Year: 2026
Type: preprint
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 $q\ge3$, each real zero $β\in(0,1)$ 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.
Source: arXiv (Cornell University)
Author: Guo, Victor Zhenyu
Institution: Xi'an Jiaotong University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223405956