Number theory · Analysis · Conjecture

Riemann hypothesis

Do all nontrivial zeros of the Riemann zeta function have real part $\frac12$ ?

Openper Clay · Millennium Problems · checked Oct 10, 2026

Literature: 3 published papers · 4 preprints · 2 projects
Formal: None linked
Computation: Certified computation
Review: Not yet reviewed

Frontier 2 reports →

Use with an agent

Earlier version 10 of this page. View current →

Overview

Known

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.

RH verified to height 3,000,175,332,800ProvedPlatt & Trudgian 2020
More than 67.25% simple zeros on the line, asymptoticallyProvedAlpöge & Furman 2026
Fewer possible zeros to the right of a fixed lineProvedGuth & Maynard 2024

Open questions

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.

Latest

OpenAI’s proposed seven-eighths half-planeOct 6, 2026Proposed
A second proof and separate counting boundsSep 8, 2026

Precise statement

Classical Riemann $\zeta(s)$ , meromorphically continued from $\sum_{n=1}^{\infty}n^{-s}$ . Nontrivial zeros are those in $0<\operatorname{Re}(s)<1$ ; the conjecture requires $\operatorname{Re}(s)=\frac12$ 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.

Definitions

Critical line: The vertical line Re(s) = 1/2. RH concerns every nontrivial zero, whereas a positive asymptotic proportion or finite-height check covers a different scope.
Bombieri · official RH description →

How to read this question

Every zero, not almost every zero

RH is a universal statement at unbounded height. A verified initial interval, a large asymptotic on-line proportion and a sparse exception bound answer different restricted questions.

Limit Even density one could allow isolated off-line zeros. The selected counts do not estimate proximity to a proof.

Classical RH holds (Open)
RH verified to height 3,000,175,332,800 (Proved)
More than 67.25% simple zeros on the line, asymptotically (Proved)

Sources & review

Supporting statements

Bombieri · official RH description ↗§I, PDF page 1
Platt–Trudgian · rigorous finite verification ↗Theorem 1, p. 2; §2 interval/ball arithmetic and Turing counting
Alpöge–Furman · simple zeros on the line ↗Theorem A, pp. 1–2; Montgomery–Taylor window; Remark 6.1

Source-grounded explanation · independent explanation review: pending.

Read the numerator and denominator first

Simple and on-line, distinct, and simple or on-line are different numerators. The selected asymptotic proportion theorems use total zeros counted with multiplicity as denominator.

Limit The 83.625% distinct and 88.762% union bounds cannot be substituted for the simple-on-line proportion.

At least 83.625% distinct zeros, asymptotically (Proved)
At least 88.762% simple OR on-line zeros, asymptotically (Proved)

Sources & review

Supporting statements

Alpöge–Furman · simple zeros on the line ↗Theorem A(ii), Montgomery–Taylor refinement
Lamzouri · another proof and further proportions ↗Theorem 1.1; §1 definition of N_{s∪0}

Source-grounded explanation · independent explanation review: pending.

Question paraphrased by OpenMath. Original wording is available in the sources.

Landscape

Figures and explorable examples. Selecting a statement opens its exact wording.

Where zeros can and cannot be

Height t on a logarithmic scale. Proved regions are solid, the proposed region is dashed. RH concerns every height, above any verified range.

VerifiedZero-freeNo zeros for σ > 7/8RH

RH asks for every nontrivial zero on Re(s) = ½.
Every zero with 0 < t ≤ 3,000,175,332,800 is verified on the line.
No zeros right of the explicit boundary.
Proposed: no zeros with Re(s) > 7/8.

Dataset zeta-zero-regions v1 · Boundary σ(t) = 1 − 1/(4.896 ln t) evaluated at 201 heights equally spaced in log10 t from log10 3 to 20; other layers are the stated constants. Software: python 3.12.13. Boundary rounded to 6 decimal places. SHA-256 c49d0d66d77c… · Download JSON

Labelled schematic and text version

Visual setup

The question in the critical strip

RH asks whether every nontrivial zero lies on Re(s) = ½, at every height. This is a schematic; no zero locations are plotted.

Reading boundary. Finite-height verification cannot exclude an off-line zero above its range. The shaded strip is the domain of the question, not a proved zero-free region. Simplicity is a separate assertion, and proposed improvements are not shown as established.

