View source for The quasi-Riemann hypothesis

'''The quasi-Riemann hypothesis''' is a group of manuscripts dated 30 September to 5 October 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]). This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== The original result ==
* '''Claim.''' Every Dirichlet ''L''-function, including the Riemann zeta function ζ(''s''), has no zeros in the half-plane Re ''s'' &gt; 7/8. The same half-plane is zero-free for every finite-order Hecke ''L''-function over '''Q'''(√−3). A companion manuscript gives a different proof of the zero-free half-plane Re ''s'' &gt; 11/12. A further manuscript gives a uniform exclusion of Landau–Siegel zeros (summary in the collection's contents list).
* '''Sources''' (openai/math repository, commit <code>adc7f12</code>):
** [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026 ''The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s) &gt; 7/8''] (30 September 2026);
** [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026 ''Uniform exclusion of Landau–Siegel zeros''] (1 October 2026);
** [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026 ''The Quasi-Riemann Hypothesis''] (5 October 2026; the 11/12 companion).
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/003.md lean/docs/003.md] states that the Lean formalization gives the 7/8 half-plane for ζ(''s'') and for every Dirichlet ''L''-function, uniformly over all moduli and characters, and for finite-order Hecke ''L''-functions over '''Q'''(√−3), with the principal-character poles at ''s'' = 1 excluded. It also gives a uniform gap 1 − β ≥ ''c''/log ''q'' for real zeros of real primitive characters, with no explicit value of ''c''. The paper's later applications are not included. The comparator statements are listed in that note.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 13–14 October 2026)
|-
| 2026-10-07 || [https://github.com/eliottcassidy2000/math/blob/813a52a4912327c363fe840d282c9ee4c23a3839/05-knowledge/results/oai3_two_orbits_twos_and_threes_20261007.md Results note of an openai/math reading session] (§4), in the GitHub repository eliottcassidy2000/math || Use (conditional) || Taking the 7/8 theorem as correct: the Borwein–Choi problem is closed, that is, every positive integer except 1, 2, 4, 6, 10, 18, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330 and 462 is ''xy'' + ''yz'' + ''zx'' with ''x'', ''y'', ''z'' ≥ 1; and the lists of imaginary quadratic fields with class-group exponent 4 and 8 of Elsenhans, Klüners and Nicolae are complete. (The completeness of Euler's list of 65 idoneal numbers is already stated in OpenAI's 5 October manuscript.) || A record in a multi-agent AI research repository; its dependent results are labelled "PROVED modulo openai/math #003 (accepted per owner directive 2026-10-07)".
|-
| 2026-10-08 (latest version) || B. Yates, ''Zero-free half-planes for Hecke L-functions of nonzero infinity type over Q(√−3)'', Zenodo ([https://doi.org/10.5281/zenodo.23207672 10.5281/zenodo.23207672]) || Extension (conditional) || If OpenAI's argument for finite-order Hecke characters of '''Q'''(√−3) is correct, then the ''L''-function of every unitary Hecke character of nonzero infinity type has no zeros in Re ''s'' &gt; 11/12; consequently ''L''(''E'', ''s'') ≠ 0 for Re ''s'' &gt; 17/12 for every elliptic curve over '''Q''' with complex multiplication by an order of '''Q'''(√−3), and the analogous statement holds for newforms with complex multiplication by '''Q'''(√−3). || A note on Zenodo; the abstract states the conditional nature of the result.
|-
| 2026-10-08 || J. Guo, [https://github.com/aster2024/openai-math-followups/tree/main/cm-elliptic-curves ''A zero-free half-plane for Hecke L-functions of nonzero angular type and for the curves y² = x³ + D''] (GitHub, aster2024/openai-math-followups) || Extension || Every unitary Hecke character of '''Q'''(√−3) of finite conductor and nonzero angular type has no zeros in Re ''s'' &gt; 11/12; hence ''L''(''E''<sub>''D''</sub>, ''s'') ≠ 0 for Re ''s'' &gt; 17/12 for every curve ''E''<sub>''D''</sub>: ''y''² = ''x''³ + ''D''. The paper also gives sector equidistribution of Eisenstein primes and a Sato–Tate error term ''O''(''x''<sup>11/12+ε</sup>) for ''E''<sub>''D''</sub>. It extends the cubic-theta method of the 5 October manuscript. || A 36-page paper with programs checking explicit identities. The README states that it was prepared independently of Yates's note.
|-
| 2026-10-08 || B. Liu, ''Slightly improved zero-free half-planes for the quasi-Riemann hypothesis'' ([https://arxiv.org/abs/2610.12234 arXiv:2610.12234], v1) || Extension || Varying the two parameters of OpenAI's proof improves 7/8 to 34999/40000 and to (1507 − 2√921)/1653 = 0.874957069799… || arXiv v1. The abstract states that the result has been formalized in Lean.
|-
| 2026-10-08 || teal-sea, [https://github.com/teal-sea/zeta-lab zeta-lab] (hunt <code>qrh_linnik</code> and related hunts) || Use || Assuming OpenAI's 7/8 theorem: the least prime in an arithmetic progression modulo ''q'' is at most ''C''(ε)''q''<sup>7/3+ε</sup> (with the density estimate of Chen, Gupta and Li, [https://arxiv.org/abs/2507.08296 arXiv:2507.08296]), and at most ''C''(ε)''q''<sup>12/5+ε</sup> with refereed inputs (Montgomery, Huxley). Other hunts treat class numbers, nonresidues and primes between powers. || The results file is graded "candidate, pending external review"; it states that the proofs were written by a model and not yet reviewed.
