Second Moment Of Dirichlet L-Functions, Character Sums Over Subgroups And Upper Bounds On Relative Class Numbers

IF 0.6 4区 数学 Q3 MATHEMATICS Quarterly Journal of Mathematics Pub Date : 2021-01-01 DOI:10.1093/qmath/haab010
Stéphane R Louboutin;Marc Munsch
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引用次数: 4

Abstract

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.
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Dirichlet l -函数的二阶矩、子群上的字符和及相对类数上的上界
我们证明了与足够大的字符子群相关的l -函数的均方平均值的一个渐近公式。我们的证明依赖于E. Elma最近引入的某些特征和${\mathcal{A}}(p,d)$的研究,其中p≥3是素数,d≥1是p−1的任何奇因子。我们得到了${\mathcal{A}}(p,d),$的渐近公式,该公式适用于p−1的任何奇数因子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。我们的证明既依赖于对大字符和频率的估计,也依赖于均匀分布理论的技术。作为应用,在$1\leq d\leq\frac{\log p}{3\log\log p}$范围内,我们得到了导体$p\equiv 1\mod{2d}$和度$m=(p-1)/d$的虚数场相对类数在平凡界$h_{p,d}^- \ll (\frac{dp}{24} )^{m/4}$上的显著改进$h_{p,d}^- \leq 2(\frac{(1+o(1))p}{24})^{m/4}$,其中d≥1为奇。
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
36
审稿时长
6-12 weeks
期刊介绍: The Quarterly Journal of Mathematics publishes original contributions to pure mathematics. All major areas of pure mathematics are represented on the editorial board.
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