The quasi-Riemann hypothesis after OpenAI: who has improved the 7/8 bound

On 29 October 2026, Baiying Liu of Purdue University posted a paper on Hexagon claiming a quasi-Riemann bound of about 0.874468, below the 7/8 = 0.875 that OpenAI announced three weeks earlier, for every finite-order Hecke L-function over every number field. Its provenance statement says that OpenAI Codex and Anthropic Claude "assisted with drafting, checking references, the search for improved bounds, and auditing", and that "The Lean formalization is in progress".

The number 7/8 started as an AI claim. On 6 October, OpenAI released a manuscript credited to "OpenAI" and produced by an internal model, which claims that the Riemann zeta function and every Dirichlet L-function have no zeros with real part greater than 7/8. Liu's paper is the ninth follow-up we could trace in the weeks since: papers and code repositories on Zenodo, GitHub, arXiv and Hexagon that push the number down, extend it to more L-functions, or both. Mathematicians have also discussed the result on blogs, in Quanta Magazine and in a seminar talk.

Below: what the quasi-Riemann hypothesis says and why it is not the Riemann hypothesis, what OpenAI released, who has responded and improved the bound since, how each follow-up describes its use of AI, and what is formalized. Every source was checked on 30 October 2026. All the results here are preprints or repositories; none has appeared in a journal, and we do not judge whether any of them is correct.

What is the quasi-Riemann hypothesis?#

It is a weaker cousin of the Riemann hypothesis, about where the zeros of the zeta function can lie. The Riemann zeta function is ζ(s) = 1 + 2^−s + 3^−s + …, extended to complex numbers s. Apart from the "trivial" zeros at −2, −4, −6, …, its zeros lie in the critical strip, where the real part of s is between 0 and 1. They come in mirror-image pairs across the line of real part ½, and the Riemann hypothesis says they all lie on that line.

Mathematicians have known since 1896 that no zero has real part exactly 1, which gives the prime number theorem. Later work widened that zero-free zone, but every classical zero-free region gets thinner higher up the strip: the larger the imaginary part, the closer a zero may come to real part 1. As OpenAI's paper puts it, such regions "do not give uniform exclusion in a fixed half-plane."

The quasi-Riemann hypothesis asks for one fixed gap instead: a number σ₀ below 1 such that ζ(s) has no zeros with real part greater than σ₀, at any height. In the words of OpenAI's paper, "It asks for one gap valid at every height, not merely nonvanishing on ℜs = 1." Because the zeros come in mirror pairs, a bound of 7/8 confines them to the strip between 1/8 and 7/8, as Quanta put it. The Riemann hypothesis is the extreme case σ₀ = ½.

OpenAI's theorem holds for a whole family at once: every Dirichlet L-function, the relatives of ζ that count primes in arithmetic progressions, with one bound that does not depend on the character, its conductor or the height. Its 11/12 companion paper says that this "rules out the existence of Landau–Siegel zeros", the hypothetical exceptional real zeros near 1 behind what Terence Tao calls "the infamous Siegel zero problem".

Why it matters, in the words of Maksym Radziwiłł of New York University, speaking to Quanta: "In a sense, the vast majority of consequences of RH already follow from quasi-RH."

Does the 7/8 result prove the Riemann hypothesis?#

No, and the paper says so itself: "The theorem does not establish the Riemann hypothesis or its generalized versions, which place nontrivial zeros on ℜs = 1/2. The Riemann hypothesis remains open." The 11/12 companion calls the quasi-Riemann hypothesis "the weaker assertion". On 30 October the Clay Mathematics Institute still listed the Riemann hypothesis as "Unsolved".

The improvements since 6 October change the bound in the fourth decimal place. The Riemann hypothesis needs it at 0.5.

Number theorists quoted so far do not expect this route to get there. Frank Calegari wrote on 8 October: "I still believe, although perhaps with slightly less conviction, that improvements to these analytic arguments are not going to be sufficient to prove the full Riemann hypothesis." Will Sawin of Princeton told Quanta: "It seems most likely to me that once mathematicians have fully internalized the new methods and worked through their implications, we will still feel as far off from the full Riemann hypothesis as we were before. But I am not certain of this." Thomas Bloom of the University of Manchester agreed that the full hypothesis still seems out of reach, but added: "we would have said the same about quasi-RH a few weeks ago! At this point nothing would surprise me."

What did OpenAI release on 6 October?#

Two manuscripts, both credited to "OpenAI", in its public GitHub repository openai/math:

• The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane ℜs > 7/8, dated 30 September, 199 pages. Its Theorem 1.1: "Every finite-order Hecke L-function over F = Q(√−3) has no zero in ℜs > 7/8. The same holds for every Dirichlet L-function, including ζ(s)."
• The Quasi-Riemann Hypothesis, dated 5 October, 49 pages, which claims the weaker bound 11/12 by a shorter route.

