Discrete mean estimates and the Landau-Siegel zero

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.