Moments and non-vanishing of $L$-functions over thin subgroups

Abstract: We obtain an asymptotic formula for all moments of Dirichlet $L$-functions $L(1,\chi)$ modulo $p$ when averaged over a subgroup of characters $\chi$ of size $(p-1)/d$ with $\varphi(d)=o(\log p)$. Assuming the infinitude of Mersenne primes, the range of our result is optimal and improves and generalises the previous result of S. Louboutin and M. Munsch (2022) for second moments. We also use our ideas to get an asymptotic formula for the second moment of $L(1/2,\chi)$ over subgroups of characters of similar size. This leads to non-vanishing results in this family where the proportion obtained depends on the height of the smallest rational number lying in the dual group. Additionally, we prove that, in both cases, we can take much smaller subgroups for almost all primes $p$. Our method relies on pointwise and average estimates on small solutions of linear congruences which in turn leads us to use and modify some results for product sets of Farey fractions.