p-adic L-函数的Dirichlet级数展开式

IF 0.4 4区 数学 Q4 MATHEMATICS
Heiko Knospe, Lawrence C. Washington
{"title":"p-adic L-函数的Dirichlet级数展开式","authors":"Heiko Knospe,&nbsp;Lawrence C. Washington","doi":"10.1007/s12188-021-00244-0","DOIUrl":null,"url":null,"abstract":"<div><p>We study <i>p</i>-adic <i>L</i>-functions <span>\\(L_p(s,\\chi )\\)</span> for Dirichlet characters <span>\\(\\chi \\)</span>. We show that <span>\\(L_p(s,\\chi )\\)</span> has a Dirichlet series expansion for each regularization parameter <i>c</i> that is prime to <i>p</i> and the conductor of <span>\\(\\chi \\)</span>. The expansion is proved by transforming a known formula for <i>p</i>-adic <i>L</i>-functions and by controlling the limiting behavior. A finite number of Euler factors can be factored off in a natural manner from the <i>p</i>-adic Dirichlet series. We also provide an alternative proof of the expansion using <i>p</i>-adic measures and give an explicit formula for the values of the regularized Bernoulli distribution. The result is particularly simple for <span>\\(c=2\\)</span>, where we obtain a Dirichlet series expansion that is similar to the complex case.</p></div>","PeriodicalId":50932,"journal":{"name":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2021-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s12188-021-00244-0.pdf","citationCount":"4","resultStr":"{\"title\":\"Dirichlet series expansions of p-adic L-functions\",\"authors\":\"Heiko Knospe,&nbsp;Lawrence C. Washington\",\"doi\":\"10.1007/s12188-021-00244-0\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We study <i>p</i>-adic <i>L</i>-functions <span>\\\\(L_p(s,\\\\chi )\\\\)</span> for Dirichlet characters <span>\\\\(\\\\chi \\\\)</span>. We show that <span>\\\\(L_p(s,\\\\chi )\\\\)</span> has a Dirichlet series expansion for each regularization parameter <i>c</i> that is prime to <i>p</i> and the conductor of <span>\\\\(\\\\chi \\\\)</span>. The expansion is proved by transforming a known formula for <i>p</i>-adic <i>L</i>-functions and by controlling the limiting behavior. A finite number of Euler factors can be factored off in a natural manner from the <i>p</i>-adic Dirichlet series. We also provide an alternative proof of the expansion using <i>p</i>-adic measures and give an explicit formula for the values of the regularized Bernoulli distribution. The result is particularly simple for <span>\\\\(c=2\\\\)</span>, where we obtain a Dirichlet series expansion that is similar to the complex case.</p></div>\",\"PeriodicalId\":50932,\"journal\":{\"name\":\"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2021-08-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s12188-021-00244-0.pdf\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s12188-021-00244-0\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s12188-021-00244-0","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 4

摘要

我们研究了Dirichlet特征的p-adic L-函数(L_p(s,\chi))。我们证明了\(L_p(s,\chi)\)对于每个正则化参数c都有一个Dirichlet级数展开,该正则化参数是p的素和\(\chi)的导体。通过对p-adic L-函数的一个已知公式的变换和对极限行为的控制,证明了该展开式。有限数量的欧拉因子可以以自然的方式从p-adic Dirichlet级数中分解出来。我们还提供了使用p-adic测度展开的另一种证明,并给出了正则化伯努利分布值的显式公式。对于\(c=2\),结果特别简单,其中我们获得了类似于复杂情况的狄利克雷级数展开。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Dirichlet series expansions of p-adic L-functions

We study p-adic L-functions \(L_p(s,\chi )\) for Dirichlet characters \(\chi \). We show that \(L_p(s,\chi )\) has a Dirichlet series expansion for each regularization parameter c that is prime to p and the conductor of \(\chi \). The expansion is proved by transforming a known formula for p-adic L-functions and by controlling the limiting behavior. A finite number of Euler factors can be factored off in a natural manner from the p-adic Dirichlet series. We also provide an alternative proof of the expansion using p-adic measures and give an explicit formula for the values of the regularized Bernoulli distribution. The result is particularly simple for \(c=2\), where we obtain a Dirichlet series expansion that is similar to the complex case.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
0.80
自引率
0.00%
发文量
7
审稿时长
>12 weeks
期刊介绍: The first issue of the "Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg" was published in the year 1921. This international mathematical journal has since then provided a forum for significant research contributions. The journal covers all central areas of pure mathematics, such as algebra, complex analysis and geometry, differential geometry and global analysis, graph theory and discrete mathematics, Lie theory, number theory, and algebraic topology.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信