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

TL;DR

This work studies the distribution and non-vanishing of Dirichlet L-functions when averaged over thin subgroups of characters. By exploiting small solutions to linear congruences and refined Farey-fraction product-sets, the authors establish asymptotic formulas for all even moments $M_{2k}(p,m)$ and $M_{2k}^-(p,m)$ under the regime $\varphi(d)=o(\log p)$, and obtain an asymptotic second moment on the critical line along with non-vanishing results for $L(1/2,\chi)$ in these thin families. The results extend to almost-all primes, yielding significant relaxation on the permissible subgroup size (e.g., $m \ge p^{2/3+\varepsilon}$, and in some cases $m$ up to $p^{1/6-\varepsilon}$ for nonvanishing). The nonvanishing proportions are explicit in terms of the height of the smallest rational in the dual subgroup, via quantities $\vartheta(m,p)$ and $\rho(\lambda,p)$, and mollifier methods are adapted to the thin-subgroup setting, producing concrete positive lower bounds on nonvanishing frequencies with quantitative dependence on the subgroup structure. These insights connect moments, distribution, and nonvanishing in the thin-family regime and illuminate potential applications to related arithmetic questions (e.g., class numbers and Dedekind sums).

Abstract

We obtain an asymptotic formula for all moments of Dirichlet $L$-functions $L(1,χ)$ modulo $p$ when averaged over a subgroup of characters $χ$ 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 $L(1/2,χ)$ 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.

Theorems & Definitions (29)

• Theorem 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Theorem 2.5
• Corollary 2.6
• Theorem 2.7
• Remark 2.8
• Theorem 2.9
• Remark 2.10