The repository went public the same day, alongside a blog post about results "produced by an internal frontier model". Its README says the manuscripts were produced by "an unnamed internal OpenAI model", that the zero-free-region work was one of the "Exceptions to this fixed procedure", and that "the writeup for the Re(s) > 11/12 zero-free region for the Riemann zeta function was human edited for readability." It does not say who edited it. Quanta asked; OpenAI replied: "The paper is the result of work from everyone at OpenAI."

The 7/8 result comes with a Lean formalization. OpenAI's scope note says it "gives θ=7/8 for the Riemann zeta function and every Dirichlet L-function, uniformly over all positive moduli and all characters", and the same bound for Hecke L-functions over Q(√−3). The 11/12 paper has no formalization listed. As of 30 October, the repository's change log records no revision or withdrawal of either manuscript, and the 7/8 folder has not changed since its first commit on 6 October.

The Advisory Group on Mathematics and Artificial Intelligence (AGMAI), which OpenAI says it consulted on the release, wrote on 6 October that its role "should not be interpreted as a judgment of the impact of these results or an endorsement of the process by which OpenAI obtained them", and that "only the mathematical community can undertake the assessment that is needed."

How did mathematicians respond?#

8 October: Calegari's digest. Frank Calegari's blog, Persiflage, ran a brief digest of the release. His section on the quasi-Riemann hypothesis opens with "WTF." He had thought classical zero-free regions "represented something close to the limits of analytic arguments", and wrote: "Obviously, I was totally wrong, and something was clearly missed. But I have no idea of the structure of what is going on." He also named a gap that later papers took up: for applications "which concern general number fields, one really needs the corresponding result for all Dedekind L-functions rather than merely Dirichlet L-functions."

19 October: Quanta. Konstantin Kakaes's article followed the specialists working through the proof. Radziwiłł said in an email that both papers "are horribly written (in a way that would lead to immediate rejection at any mathematical journal)", so readers had turned to the shorter 11/12 paper. Peter Sarnak repurposed a course to study it. Radziwiłł "can now sketch the proof's arguments in half a page", and longer write-ups of seven to 13 pages were "circulating privately". Alexander Dunn of Georgia Tech called the proof "a surprising synthesis of well-known ideas"; a formula Dunn and Radziwiłł proved in 2021 turned out to be a crucial ingredient. Quanta reported that these experts "largely agree that the result is correct". Sarnak said he won't be satisfied that it's "a Theorem" until humans check and understand all the details.

19 October: Tao on Siegel zeros. Terence Tao's blog post is about a third paper in the same release, nine pages long, which excludes Siegel zeros directly, "almost entirely via algebraic number theory and the geometry of numbers", without proving the quasi-Riemann hypothesis. He sketches Ben Green's version of it. Of the two quasi-Riemann papers he writes that they "are still in the process of being fully digested", and that in them "it appears to be some (still poorly understood) automorphic miracle that is breaking the parity barrier."

28 October: a seminar at NYU. Radziwiłł gave a talk titled "The quasi-Riemann Hypothesis" in the Courant Institute's "Mathematics in the Age of AI" seminar, according to the department calendar. According to a post on X by Jaume de Dios, who was in the room, Radziwiłł was explaining the proof. We have not seen slides, notes or a recording.

Who has improved the 7/8 bound since?#

We traced nine follow-ups, most of them cited in Liu's 29 October paper, and opened each one. The list may not be complete.

The work runs in two strands.

Tuning the same argument. OpenAI's proof fixes several parameters. Liu's first abstract names them: "b = 1/8, ℓ = 1/6". Within two days of the release, four people had tuned parameters of the argument and posted slightly smaller bounds. Mermet's and Wei's repositories reached the same number, 4374785349/5000000000, and both come with Lean proofs. Liu's second version also varied a moment parameter, κ, and claims optimality "within the explicitly specified exponent comparison system". Xu and Sang then replaced one estimate in the argument with "an amplified first moment of the reflected numerator" and reached 0.8748867. Newton refined their parameter comparison further, "a refinement of the comparison rather than a new mean-value estimate", as Liu's paper describes it. Several say outright that they depend on OpenAI's paper. Mermet's README says its note "is only as sound as that proof"; Xu and Sang write that their conclusions "are conditional on the correctness of the analytic inputs from [OAI26], whose proofs have not been independently verified here."

