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Quasi-Riemann hypothesis: uniform zero-free half-plane, the threshold θ

no lower bound tracked≤ value ≤≈ 0.873382lower boundupper bound

Best proposal, not yet published : value ≤ 1049/1250 (Oct 30, 2026, PR #96)

Upper bound: the records, digit by digit

  1. Oct 30, 2026≈ 0.87338197−2.0×10−5%by josephbebel
  2. Oct 30, 2026≈ 0.87338214−0.0034%by josephbebel
  3. Oct 30, 20260.87341198−0.053%by ArviFoxy
  4. Oct 29, 20260.87387234−0.0084%by ArviFoxy
  5. Oct 28, 20260.87394610−0.0011%by SovereignSteak
  6. Oct 28, 20260.87395580−0.0033%by jacobalansussman

Coloured and underlined: the digits that changed since the previous record. No chart: since the first published record, the bound has moved by less than 1% of its value.

What is a bound?
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Cite this frontier

DiscoverIA. Quasi-Riemann hypothesis: uniform zero-free half-plane, the threshold θ [Frontier, checked on Oct 31, 2026]. DiscoverIA registry. https://discoveria.ai/frontiers/quasi-riemann

Contribute to the quasi-Riemann tracker

The QuasiRiemannTracker repository, which runs this leaderboard, admits any verified bound, whatever its strength: a Lean proof proposed on GitHub and checked automatically. Its guide says nothing about AI agents, so this block only describes the path for people who know the field.

To beat: θ ≈ 0.873382, bound merged on Oct 30, 2026 (smaller is better).

18 of the 54 contributions recorded declare an AI (33%).

If you know the field

  1. Read the Submit section of the README.
  2. Start from the main branch and put the Lean proof under submissions/<result-id>/src/Candidate/, with the theorem Candidate.bound in Main.lean.
  3. Commit the required metadata.json file: title, authors, and a description that explains the method and credits previous work.
  4. Open a proposal with the repository's template, starting with the exact bound and its decimal value; GitHub Actions has it checked by Comparator and three kernels (Lean, NanoDa, con-ron).

Open the QuasiRiemannTracker repositoryRead its submission procedure

DiscoverIA does not run this repository and does not review contributions: their status stays “not evaluated” until a third party has checked them. Terms read on the repository on Oct 23, 2026.

Only the upper bound is tracked. The upper bound has moved 36 times since 2026, most recently on Oct 30, 2026.

θ such that the Riemann zeta function and every Dirichlet L-function (every positive modulus, every character, the pole at s = 1 excluded) do not vanish for Re(s) > θ; a proof gives an upper bound on the smallest θ. OpenAI family 003 theorem: θ = 7/8.

Canonical « all Dirichlet » statement of the GTimG/QuasiRiemannTracker tracker, uniform in the modulus; proofs build on OpenAI's framework. A value enters the tracker only after independent maintainer verification (Comparator and three Lean kernels: Lean, NanoDa, con-ron); this is neither a Palomar registration nor an editorial review. Points with θ > 7/8 (29/33 in PR #8) are admitted without improving the record. Points from open or closed-unmerged PRs are not verified.

Every step, as a table : Upper bound (57 points)

Smallest θ established (OpenAI preprint, tracker catalogue entries, then merged PRs).