Nontrivial zeros · 0 < Re(s) < 1Question scope RH target · Re(s) = ½ at all heights○ Open Verified finite height · inspect the bound▧ Finite rigorous computation

Schematic only · no zeros plotted · shaded strip ≠ zero-free region

Text version

The horizontal axis is the real part of s and the vertical axis is its imaginary part. The critical strip lies strictly between real parts 0 and 1, with the critical line at one half. RH quantifies over all heights. A separately linked rigorous computation verifies a finite height range; the drawn window is symbolic and not to scale.

Classical Riemann zeta function and its nontrivial zeros. Real and imaginary coordinates are schematic; the vertical extent does not represent a numerical range.

Nontrivial zeros · 0 < Re(s) < 1
Open
classical RH holds
Every zero at every height; negative-even trivial zeros and the pole at 1 are excluded.
classical RH holds →
RH target · Re(s) = ½ at all heights
Open
classical RH holds
Every zero at every height; negative-even trivial zeros and the pole at 1 are excluded.
classical RH holds →
Verified finite height · inspect the bound
Proved
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.
Limit. RH still quantifies over every height. A complete check below this bound cannot exclude even one higher off-line zero.
RH verified to height 3,000,175,332,800 →

Sources & review

Authored by OpenMath curation on 2026-10-11 (AI-assisted curation). Independent review of this visual explanation is pending; linked assertions and relationships have their own assessments.

Bombieri · official RH description§I, PDF page 1PublishedOriginal passage ↗
Platt–Trudgian · rigorous finite verificationTheorem 1, p. 2; §2 interval/ball arithmetic and Turing countingPublishedOriginal passage ↗

Data

Layer	Status
Verified: all zeros on the line, t ≤ 3·10¹²	proved
Zero-free: σ > 1 − 1/(4.896 log t)	proved
Proposed: no zeros for σ > 7/8	proposed
RH: every zero on σ = ½	target

Classical Newman constant Λ

Proved bounds pin the value inside the shaded interval. Proposed bounds are dashed and not yet assessed.

≥ 0 Rodgers & Tao 2018
≤ 0.2 Platt & Trudgian 2020
≤ 0.1787854 Gomila

Data

Bound	Value	Status	Conditions
Lower	0	proved	Classical heat-flow normalization.
Upper	0.2	proved	Classical heat-flow normalization; finite verification supplies an input.
Upper	0.1787854	proposed	Proposed refinement with X = 6000000185827, t₀ = 129/800 and y₀² = 87677/2500000.

Proportion bounds

Lower bounds on proportions. Each bar is a separate statement with its own definition of the count; bars are not cumulative.

Simple zeros on the critical lineAlpöge & Furman 2026
≥ 67.25%
Distinct zerosAlpöge & Furman 2026
≥ 83.63%
Simple or on-line zerosLamzouri 2026
≥ 88.76%

Data

Quantity	Lower bound	Conditions
Simple zeros on the critical line	k₀ = 0.6725007037…	Asymptotic liminf; denominator N(T) counts all zeros with multiplicity.
Distinct zeros	0.8362503518…	Asymptotic liminf; numerator counts distinct zeros, denominator counts multiplicity.
Simple or on-line zeros	0.8876200082…	Asymptotic liminf for the union, counted with multiplicity; not the simple-on-line intersection.

Zeros on the critical line

Each sign change of Hardy’s Z(t) is a zero ½ + iγ of ζ on the critical line. This window shows the first zeros; certified verification reaches far higher.

Window up to t = 10029 nontrivial zeros in (0, 100]

All zeros up to height 3,000,175,332,800 are verified on the line.

Dataset zeta-hardy-z v1 · mpmath siegelz(t) on a 0.1 grid and zetazero(n) for every zero with imaginary part at most 100, then rounded for display. Software: mpmath 1.3.0, python 3.12.13. 25 significant digits; Z rounded to 5 and γ to 9 decimal places. SHA-256 7f73492ebbfe… · Download JSON

Data

n	γ
1	14.134725142
2	21.022039639
3	25.01085758
4	30.424876126
5	32.935061588
6	37.586178159
7	40.918719012
8	43.327073281
9	48.005150881
10	49.773832478
11	52.970321478
12	56.446247697
13	59.347044003
14	60.831778525
15	65.112544048
16	67.079810529
17	69.546401711
18	72.067157674
19	75.704690699
20	77.144840069
21	79.33737502
22	82.910380854
23	84.735492981
24	87.425274613
25	88.809111208
26	92.491899271
27	94.651344041
28	95.870634228
29	98.831194218

