Uniform exclusion of Landau–Siegel zeros

On the OpenAI proof that Landau-Siegel zeros do not existarXiv.orgWilliam D. BanksOct 11, 2026in the index

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…

A simpler proof that there are no Landau--Siegel zerosarXiv.orgBryce KerrOct 15, 2026in the index

This paper simplifies the recent OpenAI proof that there are no Landau--Siegel zeros.

A sketch of the proof of non-existence of Siegel zeroesWhat's newOct 19, 2026

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…

An experimental AI-math challenge: using tenth discriminant bounds to explore arguments ruling out Siegel zeroesWhat's newOct 20, 2026

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…

An explicit elementary proof of no Landau--Siegel zerosarXiv.orgVictor Zhenyu GuoOct 21, 2026in the index

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,…

Landau-Siegel zeros over a base field and effective Brauer-SiegelarXiv.orgWilliam D. BanksOct 22, 2026in the index

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…