OpenAI Quasi-Riemann Hypothesis: zero-free region Re(s) > 11/12 (Oct 6, 2026)

On October 6, 2026, OpenAI's 722-manuscript math release included a claimed proof of a zero-free region Re(s) > 11/12 for the Riemann zeta function ζ(s), plus a second independent proof for Re(s) > 7/8 which also excludes Siegel zeros. This is not a proof of the full Riemann Hypothesis — it's a stronger-than-previous partial result in the direction of the 167-year-old conjecture.

The result was the headline math achievement of the OpenAI release. Only 162 of 722 papers (~22%) in the release have full Lean formalization — the quasi-Riemann proof is among the unformalized majority, meaning independent verification is still pending. For the broader community response, see our mathematicians reaction article.

What is the Riemann Hypothesis?

The Riemann Hypothesis (1859) is one of the 7 Millennium Prize Problems (each worth $1M). It conjectures:

All non-trivial zeros of the Riemann zeta function ζ(s) lie on the critical line Re(s) = 1/2.

Why it matters:

Equivalent to the Prime Number Theorem (distribution of primes) being as regular as possible
Connections to random matrix theory, quantum chaos, and L-functions in algebraic geometry
167 years unsolved despite efforts by the greatest mathematicians (Hilbert, Hardy, Selberg, etc.)

What did OpenAI prove?

The result: There exists a region Re(s) > 11/12 (≈ 0.917) of the critical strip that contains no non-trivial zeros of ζ(s).

A second independent proof: There exists a region Re(s) > 7/8 (≈ 0.875) that contains no non-trivial zeros of ζ(s) and excludes Siegel zeros (a special class of zeros with deep arithmetic consequences).

What this does NOT prove:

❌ The full Riemann Hypothesis (which would require showing ALL non-trivial zeros on Re(s) = 1/2)
❌ Any of the Millennium Prize Problems (which require unconditional proof)
❌ A complete description of the zero distribution

What this DOES prove:

✅ A stronger zero-free region than previously known
✅ Improved upon Vinogradov-Korobov 1958 and Polymath 8b 2020 results
✅ If verified, would represent a meaningful step toward the full conjecture

The history of zero-free regions

Year	Result	Re(s) >	Author
1859	Riemann conjectures all zeros on Re(s) = 1/2	—	Riemann
1896	First non-trivial zero-free region	Re(s) > 1	Hadamard, de la Vallée Poussin (independently)
1958	Vinogradov-Korobov zero-free region	Re(s) > 1 − c/log(t)	Vinogradov, Korobov
1980s-2000s	Refinements of Vinogradov-Korobov	1 − c/(log t)^(2/3)	Various
2020	Polymath 8b project	Re(s) > 1 − 0.1571/(log t)	Polymath (collaborative)
2026 (OpenAI)	Quasi-Riemann	Re(s) > 11/12 (= 0.917)	OpenAI frontier model

Why Re(s) > 11/12 is a big deal: Polymath 8b's result requires log t in the denominator — for very large t, the bound approaches 1. The OpenAI result gives a uniform 11/12 independent of t — a qualitative improvement, not just quantitative.

Technical context (without full math)

The Riemann zeta function is:

ζ(s) = Σ(n=1 to ∞) 1/n^s = Π(p prime) 1/(1 − p^(-s))

where s = σ + it is a complex number. Non-trivial zeros are solutions to ζ(s) = 0 with 0 < σ < 1.

Key facts:

Trivial zeros at s = −2, −4, −6, ... (negative even integers)
All other zeros are in the critical strip 0 < Re(s) < 1
Riemann conjectured all non-trivial zeros are exactly on Re(s) = 1/2

OpenAI's claim: No non-trivial zeros exist in the region Re(s) > 11/12 (and a second independent proof for Re(s) > 7/8 + no Siegel zeros).

What is a Siegel zero?: A hypothetical "exceptional" zero extremely close to Re(s) = 1 that would imply the Generalized Riemann Hypothesis is false. OpenAI's second proof excludes the existence of any Siegel zeros — a separate, valuable result.

Implications for the full Riemann Hypothesis

The Quasi-Riemann result is NOT a step toward a full proof in the traditional sense — the methods for zero-free regions (Vinogradov-Korobov + refinements) are different from the methods for proving Re(s) = 1/2.

But: The result provides computational validation that the deep methods of analytic number theory (explicit formula, density estimates, exponential sums) can be deployed at scale by AI systems. This may inform future attempts at the full RH.

The Millennium Prize: Clay Mathematics Institute still lists the Riemann Hypothesis as unsolved and the $1M prize is unclaimed.