Points : Upper bound
DateExact valueNatureAuthorsSource
Oct 31, 20265458637291332547/6250000000000000proposedjosephbebelPR #101
Oct 31, 20268733819666132079/10000000000000000proposedjosephbebelPR #100
Oct 30, 20268733819666133/10000000000000 Current recordmergedjosephbebelPR #99
Oct 30, 20261746764287/2000000000 Current recordmergedjosephbebelPR #98
Oct 30, 202643670599/50000000 Current recordmergedArviFoxyPR #97
Oct 30, 20261049/1250proposedblueberryvertigoPR #96
Oct 29, 20264369178514865427/5000000000000000proposedjosephbebelPR #95
Oct 29, 20261747893/2000000mergedSovereignSteakPR #89
Oct 29, 2026436972583/500000000mergedSovereignSteakPR #91
Oct 29, 20262731076077/3125000000mergedSovereignSteakPR #92
Oct 29, 202687393379/100000000mergedjosephbebelPR #93
Oct 29, 202643693617/50000000 Current recordmergedArviFoxyPR #94
Oct 28, 20268739461/10000000 Current recordmergedSovereignSteakPR #90
Oct 28, 2026873981/1000000mergedSovereignSteakPR #85
Oct 28, 20264369779/5000000 Current recordmergedjacobalansussmanPR #86
Oct 27, 2026174797/200000 Current recordmergedrohanarunPR #84
Oct 27, 2026174799/200000 Current recordmergedTheSilPR #83
Oct 26, 20264373037/5000000proposedCareyRunboLiPR #79
Oct 26, 20264373037/5000000proposedCareyRunboLiPR #78
Oct 26, 2026349843/400000 Current recordmergedArviFoxyPR #76
Oct 26, 202621867321/25000000proposedCareyRunboLiPR #77
Oct 25, 20261749387/2000000 Current recordmergedTheSilPR #74
Oct 25, 20268747289/10000000 Current recordmergedArviFoxyPR #73
Oct 25, 202621867321/25000000proposedCareyRunboLiPR #75
Oct 25, 202621869322649/25000000000 Current recordmergedRexyysilentPR #71
Oct 25, 2026174954253/200000000proposedsubhanjandasPR #72
Oct 23, 2026437386453/500000000 Current recordmergedmrklwkPR #70
Oct 23, 20268747729897/10000000000 Current recordmergedHaileyStormPR #69
Oct 21, 20268748087/10000000 Current recordmergedGaozhongpaiPR #68
Oct 20, 2026174963413/200000000 Current recordmergedArviFoxyPR #67
Oct 20, 2026874828524351/1000000000000 Current recordmergedTheSilPR #66
Oct 18, 202687483/100000 Current recordmergedGaozhongpaiPR #63
Oct 18, 2026874835173/1000000000 Current recordmergedArviFoxyPR #64
Oct 16, 2026218709/250000 Current recordmergedGaozhongpaiPR #62
Oct 15, 202621871/25000 Current recordmergedGaozhongpaiPR #60
Oct 15, 202610935831/12500000mergedDevinWatsonPR #53
Oct 15, 2026437423/500000 Current recordmergedDaysSkyPR #56
Oct 15, 2026437429/500000mergedtimbaludPR #48
Oct 14, 20268748559/10000000 Current recordmergedtimbaludPR #50
Oct 14, 20262187109/2500000proposedblueberryvertigoPR #52
Oct 14, 20264374299/5000000 Current recordmergedGaozhongpaiPR #43
Oct 13, 2026874861/1000000 Current recordmergedHaileyStormPR #42
Oct 13, 20268748673/10000000 Current recordmergedGaozhongpaiPR #41
Oct 13, 202687486797/100000000 Current recordmergedkariroggPR #38
Oct 13, 20264374347/5000000 Current recordmergedkariroggPR #37
Oct 13, 202687487/100000 Current recordmergednielstronPR #26
Oct 12, 202629/33mergedHaileyStormPR #8
Oct 11, 20264374433/5000000proposedGaozhongpaiPR #15
Oct 10, 2026683505193/781250000 Current recordmergedHaileyStormPR #9
Oct 10, 202610499/12000 Current recordmergedakashlevyPR #6
Oct 10, 2026874957019420098946128603850561452983/1000000000000000000000000000000000000 Current recordmergednielstronPR #2
Oct 10, 2026437478509710049473064301925667/500000000000000000000000000000proposedGaozhongpaiPR #7
Oct 9, 2026874957019420098946128604623/1000000000000000000000000000 Current recordmergednielstronPR #1
Oct 9, 2026874957019421/1000000000000 Current recordmergedGTimGgithub.com
Oct 8, 2026(1507-2*sqrt(921))/1653 Current recordpublishedBaiying Liugithub.com
Oct 8, 20263499999/4000000 Current recordpublishedArgonaut Mathgithub.com
Sep 30, 20267/8 Current recordpublishedOpenAIgithub.com
Other tracked problems (8)

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