann-hypothesis:rh-openai-seven-eighths:v1
Gomila · proposed improved Newman bound ↗
Jude Gomila · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
1 cited entry · exact source identity
- Version
- gomila-newman-candidate@git-a74738deb6d5e0f76887cb36901da08b68dca705
- Revision
- git-a74738deb6d5e0f76887cb36901da08b68dca705
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- Unavailable
- Observed
- 2026-10-10
- Check cadence
- 30 days
- README exact parameters; release manuscript; OPEN_REVIEW_QUESTIONS ↗
riemann-hypothesis:rh-gomila-newman-candidate:v1
Independent Riemann challenge · October 2026 submission ↗
R. Arun Chandru (submitter); Independent riemannzeta.fun project (operator) · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
1 cited entry · exact source identity
- Version
- riemann-community-record@git-d95a6d7d11d7836306ec7ec1eb5be3478663718e
- Revision
- git-d95a6d7d11d7836306ec7ec1eb5be3478663718e
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- Unavailable
- Observed
- 2026-10-10
- Check cadence
- 30 days
- Submission master-of-puppets; candidate_critical_line_bound; observed scoreboard 2026-10-10 ↗
riemann-hypothesis:rh-community-october-record:v1
Publication history
- Problem ID
- problem-riemann-hypothesis
- Publication checksum
- f6edf2e2903c767dea6b3378bf8b6101f59fdb15e66a38b663fb3631dc9769ae
- Published
- 2026-10-10T14:42:30.852585+00:00
- Publication reason
- Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 1 · Sep 30, 2026
Initial checked pilot publication; source attribution, rights, wording and coverage reviewed by OpenMath staff.
- Version 2 · Oct 10, 2026
Add cited resolution scope and consolidate equivalent prime formulations and supporting Claims; mathematical evidence and execution review remain distinct.
- Version 3 · Oct 10, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 4 · Oct 10, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 5 · Oct 10, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 6 · Oct 11, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 7 · Oct 11, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 8 · Oct 11, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 9 · Oct 12, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 10 · Oct 13, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 11 · Oct 13, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 12 · Oct 14, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 13 · Oct 15, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.