Landau-Siegel Zero

A Landau-Siegel zero, also called a Siegel zero or exceptional zero, is a possible real zero near 1 of a Dirichlet L-series associated with a primitive nonprincipal number theoretic character taking only real values. A character is primitive when it is not induced from one of smaller modulus; its modulus is then its conductor . Nonprincipal means that it is not identically 1 on the integers relatively prime to .

The classical zero-free region

(1)

for a primitive Dirichlet L-series has at most one exceptional zero, which must be real and simple and belong to a real character. Here is an absolute constant (Davenport 1980).

Kimura (2026) reports the explicit zero-free bound

(2)

for every real zero of a primitive real Dirichlet L-series of conductor . Kimura (2026) reports checking the analytic bound in Lean and an independent kernel, using parts of an OpenAI formalization and substantial Claude assistance.

Banks (2026a) reports the stronger explicit bound

(3)

for every real zero of a primitive real Dirichlet L-series of conductor . The analytic argument covers , and the extension to smaller conductors uses a cited computation that Banks did not independently verify. The reported gap would exclude zeros in , not every real zero in .

Kerr (2026) reports a simpler proof of the existence of a uniform positive logarithmic gap, using the pigeonhole principle in place of a determinant argument. Both follow-ups develop the OpenAI argument. Banks states that he checked the written argument line by line, while calling for further independent scrutiny.

Guo (2026) reports the stronger explicit gap for every real zero of a primitive quadratic Dirichlet L-series with conductor . Banks (2026b) reports an effective analog for quadratic extensions of a fixed number field, with an application to the Brauer-Siegel theorem. These follow-up claims require further independent verification.