Result 003, Number theory

The quasi-Riemann hypothesis

Proves that every Dirichlet L-function, including $\zeta(s)$ , is zero-free in $\Re s\gt 7/8$ , resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over $\mathbb Q(\sqrt{-3})$ . A companion gives a different proof of the zero-free half-plane $\Re s\gt 11/12$ .

Lean formalization Proof

The bigger picture

Why it matters

The manuscripts claim that broad families of number-theoretic functions never vanish in a fixed region of the complex plane. Such a uniform exclusion would give researchers a firm boundary on where their zeros can occur.

What changes?

Dirichlet L-functions are functions of a complex variable built from periodic arithmetic patterns; they include the Riemann zeta function. The main manuscript claims none vanishes where the real part exceeds seven eighths, at any imaginary height. It claims the same for every finite-order Hecke L-function, a related family over the number field obtained by adjoining the square root of minus three to the rationals. A pole at one is allowed. The companion reports the weaker threshold eleven twelfths.

What does that help mathematicians do?

If correct, the seven-eighths boundary would force every real zero between zero and one in these families to lie at least one eighth away from one. That excludes the near-one exceptional zeros known as Landau–Siegel zeros, as the companion also claims. Researchers could therefore rule out this obstruction uniformly, rather than needing an exclusion that weakens with the function's arithmetic parameters. Zeros on the boundary remain possible.

Are there practical applications?

The immediate value is foundational. A common zero-free region would provide a shared input for arguments involving zeta, Dirichlet and the specified Hecke L-functions, replacing assumptions about zeros near one with a claimed uniform guarantee. The supplied abstracts do not establish a computational speedup or an applied implementation.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\Re s\gt 7/8$

September 30, 2026 199 pages Main result formalized in Lean

We prove that all finite-order Hecke L-functions over $\mathbb Q(\sqrt{-3})$ and all Dirichlet L-functions are zero-free in the half-plane $\Re s\gt 7/8$ , with the principal pole at s = 1 allowed. In particular, the Riemann zeta function is zero-free in this half-plane, proving the quasi-Riemann hypothesis.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:The-Quasi-Riemann-Hypothesis-September-30-2026,
  author = {{OpenAI}},
  title = {{The Quasi-Riemann Hypothesis:
            A Zero-Free Half-Plane $\mathrm{Re}(s)>7/8$}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf}{OAI:The-Quasi-Riemann-Hypothesis-September-30-2026}},
  year = {2026}
}

The Quasi-Riemann Hypothesis (alternate 11/12 proof)

October 5, 2026 49 pages

We establish the quasi-Riemann hypothesis by proving that every Dirichlet L-function, including Riemann's zeta function, has no zeros in the half-plane $\mathop{\mathrm{Re}}\nolimits s\gt 11/12$ . More generally, we prove the same zero-free half-plane for every finite-order Hecke L-function over $K=\mathbb Q(\sqrt{-3})$ . In particular, this rules out the existence of Landau–Siegel zeros.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:The-Quasi-Riemann-Hypothesis-October-5-2026,
  author = {{OpenAI}},
  title = {{The Quasi-Riemann Hypothesis}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf}{OAI:The-Quasi-Riemann-Hypothesis-October-5-2026}},
  year = {2026}
}

Uniform exclusion of Landau–Siegel zeros

October 1, 2026 9 pages Main result formalized in Lean

We prove the uniform exclusion of Landau–Siegel zeros. There is an absolute constant c > 0 such that every real zero $\beta\in(0,1)$ of every primitive nonprincipal real Dirichlet L-function of conductor q ≥ 3 satisfies $(1-\beta)\log q\ge c$ .

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026,
  author = {{OpenAI}},
  title = {{Uniform exclusion of Landau--Siegel zeros}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf}{OAI:Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/003.md.

The quasi-Riemann hypothesis

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The quasi-Riemann hypothesis asks for a fixed zero-free half-plane $\Re s>\theta$ with $\theta<1$ . The formalization gives $\theta=7/8$ for the Riemann zeta function and every Dirichlet $L$ -function, uniformly over all positive moduli and all characters. It also establishes the same bound for finite-order Hecke $L$ -functions over $\mathbb Q(\sqrt{-3})$ .

