{"id":"openai-math/Uniform-exclusion-of-Landau-Siegel-zeros","pdf":"https://github.com/openai/math/blob/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026","tags":["analytic-number-theory","dirichlet-l-functions","siegel-zeros","l-functions"],"score":0,"title":"Uniform exclusion of Landau–Siegel zeros","domain":"Number theory","family":"003","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.NT","summary":"We prove the uniform exclusion of Landau–Siegel zeros. There is an absolute constant c > 0 such that every real zero $`\\beta\\in(0,1)`$ of every primitive nonprincipal real Dirichlet L -function of conductor q ≥ 3 satisfies $`(1-\\beta)\\log q\\ge c`$.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":10,"published":"2026-10-01","reactions":{},"discussion":[{"url":"https://arxiv.org/abs/2610.15483","date":"2026-10-11","kind":"web","site":"arXiv.org","text":"A recent manuscript of OpenAI asserts that Landau-Siegel zeros do not exist. Here, we give a detailed account of its argument, arranged as a direct proof with explicit constants. There are absolute constants $q_0$ and $δ_0>0$ such that, for every primitive real Dirichlet character $χ$ of conductor…","added":"2026-10-24","title":"On the OpenAI proof that Landau-Siegel zeros do not exist","author":"William D. Banks","record":"arxiv/2610.15483"},{"url":"https://arxiv.org/abs/2610.21099","date":"2026-10-15","kind":"web","site":"arXiv.org","text":"This paper simplifies the recent OpenAI proof that there are no Landau--Siegel zeros.","added":"2026-10-24","title":"A simpler proof that there are no Landau--Siegel zeros","author":"Bryce Kerr","record":"arxiv/2610.21099"},{"url":"https://x.com/thomasfbloom/status/2111718525685334277","date":"2026-10-18","kind":"x","added":"2026-10-24","author":"Thomas Bloom","handle":"thomasfbloom"},{"url":"https://x.com/jdlichtman/status/2111619107074592993","date":"2026-10-18","kind":"x","added":"2026-10-24","author":"Jared Duker Lichtman","handle":"jdlichtman"},{"url":"https://terrytao.wordpress.com/2026/10/19/a-sketch-of-the-proof-of-non-existence-of-siegel-zeroes/#:~:text=there%20was%20a%20third%20paper%20in%20the%20same%20release%20that%20did%20not%20establish%20the%20quasi-Riemann%20hypothesis","date":"2026-10-19","kind":"web","site":"What's new","text":"there was a third paper in the same release that did not establish the quasi-Riemann hypothesis, but instead excluded Siegel zeroes directly. This paper was significantly shorter (nine pages) and proceeded almost entirely via algebraic number theory and the geometry of numbers. Recently, various…","added":"2026-10-24","title":"A sketch of the proof of non-existence of Siegel zeroes"},{"url":"https://x.com/ivanswzhang/status/2112421999800992169","date":"2026-10-20","kind":"x","added":"2026-10-24","author":"Shaowu Zhang","handle":"ivanswzhang"},{"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"},{"url":"https://arxiv.org/abs/2610.30476","date":"2026-10-21","kind":"web","site":"arXiv.org","text":"We give an elementary proof of the absence of Landau--Siegel zeros with the explicit constant $c=1/100$. For every primitive quadratic Dirichlet character $χ$ of conductor $q\\ge3$, each real zero $β\\in(0,1)$ of $L(s,χ)$ satisfies $(1-β)\\log q>c$. The proof combines integer coefficient counting,…","added":"2026-10-24","title":"An explicit elementary proof of no Landau--Siegel zeros","author":"Victor Zhenyu Guo","record":"arxiv/2610.30476"},{"url":"https://arxiv.org/abs/2610.31402","date":"2026-10-22","kind":"web","site":"arXiv.org","text":"Let $k$ be a number field. We prove that for every quadratic extension $F/k$, every real zero $β\\in(0,1)$ of $L(s,χ_{F/k})$ satisfies $(1-β)\\log d_F\\ge c(k)$ with an effective constant $c(k)>0$, and we deduce (through a reduction of Stark) that the Brauer-Siegel theorem holds effectively and…","added":"2026-10-24","title":"Landau-Siegel zeros over a base field and effective Brauer-Siegel","author":"William D. Banks","record":"arxiv/2610.31402"},{"url":"https://x.com/LukeBailey181/status/2113346230046822641","date":"2026-10-22","kind":"x","added":"2026-10-24","author":"Luke Bailey","handle":"LukeBailey181"}],"formalized":true,"family_title":"The quasi-Riemann hypothesis"}