{"title":"Second Moment Of Dirichlet L-Functions, Character Sums Over Subgroups And Upper Bounds On Relative Class Numbers","authors":"Stéphane R Louboutin;Marc Munsch","doi":"10.1093/qmath/haab010","DOIUrl":null,"url":null,"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 \n<tex>${\\mathcal{A}}(p,d)$</tex>\n 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 \n<tex>${\\mathcal{A}}(p,d),$</tex>\n 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 \n<tex>$1\\leq d\\leq\\frac{\\log p}{3\\log\\log p}$</tex>\n, we obtain a significant improvement \n<tex>$h_{p,d}^- \\leq 2(\\frac{(1+o(1))p}{24})^{m/4}$</tex>\n over the trivial bound \n<tex>$h_{p,d}^- \\ll (\\frac{dp}{24} )^{m/4}$</tex>\n on the relative class numbers of the imaginary number fields of conductor \n<tex>$p\\equiv 1\\mod{2d}$</tex>\n and degree \n<tex>$m=(p-1)/d$</tex>\n, where d ≥ 1 is odd.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 4","pages":"1379-1399"},"PeriodicalIF":0.6000,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haab010","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quarterly Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://ieeexplore.ieee.org/document/9690911/","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
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.
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.