Results

Bounds 5

Proposed 2

Conditional implications 1

Proved
Dirichlet GRH implies classical RH
A one-way specialization; a proof of classical RH supplies no proof for every other character.
Alpöge & Furman 2026
Theorem · Conditional implication
ProvedAlpöge & Furman 2026
Dirichlet GRH implies classical RH
If Dirichlet GRH holds for every primitive character, including the trivial primitive character of conductor 1, then classical RH holds.
Assumptions
- The trivial conductor-one L-function is ζ(s).
In this problem
A one-way specialization; a proof of classical RH supplies no proof for every other character.
The conductor-one case is classical zeta; this specialization has no asserted converse.
Sources
- Alpöge–Furman · simple zeros on the line →
Open statement page →

Supporting results 5

Variants and analogues 4

Connections

How the recorded statements connect

━ ⇔ equivalent
→ implies
┅ analogue or related (no logical link)
Border colour = status

6 connections · 3 consequences. Select a statement in the graph to show only its entries.

Equivalent formulations 2

classical RH holds equivalent to the square-root prime-counting error holds.
Cited equivalence
The established equivalence joins two open formulations of the same classical question.
Limit The asymptotic estimate has no specified effective constant here.
Sources & review
Relationship statement: RH is equivalent to the prime-counting error targetProved
Independent relationship review: not recorded. Link to this connection
classical RH holds equivalent to the classical Newman constant is zero.
Cited equivalence
RH gives Λ ≤ 0; the proved Λ ≥ 0 turns this into the target Λ = 0.
Limit Λ ≤ 0.2 and a proposed smaller positive upper bound do not reach zero.
Sources & review
Relationship statement: RH is equivalent to Λ = 0Proved
Independent relationship review: not recorded. Link to this connection

Implications 1

Generalized RH for Dirichlet L-functions implies classical RH holds.
Cited implication
Dirichlet GRH includes classical ζ as the conductor-one case.
Limit The direction is GRH → RH. No converse is asserted.
Sources & review
Relationship statement: Dirichlet GRH implies classical RHProved
Independent relationship review: not recorded. Link to this connection

Analogues and related questions 3

classical RH holds related to Separate conjecture: all nontrivial zeros are simple.
Scope comparison
Line membership and multiplicity are distinct questions. A zero can be on the line and repeated.
Limit The simple-and-on-line proportion theorem addresses an intersection, not all-zero simplicity.
Sources & review
- Lamzouri · another proof and further proportions ↗§1 discussion of simplicity and critical-line conjectures
- Lamzouri · another proof and further proportions ↗Theorem 1.1; definitions of N, N₀ˢ in §1
Independent relationship review: not recorded. Link to this connection
The Newman constant satisfies Λ ≥ 0 related to A cited upper bound Λ ≤ 0.2.
Scope comparison
The two proved inequalities enclose the same normalized constant in [0, 0.2].
Limit An interval containing zero is not a proof that the constant equals zero.
Sources & review
- Rodgers–Tao · Newman constant is nonnegative ↗Theorem 1.1; §1 normalization and RH equivalence
- Platt–Trudgian · rigorous finite verification ↗Corollary 2, §3.4, p. 5; Polymath15 Table 1 applied to Theorem 1
Independent relationship review: not recorded. Link to this connection
classical RH holds analogue of Proved analogue: curves over finite fields.
Analogue · different setting
The finite-field curve theorem places Frobenius eigenvalues on the expected circle, paralleling a zero-location assertion.
Limit Its function field, cohomology and variable differ; it is not proved classical RH.
Sources & review
- Bombieri · official RH description ↗§IV, pp. 6–8: curve zeta functions and Weil’s theorem
Independent relationship review: not recorded. Link to this connection

Consequences 3

Assuming classical RH holdsOpen

A uniform asymptotic error for prime counts
Conditional
A proof of classical RH would establish π(x) = Li(x) + O(√x log x), controlling deviations from the average distribution at all sufficiently large x.
Limit This selected equivalence supplies no explicit numerical constant or practical algorithm. Consequences needing GRH have a stronger premise.
- RH is equivalent to the prime-counting error targetProved
Sources & review
Supporting statements
- RH is equivalent to the prime-counting error target →
- Bombieri · official RH description ↗§II, p. 4, equation π(x) = Li(x) + O(√x log x)
Source-grounded explanation · independent explanation review: pending.
The heat-flow threshold would be exactly zero
Conditional
Together with the proved nonnegativity theorem, classical RH would identify Λ = 0, fixing the real-zero transition in the classical heat flow.
Limit The claim depends on the stated kernel and normalization; it supplies no general result for other heat-flow families.
- RH is equivalent to Λ = 0Proved
Sources & review
Supporting statements
- RH is equivalent to Λ = 0 →
- Rodgers–Tao · Newman constant is nonnegative ↗§1: RH iff Λ ≤ 0; Theorem 1.1: Λ ≥ 0
Source-grounded explanation · independent explanation review: pending.

