On the size of L(1,χ) and S. Chowla's hypothesis implying that L(1,χ)>0 for s>0 and for real characters χ

Abstract

We give explicit constants κ such that if χ is a real non-principal Dirichlet character for which L(1,χ)≤κ, then Chowla's hypothesis is not satisfied and we cannot use Chowla's method for proving that L(s,χ)>0 for s>0. These constants are larger than the previous ones κ=1−log2=0.306… and κ=0.367… we obtained elsewhere.