{"id":"hexagon/2610.00461","pdf":"https://hexagonmath.org/pdf/2610.00461v1","url":"https://hexagonmath.org/abs/2610.00461v1","tags":["msc:11M20","msc:11N05","Frobenius congruences","Landau--Siegel zeros","elementary methods","explicit estimates","quadratic characters"],"score":0,"title":"An explicit elementary proof of no Landau--Siegel zeros","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Victor Z. Guo"],"reviews":0,"subject":"math.NT","summary":"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 $\\chi$ of conductor $q\\ge3$, each real zero $\\beta\\in(0,1)$ of $L(s,\\chi)$ satisfies $(1-\\beta)\\log q>c$. 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,…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-10-22","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}