Assuming classical RH failsOpen

The square-root error target would fail
Conditional
An off-line nontrivial zero would refute RH and, by the same established equivalence, rule out the stated O(√x log x) prime-counting estimate.
Limit This does not refute the prime number theorem or quantify a particular finite error. No off-line zero is accepted by this curation.
- RH is equivalent to the prime-counting error targetProved
Sources & review
Supporting statements
- RH is equivalent to the prime-counting error target →
- Bombieri · official RH description ↗§II, p. 4, equation π(x) = Li(x) + O(√x log x)
Source-grounded explanation · independent explanation review: pending.

Approaches

How the problem is attacked, and what each line of attack cannot yet reach.

Analytic estimates restrict possible exceptions

Mollifiers and moment estimates give on-line proportions; large-value estimates give zero-density bounds; explicit estimates exclude edge regions.

Cannot yet reach These methods currently leave exceptional zeros possible. Different asymptotic and height assumptions cannot be merged into one all-zero guarantee.

More than 67.25% simple zeros on the line, asymptotically (Proved)
Fewer possible zeros to the right of a fixed line (Proved)

Sources & review

Supporting statements

Lamzouri · another proof and further proportions ↗Theorem 1.1; definitions of N, N₀ˢ in §1
Guth–Maynard · zero-density estimates ↗Theorem 1.2 and equation (1.4), p. 2
Bellotti–Trudgian–Yang · explicit zero-free region ↗Theorem 1, p. 2 (strict inequality in theorem statement)

Source-grounded explanation · independent explanation review: pending.

Finite certificates and the heat-flow target

Rigorous zero counting supplies finite-height certificates and feeds bounds on Λ. The heat-flow formulation makes the remaining target exactly zero.

Cannot yet reach Positive upper bounds remain insufficient. A reported external formalization or numerical audit must be checked against its precise analytic inputs.

RH verified to height 3,000,175,332,800 (Proved)
RH is equivalent to Λ = 0 (Proved)

Sources & review

Supporting statements

Platt–Trudgian · rigorous finite verification ↗Theorem 1, p. 2; §2 interval/ball arithmetic and Turing counting
Rodgers–Tao · Newman constant is nonnegative ↗§1: RH iff Λ ≤ 0; Theorem 1.1: Λ ≥ 0

Source-grounded explanation · independent explanation review: pending.

Geometric, positivity and spectral ideas

Bombieri describes finite-field geometry and positivity or spectral approaches as routes toward understanding zero location.

Cannot yet reach A classical proof still needs the relevant object or positivity statement. The finite-field theorem does not transfer automatically.

Proved analogue: curves over finite fields (Proved)

Sources & review

Supporting statements

Proved analogue: curves over finite fields →

Bombieri · official RH description ↗§IV finite-field geometry and §V approaches

Source-grounded explanation · independent explanation review: pending.

