{"id":"openai-math/The-Quasi-Riemann-Hypothesis","pdf":"https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026","tags":["analytic-number-theory","l-functions","zeta-function"],"score":1,"title":"The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $`\\Re s\\gt 7/8`$","domain":"Number theory","family":"003","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.NT","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":16,"published":"2026-09-30","reactions":{"up":1},"discussion":[{"url":"https://galoisrepresentations.org/2026/10/08/a-brief-digest-of-some-new-openai-results/#:~:text=analytic%20techniques%20are%20not%20the%20right%20way%20of%20thinking%20about%20the%20problem","kind":"web","site":"galoisrepresentations.org","text":"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…","added":"2026-10-24","title":"A Brief Digest of Some New OpenAI Results | Persiflage","author":"Persiflage"},{"url":"https://gtimg.github.io/QuasiRiemannTracker/","kind":"web","site":"gtimg.github.io","text":"Bounds for the quasi-Riemann hypothesis, with Lean proofs, sources and verification status.","added":"2026-10-24","title":"QRH Leaderboard"},{"url":"https://x.com/AlexKontorovich/status/2107609087902941646","date":"2026-10-06","kind":"x","added":"2026-10-24","author":"Alex Kontorovich","handle":"AlexKontorovich"},{"url":"https://arxiv.org/abs/2610.12803","date":"2026-10-08","kind":"web","site":"arXiv.org","text":"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 $1/5$.…","added":"2026-10-24","title":"A conditional bound $\\Lambda\\le0.13963$ for the de Bruijn--Newman constant","author":"Cetin Hakimoglu-Brown","record":"arxiv/2610.12803"},{"url":"https://arxiv.org/abs/2610.12234","date":"2026-10-08","kind":"web","site":"arXiv.org","text":"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,…","added":"2026-10-24","title":"Improved zero-free half-planes for the quasi-Riemann hypothesis","author":"Baiying Liu","record":"arxiv/2610.12234"},{"url":"https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/#:~:text=But%20when%20I%20saw%20the%207%2F8%20bound%20for%20the%20zero-free%20region%20of%20the%20Riemann%20zeta%20function%20among%20the%20released%20problems","date":"2026-10-08","kind":"web","site":"Proofs and Prompts","text":"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.…","added":"2026-10-24","title":"100+ reactions to 100+ solutions"},{"url":"https://mathoverflow.net/a/515854","date":"2026-10-08","kind":"web","site":"mathoverflow.net","text":"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.","added":"2026-10-24","title":"Answer to: Which of the results released by OpenAI went against the conventional mathematical wisdom?","author":"Pace Nielsen"},{"url":"https://arxiv.org/abs/2610.15291","date":"2026-10-11","kind":"web","site":"arXiv.org","text":"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…","added":"2026-10-24","title":"Amplified moments and a refinement of the quasi-Riemann hypothesis","author":"Yuanda Xu, Hejian Sang","record":"arxiv/2610.15291"},{"url":"https://mathoverflow.net/questions/515985/why-cubic-gauss-sums","date":"2026-10-12","kind":"web","site":"mathoverflow.net","text":"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…","added":"2026-10-24","title":"Why cubic Gauss sums?","author":"Stopple"},{"url":"https://mathoverflow.net/questions/515982/group-collaborative-project-on-quasi-riemann-hypothesis-proof","date":"2026-10-12","kind":"web","site":"mathoverflow.net","text":"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…","added":"2026-10-24","title":"Group collaborative project on Quasi Riemann Hypothesis proof","author":"Benjamin Durkan"},{"url":"https://arxiv.org/abs/2610.18491","date":"2026-10-13","kind":"web","site":"arXiv.org","text":"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…","added":"2026-10-24","title":"Some applications of the quasi-Riemann hypothesis to class groups and small generators of number fields","author":"Christopher Frei, Efthymios Sofos, Martin Widmer","record":"arxiv/2610.18491"},{"url":"https://arxiv.org/abs/2610.20058","date":"2026-10-14","kind":"web","site":"arXiv.org","text":"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)…","added":"2026-10-24","title":"Primes between 8th powers?","author":"Matt Visser","record":"arxiv/2610.20058"},{"url":"https://www.youtube.com/watch?v=QNSbI3mrrhc","kind":"web","site":"youtube.com","text":"Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube.","added":"2026-10-24","title":"OpenAI Claimed 722 Math Breakthroughs. Did Any Hold Up?","author":"Nontrivial"},{"url":"https://terrytao.wordpress.com/2026/10/19/a-sketch-of-the-proof-of-non-existence-of-siegel-zeroes/#:~:text=These%20arguments%20are%20analytic%20in%20nature%2C%20using%20identities%20relating%20the%20M%C3%B6bius%20function","date":"2026-10-19","kind":"web","site":"What's new","text":"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…","added":"2026-10-24","title":"A sketch of the proof of non-existence of Siegel zeroes"},{"url":"https://www.quantamagazine.org/ai-proved-the-quasi-riemann-hypothesis-what-happens-now-20261019/#:~:text=the%2011%2F12%20result%20is%20the%20only%20one%20of%20the%20722%20OpenAI%20papers%20released%20on%20October%206%20to%20have%20been%20edited%20by%20a%20person","date":"2026-10-19","kind":"web","site":"Quanta Magazine","text":"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…","added":"2026-10-24","title":"AI Proved the Quasi-Riemann Hypothesis. What Happens Now? | Quanta Magazine"},{"url":"https://terrytao.wordpress.com/2026/10/20/an-experimental-ai-math-challenge-using-tenth-discriminant-bounds-to-explore-arguments-ruling-out-siegel-zeroes/#:~:text=new%20%28and%20effective%29%20ways%20to%20rule%20out%20Siegel%20zeroes%2C%20either%20by%20establishing%20a%20quasi-Riemann%20hypothesis","date":"2026-10-20","kind":"web","site":"What's new","text":"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…","added":"2026-10-24","title":"An experimental AI-math challenge: using tenth discriminant bounds to explore arguments ruling out Siegel zeroes"}],"formalized":true,"family_title":"The quasi-Riemann hypothesis"}