PreprintCC BY 4.0arXiv
An explicit elementary proof of no Landau--Siegel zeros
Abstract
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≥3, each real zero β∈(0,1) of L(s,χ) satisfies (1−β)log q>c. The proof combines integer coefficient counting, polynomial congruences, a paired-tail hyperbola estimate, and a logarithmically weighted comparison of positive sums. All required constants are explicit, and one choice of parameters covers every conductor. The argument uses elementary arithmetic and real summation and integration, without separate zero computations for small conductors.
Related
- A simpler proof that there are no Landau--Siegel zeros
- On the OpenAI proof that Landau-Siegel zeros do not exist
- Diophantine approximation by primes and Landau--Siegel zeros
- Landau-Siegel zeros over a base field and effective Brauer-Siegel
- Surjective Stability of Dickson-Siegel-Eichler-Roy Elementary Orthogonal Group over Projective Modules
BibTeX
@misc{guo2026explicit,
title = {{An explicit elementary proof of no Landau--Siegel zeros}},
author = {Victor Zhenyu Guo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30476},
eprint = {2610.30476},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30476},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An explicit elementary proof of no Landau--Siegel zeros
AU - Victor Zhenyu Guo
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30476
UR - https://arxiv.org/abs/2610.30476
AB - 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≥3, each real zero β∈(0,1) of L(s,χ) satisfies (1−β)log q>c. The proof combines integer coefficient counting, polynomial congruences, a paired-tail hyperbola estimate, and a logarithmically weighted comparison of positive sums. All required constants are explicit, and one choice of parameters covers every conductor. The argument uses elementary arithmetic and real summation and integration, without separate zero computations for small conductors.
ER -