The principal-character poles at $s=1$ are excluded in the Dirichlet and Hecke statements. The paper's later applications are not included.

The formalized result gives a uniform logarithmic exclusion region for Landau–Siegel zeros. There is one constant $c>0$ such that every primitive nonprincipal real Dirichlet character of conductor $q\ge3$ and every real zero $0<\beta<1$ of its $L$ -function satisfy $1-\beta\ge c/\log q$ .

Both character parities are included. No explicit value of $c$ is given. This excludes real zeros in $1-c/\log q<\beta<1$ , but does not rule out real zeros elsewhere in $(0,1)$ .

Comparator links

Result	Comparator statement
Riemann zeta $7/8$ bound	QuasiRiemannHypothesis.lean
Dirichlet $L$ -function $7/8$ bound	DirichletSevenEighths.lean
Finite-order Hecke $L$ -function $7/8$ bound	HeckeSevenEighths.lean
Uniform real-zero gap	SiegelZeros.lean

Posts about this result

Alex Kontorovich @AlexKontorovich, Distinguished Professor of Mathematics, Rutgers

Quasi-RH?!?!???! Are you kidding me? If a human did this, it would be an instant Fields Medal, no questions asked. RH says zeta has no zeros in Re(s)>1/2. The best we had until a second ago was a region that got thinner and thinner the higher up the imaginary axis you go. I thought maybe they’d fatten that up a bit, that’d be a massive breakthrough. But no. They got a zero free strip!!!! Insane

Quoting @OpenAI: We’re releasing a broad range of new mathematical results produced by an internal frontier model. We’ve been consulting with the independent Advisory Group on Mathematics and Artificial Intelligence at the Institute f...

Oct 6, 2026, 7:08 PM ET

Jared Duker Lichtman @jdlichtman, Assistant Professor of Mathematics, Stanford

While reading their proof, important note: The cubic family of L-functions appear to be essential for the proof, even for the corollary to the Riemann zeta function

Quoting @jdlichtman: OpenAI has just released floodgates for 300+ solutions to math problems. Among them include a proof of: -- a zero-free strip for the Riemann Zeta function -- a proof of the Hodge conjecture for CM abelian varieties ht...

Oct 6, 2026, 7:37 PM ET

Acer @AcerFur, Cambridge mathematics student, formerly OpenAI

It will always be the highlight of my career to have been able to see firsthand a model make progress towards the problem that got me interested in mathematics a decade ago. It was a huge honour to have been part of this team. A uniform zero-free strip. github.com/openai/math/blob/main/preprints/Th...

Oct 6, 2026, 6:24 PM ET

Acer @AcerFur, Cambridge mathematics student, formerly OpenAI

I hope this is a gift of knowledge to humanity. I expect a renaissance in some fields of mathematics thanks to this. Polymath project bringing down 7/8 towards 1/2 when? (FWIW the method has a barrier at 3/4, so it needs a nice new idea)

Oct 6, 2026, 6:53 PM ET

At a glance

Subject: Number theory
Manuscripts: 3
Pages: 257
Dated: Sep 30 to Oct 5, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Quasi-Riemann hypothesisProved

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

Most similar results

022, Number theoryThe weak inhomogeneous Duffin–Schaeffer conjecture
004, Number theoryHilbert’s tenth problem over ℚ
010, Number theoryUnrestricted pro-modularity at the prime two
007, Number theoryOrdinary two-point correlations and the corrected Elliott conjecture
014, Number theoryRestricted geometric Langlands, global Arthur enhancements, and generic Ramanujan
015, Number theoryTorus-packet equidistribution in prime, quartic, and sextic degrees

By similarity of summaries and abstracts.

On the map

This result, outlined, and its most similar results among all 372. Open the full map.

Why it matters

What changes?

What does that help mathematicians do?

Are there practical applications?

3 manuscripts

The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane ℜs>7/8\Re s\gt 7/8ℜs>7/8

The Quasi-Riemann Hypothesis (alternate 11/12 proof)

Uniform exclusion of Landau–Siegel zeros

Lean formalization

The quasi-Riemann hypothesis

Scope

Comparator links

Posts about this result

The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\Re s\gt 7/8$