{"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":25,"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":"Frank Calegari"},{"url":"https://alexkontorovich.github.io/quasi-rh/index.html","kind":"web","site":"alexkontorovich.github.io","text":"A twelve-beat blueprint of the Lean proof that the Riemann zeta function is nonzero for Re(s) > 11/12.","added":"2026-11-03","title":"The 11/12 argument · Lean blueprint","author":"Alex Kontorovich"},{"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/2610.12234","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","author":"Jakob Glas"},{"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://x.com/ShivaKintali/status/2107985377973920121","date":"2026-10-08","kind":"x","added":"2026-11-03","author":"Shiva Kintali","handle":"ShivaKintali"},{"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://mathoverflow.net/questions/515968/consequences-of-quasi-rh","date":"2026-10-12","kind":"web","site":"mathoverflow.net","text":"It is common knowledge that there are many consequences of the (generalized) Riemann hypothesis. Some of those consequences would also follow from a tighter critical strip. Now that we know $7/8$ works (see this tracker for the current best verified bound), what follows? What still needs a better…","added":"2026-11-04","title":"Consequences of quasi-RH","author":"Pace Nielsen"},{"url":"https://mathoverflow.net/a/515971","date":"2026-10-12","kind":"web","site":"mathoverflow.net","text":"The first example I would give is the error term in the Prime Number Theorem. If one has a zero free region $\\Re(s)>\\theta$ , then the error term in the PNT is $O(x^{\\theta}\\log x)$ (due to von Koch). Naturally this error will improve upon taking $\\theta$ closer to $1/2$ . I'll briefly justify the…","added":"2026-11-04","title":"Answer to: Consequences of quasi-RH","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://mathoverflow.net/a/516001","date":"2026-10-13","kind":"web","site":"mathoverflow.net","text":"One important consequence (present in the Astra-generated mega document that user GH from MO links to in their answer below, but worth highlighting here) is that the quasi-RH implies essentially optimal cancellation for short character sums (an improvement of Pólya-Vinogradov): For every…","added":"2026-11-04","title":"Answer to: Consequences of quasi-RH","author":"Thomas Bloom"},{"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://x.com/ObhishekSaha/status/2110275632223453627","date":"2026-10-14","kind":"x","added":"2026-11-03","author":"Abhishek Saha","handle":"ObhishekSaha"},{"url":"https://mathoverflow.net/a/516041","date":"2026-10-14","kind":"web","site":"mathoverflow.net","text":"The very recent preprint of Althöfer and Pintz shows that, up to $X$ , all but $X^{7/12+o(1)}$ even numbers can be written as a sum of two primes. This is a consequence of the QRH for Dirichlet $L$ -functions, now a theorem. Under GRH for Dirichlet $L$ -functions, $7/12$ would improve to $1/2$ .…","added":"2026-11-04","title":"Answer to: Consequences of quasi-RH","author":"GH from MO"},{"url":"https://mathoverflow.net/a/516109","date":"2026-10-16","kind":"web","site":"mathoverflow.net","text":"Matt Visser used the quasi-Riemann hypothesis to prove that for all positive integers $n$ there is a prime $p$ satisfying $n^8 < p \\leq n^8+n^7$ . In particular, there is a prime between every consecutive $8$ -th powers. Previous unconditional results either require $86$ -th powers, or are known…","added":"2026-11-04","title":"Answer to: Consequences of quasi-RH","author":"Bogdan Grechuk"},{"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","author":"Terence Tao"},{"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","author":"Konstantin Kakaes"},{"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","author":"Terence Tao"},{"url":"https://x.com/ObhishekSaha/status/2112415591952720073","date":"2026-10-20","kind":"x","added":"2026-11-03","author":"Abhishek Saha","handle":"ObhishekSaha"},{"url":"https://bsky.app/profile/johncarlosbaez.bsky.social/post/3myd7hu2irs2u","date":"2026-10-20","kind":"web","site":"Bluesky Social","text":"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/","added":"2026-11-03","title":"John C. Baez (@johncarlosbaez.bsky.social)","author":"John C. Baez"},{"url":"https://math.stackexchange.com/a/5153070","date":"2026-10-28","kind":"web","site":"math.stackexchange.com","text":"The Bound Formula column already has your answer: when $1/2 \\leq \\sigma < 1$ , if all Dirichlet $L$ -functions are zero-free for ${\\rm Re}(s) > \\sigma$ , then $n(p) \\ll (\\log p)^{1/(1-\\sigma)}$ . Let $\\sigma$ be whatever you want in the interval $[1/2,1)$ and you get a corresponding upper bound.…","added":"2026-11-04","title":"Answer to: Is the exponent $8$ optimal for the least quadratic nonresidue under QRH?","author":"KCd"},{"url":"https://mathoverflow.net/questions/516517/is-millers-original-primality-test-now-deterministic","date":"2026-10-28","kind":"web","site":"mathoverflow.net","text":"In the 1970's, Gary L. Miller obtained a conditionally deterministic primality test that runs in polynomial time. It relies on some form of GRH. My understanding is that only GRH for Dirichlet $L$ -functions are needed, which is covered under the recent breakthrough on Quasi-RH. Note that…","added":"2026-11-04","title":"Is Miller's original primality test now deterministic?","author":"Stanley Yao Xiao"}],"formalized":true,"family_title":"The quasi-Riemann hypothesis"}