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

no lower bound tracked≤ value ≤87483/100000lower boundupper bound

Upper bound: the records, digit by digit

  1. Oct 18, 20260.8748300−5.9×10−4%by Gaozhongpai
  2. Oct 18, 2026≈ 0.8748352−9.5×10−5%by ArviFoxy
  3. Oct 16, 20260.8748360−4.6×10−4%by Gaozhongpai
  4. Oct 15, 20260.8748400−6.9×10−4%by Gaozhongpai
  5. Oct 15, 20260.8748460−0.0011%by DaysSky
  6. Oct 14, 20260.8748559−4.5×10−4%by timbalud

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.

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Cite this frontier

DiscoverIA. Quasi-Riemann hypothesis: uniform zero-free half-plane, the threshold θ [Frontier, checked on Oct 18, 2026]. DiscoverIA registry. https://discoveria.ai/en/frontieres/quasi-riemann/

Only the upper bound is tracked. The upper bound has moved 19 times since 2026, most recently on Oct 18, 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 (26 points)

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

Points : Upper bound
DateExact valueNatureAuthorsSource
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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