We prove an asymptotic formula for the mean-square average of L-functions associated with subgroups of characters of sufficiently large size. Our proof relies on the study of certain character sums ${\mathcal{A}}(p,d)$ recently introduced by E. Elma, where p ≥ 3 is prime and d ≥ 1 is any odd divisor of p − 1. We obtain an asymptotic formula for ${\mathcal{A}}(p,d),$ which holds true for any odd divisor d of p − 1, thus removing E. Elma’s restrictions on the size of d. This answers a question raised in Elma’s paper. Our proof relies on both estimates on the frequency of large character sums and techniques from the theory of uniform distribution. As an application, in the range $1\leq d\leq\frac{\log p}{3\log\log p}$, we obtain a significant improvement $h_{p,d}^- \leq 2(\frac{(1+o(1))p}{24})^{m/4}$ over the trivial bound $h_{p,d}^- \ll (\frac{dp}{24} )^{m/4}$ on the relative class numbers of the imaginary number fields of conductor $p\equiv 1\mod{2d}$ and degree $m=(p-1)/d$, where d ≥ 1 is odd.

中文翻译：

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狄利克雷 L 函数的二阶矩、子群上的字符总和以及相对类数的上界

我们证明了与足够大的字符子组相关的 L 函数的均方平均值的渐近公式。我们的证明依赖于对 E. Elma 最近引入的某些字符和 ${\mathcal{A}}(p,d)$ 的研究，其中 p ≥ 3 是素数，d ≥ 1 是 p − 1 的任何奇数除数。我们得到 ${\mathcal{A}}(p,d),$ 的渐近公式，它适用于 p − 1 的任何奇数除数 d，从而消除了 E. Elma 对 d 大小的限制。这回答了 Elma 论文中提出的一个问题。我们的证明依赖于对大字符和频率的估计和来自均匀分布理论的技术。作为一个应用，在 $1\leq d\leq\frac{\log p}{3\log\log p}$ 范围内，我们获得了显着的改进 $h_{p,d}^- \leq 2(\frac {(1+o(1))p}{24})^{m/4}$ 超过平凡界限 $h_{p,