Timeline

Oct 6, 2026OpenAI’s proposed seven-eighths half-planeProposal
The company release links a manuscript and formal evidence for a fixed zero-free half-plane Re(s) > 7/8.
Independent acceptance and proof-chain correspondence are unassessed here. Even a valid bound would leave the region between 1/2 and 7/8.
- A fixed zero-free half-plane Re(s) > 7/8Proposed
- OpenAI · reported seven-eighths half-plane ↗Theorem 1.1; formalization scope family 003; comparator OAI.riemannZeta_ne_zero_of_seven_eighths_lt_re
Also on the Frontier: Quasi-Riemann hypothesis
Sep 8, 2026A second proof and separate counting boundsBound established
Lamzouri v2 proves the same simple-on-line statement by a different argument and records further union and average bounds.
A second proof does not count as a second increment of coverage. Simple OR on-line is not simple AND on-line.
- More than 67.25% simple zeros on the line, asymptoticallyProved
- At least 88.762% simple OR on-line zeros, asymptoticallyProved
- Lamzouri · another proof and further proportions ↗Theorem 1.1; definitions of N, N₀ˢ in §1
- Lamzouri · another proof and further proportions ↗Theorem 1.1; §1 definition of N_{s∪0}
Aug 19, 2026Above 67.25% simple zeros on the critical lineBound established
Alpöge–Furman v2 presents the Claude-discovered argument, with human mathematical responsibility and reported Lean formalization.
An asymptotic simple-and-on-line fraction over total multiplicity is not a percentage of RH solved. OpenMath has not executed the Lean code.
- More than 67.25% simple zeros on the line, asymptoticallyProved
- Alpöge–Furman · simple zeros on the line ↗Theorem A, pp. 1–2; Montgomery–Taylor window; Remark 6.1
- Alpöge–Furman · simple zeros on the line ↗Pinned Anthropic README, Solution.lean and formalization.yaml; Appendix A
Aug 13, 2026Zeta zeros on the critical lineImproved bound
A lower bound about 67.250% for simple critical-line zeros; a related distinct-zero bound about 83.625% is also reported.
Open the Frontier report →
Mar 23, 2026An explicit logarithmic zero-free regionBound established
Bellotti–Trudgian–Yang excludes zeros near Re(s) = 1 with an explicit constant and t ≥ 3.
The shrinking edge region does not cover the whole half-plane to the right of 1/2.
- An explicit zero-free region near Re(s) = 1Proved
- Bellotti–Trudgian–Yang · explicit zero-free region ↗Theorem 1, p. 2 (strict inequality in theorem statement)
May 31, 2024Guth–Maynard zero-density estimateBound established
The first preprint introduced the improved exception-count estimate; the selected v2 and 2026 journal publication support the displayed statement.
For fixed σ ≥ 3/4 this bounds possible off-line zeros; it does not exclude every exception.
- Fewer possible zeros to the right of a fixed lineProved
- Guth–Maynard · zero-density estimates ↗Theorem 1.2 and equation (1.4), p. 2
Apr 21, 2020Certified verification to height 3,000,175,332,800Bound established
Platt–Trudgian uses rigorous arithmetic and counting to cover every zero through the specified ordinate; it also gives Λ ≤ 0.2.
Height is not the number of checked zeros; no assertion about every higher zero follows.
- RH verified to height 3,000,175,332,800Proved
- A cited upper bound Λ ≤ 0.2Proved
- Platt–Trudgian · rigorous finite verification ↗Theorem 1, p. 2; §2 interval/ball arithmetic and Turing counting
- Platt–Trudgian · rigorous finite verification ↗Corollary 2, §3.4, p. 5; Polymath15 Table 1 applied to Theorem 1
Jan 18, 2018Newman’s nonnegativity conjectureBound established
Rodgers–Tao establishes Λ ≥ 0. The selected corrected paper preserves this theorem.
The first submission date is historical; the cited v5 has a later correction. RH still requires Λ ≤ 0.
- The Newman constant satisfies Λ ≥ 0Proved
- Rodgers–Tao · Newman constant is nonnegative ↗Theorem 1.1; §1 normalization and RH equivalence
1859Riemann connects zeros to prime distributionFoundation
The all-zero hypothesis arose alongside an explicit formula for counting primes.
The conjectured all-zero location is not established by the explicit formula.
- Classical RH holdsOpen
- Bombieri · official RH description ↗§I, PDF page 1: Riemann’s 1859 memoir
Date unavailableFrom infinitely many zeros to positive proportionsFoundation
Bombieri’s historical account follows the Hardy, Selberg, Levinson and Conrey line of work.
This is a grouped historical bridge with no single supported event date; the modern denominator and simplicity convention must still be read separately.
- Bombieri · official RH description ↗§III: Hardy, Selberg, Levinson and Conrey; compare current AF Theorem A introduction
Date unavailableAn individual numerical Newman-bound candidateProposal
The pinned audit package proposes Λ ≤ 0.1787854 and attributes human–AI collaboration.
The selected package has no independently established result date or accepted recompile here; the statement remains proposed.
- Λ ≤ 0.1787854Proposed
- Gomila · proposed improved Newman bound ↗README exact parameters; release manuscript; OPEN_REVIEW_QUESTIONS

Full history with sources →

Evidence

Evidence by kind
Literature	3 published papers · 4 preprints · 2 projects Status per Clay · Millennium Problems · checked Oct 10, 2026. Open: Clay · Millennium Problems · Unsolved ↗ · source date not recorded · checked Oct 10, 2026The cited source item and version only; this is not an independent OpenMath assessment.
Formal	None linked Reported in sources: 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.
Computation	Certified computation Finite-height verification: RH verified to height 3,000,175,332,800 · Platt & Trudgian 2020
Review	Not yet reviewed Collection last checked Oct 10, 2026.

