Discrete mean estimates and the Landau-Siegel zero

Abstract

Let $\chi$ be a real primitive character to the modulus $D$. It is proved that $$L(1,\chi )\gg (\log D{)}^{-2022}$$ where the implied constant is absolute and effectively computable. In the proof, the lower bound for $L(1,\chi )$ is first related to the distribution of zeros of a family of Dirichlet $L$-functions in a certain region, and some results on the gaps between consecutive zeros are derived. Then, by evaluating certain discrete means of the large sieve type, a contradiction can be obtained if $L(1,\chi )$ is too small.