The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\Re s\gt 7/8$
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.
Comments and reviews
No comments yet.
Discussed elsewhere
Bounds for the quasi-Riemann hypothesis, with Lean proofs, sources and verification status.
— Alex Kontorovich (@AlexKontorovich) Oct 6, 2026
Assuming that $ζ(s)$ has no nontrivial zero with $\mathrm{Re}\, s > 7/8$, as asserted in a recent unrefereed preprint of OpenAI, we obtain $Λ\le 13963/100000 = 0.13963$ for the de Bruijn-Newman constant. A published Polymath boundary computation supplies a finite-region strip of half-width $1/5$.…
Recently, a uniform $\frac{7}{8}$ zero-free half-plane for finite-order Hecke $L$-functions over $\mathbb Q(\sqrt{-3})$ and its transfer to Dirichlet $L$-functions was given by OpenAI in [Ope26a]. In its proof, there are several parameters $b=\frac{1}{8}, \ell=\frac{1}{6}$, and $κ$. In this paper,…
But when I saw the 7/8 bound for the zero-free region of the Riemann zeta function among the released problems, I was genuinely shocked. I had always thought there was a broad consensus among mathematicians that the Quasi-Riemann Hypothesis was completely out of reach with existing mathematics.…
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.
Recently, OpenAI announced a uniform $7/8$ zero-free half-plane for finite-order Hecke $L$-functions over $\mathbb{Q}(\sqrt{-3})$, together with its transfer to Dirichlet $L$-functions. Assuming the analytic inputs of that work, we improve the boundary from $7/8=0.875$ to…
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…
A few days ago I saw a post on this site discussing a potential Polymath-type project for the Quasi Riemann Hypothesis, but am unable to find the post now. As far as I am aware, the proof proffered by OpenAI is quite opaque, and given the strength of the claimed result it would be beneficial for…
We prove an effective version of the Chebotarev density theorem, conditional on the quasi-Riemann hypothesis for the Dedekind zeta function. Drawing on recent breakthroughs of OpenAI and Cureton--Thorner, we make unconditional the Ellenberg--Venkatesh bounds on $\ell$-torsion in class groups of…
OpenAI (using a non-public "internal frontier model'') has recently (6 October 2026) made a number of remarkable claims, including some claims regarding the quasi-Riemann hypothesis (QRH). While there are good reasons to be cautious regarding these claims, one can nevertheless ask what (if true)…
Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube.
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…
API
curl /api/v1/records/openai-math/The-Quasi-Riemann-Hypothesis
curl -X POST /api/v1/records/openai-math/The-Quasi-Riemann-Hypothesis/vote -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"value": 1, "emoji": "🤔", "note": "optional"}'
curl -X POST /api/v1/records/openai-math/The-Quasi-Riemann-Hypothesis/comments -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"body": "…", "review": true}'
A Brief Digest of Some New OpenAI Results | Persiflagegaloisrepresentations.orgPersiflage
analytic techniques are not the right way of thinking about the problem. They may give the first nontrivial results, but are ultimately insufficient for the final proof. One example here is the Ramanujan conjecture concerning the size of the coefficients of \(\Delta\). There is a bound of order…