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 $q\ge3$, 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.