More number fields. This is the gap Calegari pointed to. OpenAI's theorem covers Hecke L-functions over one field, Q(√−3), and, through them, Dirichlet L-functions. Cureton and Thorner claimed a zero-free half-plane at 31/32 for the Dedekind zeta function of every number field, "Following their strategy", meaning the approach of OpenAI's related paper on Artin's primitive root conjecture. Liu's 19 October paper extends the tuned argument to every number field, at about 0.8748866, and his 29 October paper claims a lower bound. Liu notes that the two settings are not interchangeable: "a bound over the Eisenstein field alone does not imply the same bound over every number field."

For scale: the newest bound sits about 0.00053 below 7/8, against about 0.00011 for the best bound before it.

What does Liu's 29 October paper claim?#

The paper, "An improved zero-free half-plane for Hecke L-functions over number fields and Quasi-Riemann Hypothesis", is 64 pages long, dated 27 October and deposited on Hexagon on 29 October under a CC BY 4.0 license. Hexagon is a repository for research in the mathematical sciences and theoretical computer science. Liu is the only author.

Its main theorem: "Let K be a number field and let χ be a finite-order Hecke character of K. Then L_K(s, χ) has no zeros in ℜs > B∗." The bound B∗ = 0.87446830769353045955… is "an explicitly defined algebraic number", built from the root of a cubic equation. Since the rational numbers are a number field too, the statement covers ζ(s) and Dirichlet L-functions. The paper is careful about its edges: "The boundary is independent of K and χ. No assertion is made on the line ℜs = B∗."

In the abstract's words, "The proof refines the cubic-theta comparison." It builds on what came before. "We use the earlier results cited in Section 2 as established theorems," it says. Those results come chiefly from OpenAI's 7/8 paper, a second OpenAI paper and Liu's own 19 October paper, along with classical work. Its optimality claim is limited to that framework: "It does not assert the best possible zero-free region for a field."

What it says about AI. The Hexagon record carries two declarations: the text is "A mix of human-written and AI-generated", and on human understanding, "Some parts of this contribution, but not all, have been checked and understood by a human." The provenance statement, in full:

OpenAI Codex (GPT 6.1 Sol Medium-Ultra) and Anthropic Claude (Opus 5.5 Medium) assisted with drafting, checking references, the search for improved bounds, and auditing. The Lean formalization is in progress, based on the Lean files in [Liu26c] and in [CTLean].

The model names are the author's own. [Liu26c] is Liu's GitHub repository, and [CTLean] is Axiom Math's Lean formalization of the Cureton–Thorner result.

What the repository holds. Liu's GitHub repository holds the Lean files for his earlier arXiv bound, 0.874957…, not for the new one. Its README reports "Verification status: PASS (11 October 2026, Hong Kong)", and says the files "passed fresh strict replay with Lean 4.34.1". That is the author's own report of a check on the earlier result. For the 29 October bound, the Lean work is, in the paper's words, "in progress".

How did AI tools figure in the follow-ups?#

On both sides: producing results and helping people read them. Each work describes it in its own words:

The Zenodo record pages for Chakhkiev's and Newton's papers say nothing about AI. Zenodo blocked our downloads, so we could not read the PDFs themselves.

The declarations range from a one-line README to Hexagon's structured labels. Only some name a model and version, and every name is self-reported. The human part is stated in different ways too: Cureton and Thorner "directed the research and made the mathematical judgments"; Xu and Sang write that the key idea "is due to the author"; Liu lists the tools as having "assisted".

Which of these results are formalized in Lean?#

Some, and the formal ones lean on each other.

• Formalized, by the source's own account: OpenAI's 7/8; the Mermet and Wei refinements; Liu's arXiv bound of 12 October; and Cureton and Thorner's 31/32, whose README says the proof "uses only Lean's three standard axioms".
• In progress: Liu's 19 October and 29 October papers.
• Not formalized, as far as their pages say: the 11/12 companion; Chakhkiev's and Newton's conditional bounds; and Xu and Sang's, which the later Liu papers build on.

The two Lean refinements we read most closely do not take OpenAI's result on trust; they rebuild its proof. Wei's repository keeps OpenAI's 2,926 Lean modules for the 7/8 proof "unchanged". Liu's merges 2,925 of them into eight files and replays them before its own proofs; its README says the OpenAI foundation "is source material, not an axiom replacing its theorems."

A Lean build checks that the formal statements follow from the axioms and the imported libraries. It does not check that those statements are the theorem mathematicians mean; people do that comparison. A follow-up without Lean is as strong as its inputs, as Xu and Sang's note says.

What three weeks of uptake look like#

In 23 days the claim went from a GitHub release to a blog digest, a news feature, a seminar talk and nine follow-ups in four venues, several of them revised once already. Seven of the nine declare AI help on the pages we could read, and six of those say AI took part in producing the proof or searching for the bound.

