The Quasi-Riemann Hypothesis (alternate 11/12 proof)
Source: github.com/openai/math/tree/main/preprints/The-Quasi-Rieman…
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.
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IMO, the biggest thing that goes against conventional wisdom is that there would be 3(!) proofs using conventional technology , to show the critical strip for the Riemann zeta-function can be narrowed.
Back in 2012 I overheard Andrew Wiles commenting on the development of IUTT in Mochizuki's purported proof of the ABC conjecture: "It's like being present at the invention of Galois Theory". One had the sense of a new tool with potentially immense power. Looking at OpenAI's paper on the…
I can give a starting point for an answer to this question. The key identity uses, in notation matching OpenAI's writeup, the Gauss sum $$ \gamma_j(n) = \frac{1}{ \sqrt{N_{K/\mathbb Q}(n)}}\sum_{x \bmod n} \chi_n(x)^je(x/n).$$ The identity is $$\mu(n) \gamma_{-1}(n)= \chi_n(-1)^{-1} G(n)^{-1}…
— Abhishek Saha (@ObhishekSaha) Oct 14, 2026
These arguments are analytic in nature, using identities relating the Möbius function to the Fourier coefficients of certain automorphic forms; they are still in the process of being fully digested, and I will not be discussing them further here. However, there was a third paper in the same…
the 11/12 result is the only one of the 722 OpenAI papers released on October 6 to have been edited by a person. OpenAI has not addressed inquiries about who that person is, or why this is the only paper to have had the benefit of human expertise before being released. In response to questions…
new (and effective) ways to rule out Siegel zeroes, either by establishing a quasi-Riemann hypothesis, or by some version of the argument in this previous post . In principle, each version of these arguments should also give a bound on the tenth discriminant (without using Heegner’s solution to…
This article says the implications of the newly proved quasi-Riemann hypothesis are "staggering" - but it doesn't mention any. So what are they? https://www.quantamagazine.org/ai-proved-the-quasi-riemann-hypothesis-what-happens-now-20261019/
— Ricardo Pérez-Marco (@rperezmarco) Oct 23, 2026
Inspired by Will Sawin's answer, I decided to track down the proof of the relation between Gauss sums and the Möbius function. OpenAI references Hasse's Vorlesungen über Zahlentheorie , but my German is not good enough to get anything out of that, and also Heath-Brown's paper 'Kummer’s conjecture…
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The 11/12 argument · Lean blueprintalexkontorovich.github.ioAlex Kontorovich
A twelve-beat blueprint of the Lean proof that the Riemann zeta function is nonzero for Re(s) > 11/12.