An elementary bound on Siegel zeroes

Abstract

We consider Dirichlet $L$-functions $L(s, χ)$ where $χ$ is a real, non-principal character modulo $q$. Using Pintz's refinement of Page's theorem, we prove that for $q\geq 3$ the function $L(s, χ)$ has at most one real zero $β$ with $1- 1.011/\log q < β< 1$.

8 pages