{"title":"On Sums of Sums Involving the Von Mangoldt Function","authors":"Isao Kiuchi, Wataru Takeda","doi":"10.1007/s00025-024-02276-3","DOIUrl":null,"url":null,"abstract":"<p>Let <span>\\(\\Lambda \\)</span> denote the von Mangoldt function, and (<i>n</i>, <i>q</i>) be the greatest common divisor of positive integers <i>n</i> and <i>q</i>. For any positive real numbers <i>x</i> and <i>y</i>, we shall consider several asymptotic formulas for sums of sums involving the von Mangoldt function; <span>\\( S_{k}(x,y):=\\sum _{n\\le y}\\left( \\sum _{q\\le x}\\right. \\left. \\sum _{d|(n,q)}d\\Lambda \\left( \\frac{q}{d}\\right) \\right) ^{k} \\)</span> for <span>\\(k=1,2\\)</span>.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.1000,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Results in Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00025-024-02276-3","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let \(\Lambda \) denote the von Mangoldt function, and (n, q) be the greatest common divisor of positive integers n and q. For any positive real numbers x and y, we shall consider several asymptotic formulas for sums of sums involving the von Mangoldt function; \( S_{k}(x,y):=\sum _{n\le y}\left( \sum _{q\le x}\right. \left. \sum _{d|(n,q)}d\Lambda \left( \frac{q}{d}\right) \right) ^{k} \) for \(k=1,2\).
期刊介绍:
Results in Mathematics (RM) publishes mainly research papers in all fields of pure and applied mathematics. In addition, it publishes summaries of any mathematical field and surveys of any mathematical subject provided they are designed to advance some recent mathematical development.