{"title":"加泰隆常数及其相关常数的计算","authors":"Wadim Zudilin","doi":"10.1007/s12188-019-00203-w","DOIUrl":null,"url":null,"abstract":"<div><p>We prove that at least one of the six numbers <span>\\(\\beta (2i)\\)</span> for <span>\\(i=1,\\ldots ,6\\)</span> is irrational. Here <span>\\(\\beta (s)=\\sum _{k=0}^{\\infty }(-1)^k(2k+1)^{-s}\\)</span> denotes Dirichlet’s beta function, so that <span>\\(\\beta (2)\\)</span> is Catalan’s constant.</p></div>","PeriodicalId":50932,"journal":{"name":"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2019-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12188-019-00203-w","citationCount":"1","resultStr":"{\"title\":\"Arithmetic of Catalan’s constant and its relatives\",\"authors\":\"Wadim Zudilin\",\"doi\":\"10.1007/s12188-019-00203-w\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We prove that at least one of the six numbers <span>\\\\(\\\\beta (2i)\\\\)</span> for <span>\\\\(i=1,\\\\ldots ,6\\\\)</span> is irrational. Here <span>\\\\(\\\\beta (s)=\\\\sum _{k=0}^{\\\\infty }(-1)^k(2k+1)^{-s}\\\\)</span> denotes Dirichlet’s beta function, so that <span>\\\\(\\\\beta (2)\\\\)</span> is Catalan’s constant.</p></div>\",\"PeriodicalId\":50932,\"journal\":{\"name\":\"Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2019-05-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1007/s12188-019-00203-w\",\"citationCount\":\"1\",\"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/s12188-019-00203-w\",\"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/s12188-019-00203-w","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
Arithmetic of Catalan’s constant and its relatives
We prove that at least one of the six numbers \(\beta (2i)\) for \(i=1,\ldots ,6\) is irrational. Here \(\beta (s)=\sum _{k=0}^{\infty }(-1)^k(2k+1)^{-s}\) denotes Dirichlet’s beta function, so that \(\beta (2)\) is Catalan’s constant.
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.
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