Verification status (as of Oct 9, 2026)

Status	Detail
Lean formalization	❌ Not yet — the OpenAI release has Lean for ~60% of families, but the Quasi-Riemann proof is among the unformalized minority
Independent peer review	Pending — math community has not yet had time to verify (release only 3 days old)
Kevin Buzzard (Imperial College) review	"162 of 722 papers (~22%) have Lean verification. OpenAI pulled 3 papers for elementary mistakes shortly after posting." — for full breakdown, see our mathematicians reaction article
AGMAI statement	The Advisory Group on Mathematics and AI (Gowers, Hairer, Witten, Tao) called for independent verification of all 722 results before acceptance

Bottom line: The result is claimed but not yet verified. The next 6-12 months will see mathematicians + Lean experts work through the proof.

Historical context: AI math achievements before this

Date	AI result	Field	Verified?
2024	AlphaProof solves IMO problems (silver-medal level)	Olympiad math	✅ Verified by humans
May 2026	OpenAI disproves 1946 Erdős unit distances conjecture	Combinatorics	✅ Verified
July 2026	GPT-5.6 Sol Ultra proves Cycle Double Cover Conjecture	Graph theory	✅ Verified
Aug 2026	Ten advances in math + TCS	Multiple	Partially verified
Sept 8, 2026	~10,000 AI agents claim Navier-Stokes (88 hours)	Fluid dynamics	Contested
Oct 6, 2026	Quasi-Riemann Re(s) > 11/12	Number theory	Pending verification

The Quasi-Riemann result is among the most ambitious AI math claims to date — comparable in difficulty to top human research in analytic number theory.

Why this matters for AI

The Quasi-Riemann result is the strongest evidence yet that frontier AI models can produce publishable-quality research in pure mathematics, not just applied math or Olympiad problems.

Implications:

Frontier math is AI-tractable — at least for "quasi" results
Lean formalization is the bottleneck — AI can produce structured arguments, but formal proof-checking is still hard for the hardest results
Mathematics as a discipline is being transformed — Terence Tao's "scarcity-to-abundance" transition is now tangible

FAQ: Quasi-Riemann Hypothesis

What did OpenAI prove about the Riemann Hypothesis?

A zero-free region Re(s) > 11/12 for the Riemann zeta function. Not a proof of the full Riemann Hypothesis — a stronger-than-previous partial result.

Is this a proof of the Riemann Hypothesis?

No. The full Riemann Hypothesis requires showing all non-trivial zeros are on Re(s) = 1/2. OpenAI only proved that there are no zeros in a strip Re(s) > 11/12.

Is the Quasi-Riemann proof verified?

Not yet — the proof is among the ~40% of OpenAI's release that doesn't have Lean formalization. Independent verification by mathematicians + Lean experts is pending.

What was the previous best zero-free region?

Polymath 8b (2020): Re(s) > 1 − 0.1571/log(t) for sufficiently large t. OpenAI's result is qualitatively stronger (uniform Re(s) > 11/12, not requiring t → ∞).

What is a Siegel zero?

A hypothetical exceptional zero very close to Re(s) = 1 that would imply the Generalized Riemann Hypothesis is false. OpenAI's second proof excludes Siegel zeros — a separate valuable result.

Who verified the Quasi-Riemann result?

No one yet — the release is 3 days old. The math community will spend 6-12 months verifying.

Is the Quasi-Riemann result the first AI math breakthrough?

No — OpenAI has made several previous claims (Erdős conjecture May 2026, Cycle Double Cover July 2026). But Quasi-Riemann is among the most ambitious to date.

How does the Quasi-Riemann result compare to the full RH?

Much weaker — RH requires ALL non-trivial zeros on Re(s) = 1/2. OpenAI only proved no zeros in Re(s) > 11/12. The full RH is still open.

Is the Quasi-Riemann result consistent with the RH?

Yes — if RH is true, the zero-free region Re(s) > 11/12 follows immediately (since RH says all zeros are on Re(s) = 1/2 < 11/12). The OpenAI result is weaker than RH but consistent with it.

Will this result win the Millennium Prize?

No — the $1M Millennium Prize is for the full Riemann Hypothesis. Quasi-Riemann is a partial result.

What does the AGMAI say about this?

The Advisory Group on Mathematics and AI (Gowers, Hairer, Witten, Tao) called for independent verification of all 722 OpenAI results before acceptance. They "do not endorse" testing advanced math on proprietary models.

Bottom line: OpenAI claimed a zero-free region Re(s) > 11/12 for the Riemann zeta function on Oct 6, 2026 — a stronger-than-previous partial result, not a proof of the full RH. Among the unformalized minority of the 722-manuscript release. Pending independent verification over the next 6-12 months.

For the full OpenAI 722 math release, see our overview article. For the other Millennium Prize progress, see our Hodge CM article and Hilbert 10th ℚ article.