{"title":"Mean-square values of the Riemann zeta function on arithmetic progressions","authors":"Hirotaka Kobayashi","doi":"10.1007/s00605-024-01996-6","DOIUrl":null,"url":null,"abstract":"<p>We obtain asymptotic formulae for the second discrete moments of the Riemann zeta function over arithmetic progressions <span>\\(\\frac{1}{2} + i(a n + b)\\)</span>. It reveals noticeable relation between the discrete moments and the continuous moment of the Riemann zeta function. Especially, when <i>a</i> is a positive integer, main terms of the formula are equal to those for the continuous mean value. The proof requires the rational approximation of <span>\\(e^{\\pi k/a}\\)</span> for positive integers <i>k</i>.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":"32 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte für Mathematik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00605-024-01996-6","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We obtain asymptotic formulae for the second discrete moments of the Riemann zeta function over arithmetic progressions \(\frac{1}{2} + i(a n + b)\). It reveals noticeable relation between the discrete moments and the continuous moment of the Riemann zeta function. Especially, when a is a positive integer, main terms of the formula are equal to those for the continuous mean value. The proof requires the rational approximation of \(e^{\pi k/a}\) for positive integers k.
我们得到了黎曼zeta函数在算术级数 \(\frac{1}{2} 上的第二离散矩的渐近公式。+ i(a n + b))。它揭示了黎曼zeta函数离散矩与连续矩之间的明显关系。特别是当 a 为正整数时,公式的主要项等于连续均值的主要项。证明需要对正整数 k 进行 \(e^{\pi k/a}\) 的有理逼近。