A Conditional Refinement of Page's Theorem on zeros of Dirichlet $L$-functions

Landau–Siegel zeros are hypothetical zeros of Dirichlet $L$-functions that are close to the point $s=1$. A classic theorem of Page shows at most one such zero can exist among all Dirichlet $L$-functions with conductor $\le Q$. We show that one can significantly refine Page’s theorem under the assumption that all non-real zeros of Dirichlet $L$-functions lie outside a shrinking neighborhood of $s=1$.