Sources

Choose a starting point below. Each result links to its exact statement, cited passages and checks.

Start here

Introduction
Bombieri · official RH description →
Read the setting and notation before comparing the results.
Proof source
Alpöge–Furman · simple zeros on the line →
Read the mathematical argument; inspect its statement for assessment and scope.
Proof source
Lamzouri · another proof and further proportions →
Read the mathematical argument; inspect its statement for assessment and scope.

Bibliography 11

Bombieri · official RH description →
Enrico Bombieri · Published
Clay · Millennium Problems →
Clay Mathematics Institute
Platt–Trudgian · rigorous finite verification →
Dave Platt · Tim Trudgian · Published
Alpöge–Furman · simple zeros on the line →
Levent Alpöge · Ralph Furman · Preprint
Lamzouri · another proof and further proportions →
Youness Lamzouri · Preprint
Guth–Maynard · zero-density estimates →
Larry Guth · James Maynard · Published
Bellotti–Trudgian–Yang · explicit zero-free region →
Chiara Bellotti · Timothy S. Trudgian · Andrew Yang · Preprint
Rodgers–Tao · Newman constant is nonnegative →
Brad Rodgers · Terence Tao · Published
OpenAI · reported seven-eighths half-plane →
OpenAI · Preprint
Gomila · proposed improved Newman bound →
Jude Gomila · Project documentation
Independent Riemann challenge · October 2026 submission →
R. Arun Chandru (submitter) · Independent riemannzeta.fun project (operator) · Project documentation

Research activity · 6 entries

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.

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.

Related statements · context, not additional proved progress

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

Independent Riemann challenge · October 2026 submission · sources & artifacts →

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.

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.

Related statements · context, not additional proved progress

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

Alpöge–Furman · simple zeros on the line · sources & artifacts →

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.

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.

Related statements · context, not additional proved progress

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

Lamzouri · another proof and further proportions · sources & artifacts →

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.

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

Related statements · context, not additional proved progress

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

OpenAI · reported seven-eighths half-plane · sources & artifacts →

Gomila · proposed improved Newman bound · sources & artifacts →

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.

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.

Related statements · context, not additional proved progress

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

Bombieri · official RH description · sources & artifacts →

Alpöge–Furman · simple zeros on the line · sources & artifacts →

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.

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 · sources & artifacts →

Selected literature; source observations and coverage limits are retained in the publication details.

Contributors

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.

Authors & roles

Bombieri · official RH description →

Enrico Bombieri

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

Platt–Trudgian · rigorous finite verification →

Dave Platt · Tim Trudgian

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

Alpöge–Furman · simple zeros on the line →

Levent Alpöge · Ralph Furman

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.

Lamzouri · another proof and further proportions →

Youness Lamzouri

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.

Guth–Maynard · zero-density estimates →

Larry Guth · James Maynard

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

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

Chiara Bellotti · Timothy S. Trudgian · Andrew Yang

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

Rodgers–Tao · Newman constant is nonnegative →

Brad Rodgers · Terence Tao

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

OpenAI · reported seven-eighths half-plane →

OpenAI

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.

Gomila · proposed improved Newman bound →

Jude Gomila

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.

Independent Riemann challenge · October 2026 submission →

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

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.

These are source credits; they do not imply OpenMath accounts, endorsements or independent mathematical review.

Record details

Record version: 10
Classification: MSC2020 11M26
Prerequisite level: specialist

Source observations & version history →

Overview

Known

Open questions

Latest

Definitions

How to read this question

Every zero, not almost every zero

Read the numerator and denominator first

Landscape

Where zeros can and cannot be

The question in the critical strip

classical RH holds

classical RH holds

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

Classical Newman constant Λ

Proportion bounds

Zeros on the critical line

Results

The question 2

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Equivalent formulations 2

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Computations 1

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Bounds 5

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Proof outlines

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Proposed 2

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Conditional implications 1

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Supporting results 5

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