An elementary bound on Siegel zeroes

Abstract

We consider Dirichlet $L$-functions $L(s,\ddot{I}‡)$ where $\ddot{I}‡$ is a real, non-principal character modulo $q$. Using Pintz's refinement of Page's theorem, we prove that for $q\ge 3$ the function $L(s,\ddot{I}‡)$ has at most one real zero ${\hat{I}}^2$ with $1-1.011/\log q<{\hat{I}}^2<1$.

8 pages