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 (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 > 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 > 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
adc7f12):- The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s) > 7/8 (30 September 2026);
- Uniform exclusion of Landau–Siegel zeros (1 October 2026);
- The Quasi-Riemann Hypothesis (5 October 2026; the 11/12 companion).
- Formalization, as described by the repository. The scope note 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
| Date | Work | Relation | Public claim | Public status (checked 13 October 2026) |
|---|---|---|---|---|
| 2026-10-08 (latest version) | B. Yates, Zero-free half-planes for Hecke L-functions of nonzero infinity type over Q(√−3), Zenodo (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 > 11/12; consequently L(E, s) ≠ 0 for Re s > 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, 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 > 11/12; hence L(ED, s) ≠ 0 for Re s > 17/12 for every curve ED: y² = x³ + D. The paper also gives sector equidistribution of Eisenstein primes and a Sato–Tate error term O(x11/12+ε) for ED. 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 (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, zeta-lab (hunt qrh_linnik and related hunts) |
Use | Assuming OpenAI's 7/8 theorem: the least prime in an arithmetic progression modulo q is at most C(ε)q7/3+ε (with the density estimate of Chen, Gupta and Li, arXiv:2507.08296), and at most C(ε)q12/5+ε 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 (arXiv:2610.12803, v1) | Use (conditional) | Assuming that ζ(s) has no zero with Re s > 7/8, the de Bruijn–Newman constant satisfies Λ ≤ 0.13963, using the published Polymath boundary computation. | arXiv v1. |
| 2026-10-11 | S. Bhattacharya, 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, A quasi-Riemann hypothesis for Dedekind zeta functions, Hexagon 2610.00200v1 | Related extension | For every number field K, the Dedekind zeta function ζK(s) has no zeros in Re s > 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 | B. Hu, 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, LF(s, r) has no zeros in Re s > 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 Rogerhu12/openai-math-extensions. |
| 2026-10-13 | M. Rubinstein, M. Restrepo Borrero, 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(Ed, s) ≠ 0 for Re s > 17/12; adapting the 7/8 argument to the infinite-order characters attached to these curves gives L(Ed, s) ≠ 0 for Re s > 11/8, for every curve Ed: y² = x³ + d. | Posted on Hexagon (v1), with source code at 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.