{"title":"p-adic L-函数的Dirichlet级数展开式","authors":"Heiko Knospe, 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, 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}
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.
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.