The bookkeeping is being done by hand, in reference lists. Liu's 29 October paper pins its sources: the Axiom formalization "revision 09afbb4ce6570c1615d58509dc197fc9f0956b7d, accessed 15 October 2026"; the two GitHub refinements "accessed 28 October 2026"; OpenAI's paper with "All result and equation locators … refer to this dated version." When results build on results this quickly, which version said what, and when, becomes part of the mathematics.

Two questions stay open. If a follow-up uses an AI-produced paper "as established theorems", what happens to it if the original is revised? And who reads the 199-page original, now that specialists study the shorter 11/12 paper instead?

Where to follow it on Papex#

The Riemann hypothesis page on Papex lists OpenAI's 7/8 claim and the responses to it, as their sources state them. With this post, Papex adds Liu's 29 October paper there as a new partial-result claim. It also appears on /ai-claims. Listing a claim is not a judgment of it. The Riemann hypothesis stays open, as the Clay Mathematics Institute lists it. For where the other Millennium Prize Problems stand, see Has AI solved a Millennium Prize Problem?

Sources#

• An improved zero-free half-plane for Hecke L-functions over number fields and Quasi-Riemann Hypothesis, hexagon:2610.00655v1 (Baiying Liu, Hexagon, 29 October 2026), record page and PDF, checked 30 October 2026
• Improved-zero-free-half-planes-for-the-quasi-Riemann-hypothesis (Baiying Liu, GitHub), README and README_on_lean, checked 30 October 2026
• Improved zero-free half-planes for the quasi-Riemann hypothesis, arXiv:2610.12234 (Baiying Liu, v1 8 October 2026, v2 12 October 2026), checked 30 October 2026
• Zero-free half-planes for Hecke L-functions over number fields and Quasi-Riemann Hypothesis, arXiv:2610.26835 (Baiying Liu, v1 19 October 2026, v2 29 October 2026), checked 30 October 2026
• Amplified moments and a refinement of the quasi-Riemann hypothesis, arXiv:2610.15291 (Yuanda Xu and Hejian Sang, v1 11 October 2026, v2 15 October 2026), checked 30 October 2026
• A quasi-Riemann hypothesis for Dedekind zeta functions, hexagon:2610.00200v2 (Dean Cureton and Jesse Thorner, Hexagon, v1 12 October 2026, v2 14 October 2026), checked 30 October 2026
• Dedekind Zeta Zero-Free Region (Axiom Math, GitHub), README, checked 30 October 2026
• The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane ℜ(s) > 0.8749570698 (manuscript dated 8 October 2026, GitHub repository CrossNox/quasi-riemann-hypothesis-extension), README and manuscript source, checked 30 October 2026
• Quasi-Riemann boundary refinement in Lean (GitHub repository LW56e2/quasi-riemann-improved), README, checked 30 October 2026
• A conditional improvement of the seven-eighths zero-free threshold (Khamzat Chakhkiev, Zenodo, 7 October 2026), checked 30 October 2026
• Prime Errors and the Quasi-Riemann Frontier: A Certified Conditional Boundary and Its Parameter-Optimization Barrier (Tony Newton, Zenodo, 16 October 2026), checked 30 October 2026
• The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane ℜs > 7/8 (OpenAI, 30 September 2026), checked 30 October 2026
• The Quasi-Riemann Hypothesis, 11/12 (OpenAI, 5 October 2026), checked 30 October 2026
• Lean scope note 003, the quasi-Riemann hypothesis (openai/math, GitHub), checked 30 October 2026
• openai/math README and history (GitHub), checked 30 October 2026
• Sharing AI progress in mathematics (OpenAI, 6 October 2026), checked 30 October 2026 through the Internet Archive copy of 6 October 2026; the live page blocked automated access
• On OpenAI's Release of Mathematical Results (AGMAI, 6 October 2026), checked 30 October 2026
• A Brief Digest of Some New OpenAI Results (Frank Calegari, Persiflage, 8 October 2026), checked 30 October 2026
• AI Proved the Quasi-Riemann Hypothesis. What Happens Now? (Konstantin Kakaes, Quanta Magazine, 19 October 2026), checked 30 October 2026
• A sketch of the proof of non-existence of Siegel zeroes (Terence Tao, What's new, 19 October 2026), checked 30 October 2026
• The quasi-Riemann Hypothesis, Mathematics in the Age of AI seminar (NYU Courant, 28 October 2026), checked 30 October 2026
• Post by Jaume de Dios (X, 28 October 2026), read on 30 October 2026 through a public read-only copy; x.com itself did not open without signing in
• Riemann Hypothesis (Clay Mathematics Institute), checked 30 October 2026