|-
| 2026-10-08 || C. Hakimoglu-Brown, ''A conditional bound Λ ≤ 0.13963 for the de Bruijn–Newman constant'' ([https://arxiv.org/abs/2610.12803 arXiv:2610.12803], v1) || Use (conditional) || Assuming that ζ(''s'') has no zero with Re ''s'' &gt; 7/8, the de Bruijn–Newman constant satisfies Λ ≤ 0.13963, using the published Polymath boundary computation. || arXiv v1.
|-
| 2026-10-11 || S. Bhattacharya, [https://hexagonmath.org/2610.00178v1 ''Unconditional implications of the Quasi-Riemann Hypothesis''], Hexagon 2610.00178v1 || Use (survey of consequences) || Identifies published conclusions formerly obtained under RH or GRH whose proofs need only a fixed zero-free half-plane for Dirichlet ''L''-functions, taking the 7/8 theorem as input, with an accompanying audit of sources and rejected candidates. || Posted on Hexagon (v1). The page states that it was generated by a single prompt to GPT-6 Astra and that "No human involvement or scrutiny has been performed".
|-
| 2026-10-12 || D. Cureton, J. Thorner, [https://hexagonmath.org/2610.00200v1 ''A quasi-Riemann hypothesis for Dedekind zeta functions''], Hexagon 2610.00200v1 || Related extension || For every number field ''K'', the Dedekind zeta function ζ<sub>''K''</sub>(''s'') has no zeros in Re ''s'' &gt; 1 − 1/6000, and the same holds for finite-order Hecke ''L''-functions over ''K''. The abstract describes the method as following the strategy of recent work on Artin's primitive root conjecture; the references include the OpenAI quasi-Riemann manuscripts and OpenAI's ''Primitive Roots for Every Admissible Integer Base''. || Posted on Hexagon (v1). The page states that the proofs and the Lean formalization were generated by AxiomProver, with human understanding of "some parts".
|-
| 2026-10-13 || data1493, [https://github.com/data1493/quasi-rh-audit ''External Audit of OpenAI's Quasi-Riemann Hypothesis Manuscript (Re(s) &gt; 7/8)''] (GitHub) || External audit (partial) || Reconstructs the 30 September manuscript layer by layer and tabulates its loss budgets. Scope-limited verdict: the paper's own exponent ledgers close with a positive remainder under the losses the paper explicitly charges (a Stage 1 saving of 1/4800 and a Stage 2 endpoint constant 49/440640). Residual risk: the margins are small, and an intermediate lemma could carry an unlisted loss; the 11/12 companion and the Landau–Siegel manuscript are not covered. || The README states that this is "not an independent re-proof of the analytic estimates", that the audit is AI-assisted (SuperGrok Heavy agents) and that "no professional number theorist has reviewed this work yet"; it invites specialist review through issues.
|-
| 2026-10-13 || B. Hu, [https://hexagonmath.org/2610.00216v1 ''A seven-eighths zero-free half-plane for unitary Weil L-functions''], Hexagon 2610.00216v1 || Extension || For every number field ''F'' and every finite-dimensional continuous unitary representation ''r'' of the global Weil group, ''L''<sub>''F''</sub>(''s'', ''r'') has no zeros in Re ''s'' &gt; 7/8. The rank-one core covers every unitary Hecke character of every number field, including characters of infinite order; stated consequences include the same half-plane for all Artin ''L''-functions and an effective Chebotarev theorem. The abstract describes the method as extending the cubic-theta method of the OpenAI preprint. || Posted on Hexagon (v1). The page states that the arguments and text were generated predominantly by AI language models and that "The complete proof has not been independently verified by a human"; it links an earlier draft at [https://github.com/Rogerhu12/openai-math-extensions Rogerhu12/openai-math-extensions].
|-
| 2026-10-13 || M. Rubinstein, M. Restrepo Borrero, [https://hexagonmath.org/2610.00220v1 ''A zero-free strip for the L-functions of the elliptic curves y² = x³ + d''], Hexagon 2610.00220v1 || Extension || Adapting OpenAI's 11/12 argument gives ''L''(''E''<sub>''d''</sub>, ''s'') ≠ 0 for Re ''s'' &gt; 17/12; adapting the 7/8 argument to the infinite-order characters attached to these curves gives ''L''(''E''<sub>''d''</sub>, ''s'') ≠ 0 for Re ''s'' &gt; 11/8, for every curve ''E''<sub>''d''</sub>: ''y''² = ''x''³ + ''d''. || Posted on Hexagon (v1), with source code at [https://github.com/mrubinst/L-function_zero_free_strips mrubinst/L-function_zero_free_strips]. The page states that much of the development, writing and test code was produced with Claude Opus 5.5 and OpenAI Astra/Codex, and that the authors "do not claim that the complete proof has been independently checked and understood by a human mathematician".
|}

== Notes ==
* Several follow-ups state substantial AI assistance in producing their proofs and texts, as recorded in the table.
* Results marked conditional assume the correctness of the OpenAI argument.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]