{"id":"openai-math/The-Quasi-Riemann-Hypothesis-October-5-2026__paper2","pdf":"https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026","tags":["analytic-number-theory","l-functions","zeta-function"],"score":0,"title":"The Quasi-Riemann Hypothesis (alternate 11/12 proof)","domain":"Number theory","family":"003","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.NT","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":11,"published":"2026-10-05","reactions":{},"discussion":[{"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://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://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/a/515992","date":"2026-10-12","kind":"web","site":"mathoverflow.net","text":"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}…","added":"2026-11-04","title":"Answer to: Why cubic Gauss sums?","author":"Will Sawin"},{"url":"https://x.com/ObhishekSaha/status/2110275632223453627","date":"2026-10-14","kind":"x","added":"2026-11-03","author":"Abhishek Saha","handle":"ObhishekSaha"},{"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://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://x.com/rperezmarco/status/2113515394841428306","date":"2026-10-23","kind":"x","added":"2026-11-03","author":"Ricardo Pérez-Marco","handle":"rperezmarco"},{"url":"https://mathoverflow.net/a/516520","date":"2026-10-29","kind":"web","site":"mathoverflow.net","text":"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…","added":"2026-11-04","title":"Answer to: Why cubic Gauss sums?","author":"Stopple"}],"formalized":false,"family_title":"The quasi-Riemann hypothesis"}