On the OpenAI proof that Landau-Siegel zeros do not exist
Abstract
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 q0 and δ0>0 such that, for every primitive real Dirichlet character χ of conductor q≥ q0, every real zero β∈(0,1) of L(s,χ) satisfies (1−β)log q≥δ0, and one can take q0=1010 and δ0=3.0× 10−3. Combined with a recent computation of Lu, Zaman, and Zhao, this result yields the bound (1−β)log q≥ 3.0× 10−3 for every conductor q≥ 3. The proof is an interpolation determinant argument in the biquadratic field generated by √2 over the quadratic field attached to χ. A real zero close to one forces χ(p)=−1 for most primes p, and the Frobenius congruence at these primes makes a certain nonzero determinant divisible by higher powers of these primes than its Archimedean size allows. The underlying ideas are those of the OpenAI manuscript. The explicit constants, and the presentation, are new.
Related
BibTeX
@misc{banks2026openai,
title = {{On the OpenAI proof that Landau-Siegel zeros do not exist}},
author = {William D. Banks},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15483},
eprint = {2610.15483},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15483},
}
RIS
TY - UNPB
TI - On the OpenAI proof that Landau-Siegel zeros do not exist
AU - William D. Banks
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15483
UR - https://arxiv.org/abs/2610.15483
AB - 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 q≥ q_0, every real zero β∈(0,1) of L(s,χ) satisfies (1−β)log q≥δ_0, and one can take q_0=10^10 and δ_0=3.0× 10^−3. Combined with a recent computation of Lu, Zaman, and Zhao, this result yields the bound (1−β)log q≥ 3.0× 10^−3 for every conductor q≥ 3. The proof is an interpolation determinant argument in the biquadratic field generated by √2 over the quadratic field attached to χ. A real zero close to one forces χ(p)=−1 for most primes p, and the Frobenius congruence at these primes makes a certain nonzero determinant divisible by higher powers of these primes than its Archimedean size allows. The underlying ideas are those of the OpenAI manuscript. The explicit constants, and the presentation, are new.
ER -