{"title":"论涉及小算术函数的一些和","authors":"Wen Guang Zhai","doi":"10.1007/s10114-024-2129-y","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>f</i> be any arithmetic function and define <span>\\({S_{f}}(x):=\\sum\\nolimits_{{n \\le x}}f([x/n])\\)</span>. If the function <i>f</i> is small, namely, <i>f</i>(<i>n</i>) ≪ <i>n</i><sup><i>ε</i></sup>, then the error term <i>E</i><sub><i>f</i></sub>(<i>x</i>) in the asymptotic formula of <i>S</i><sub><i>f</i></sub>(<i>x</i>) has the form <i>O</i>(<i>x</i><sup>1/2+<i>ε</i></sup>). In this paper, we shall study the mean square of <i>E</i><sub><i>f</i></sub>(<i>x</i>) and establish some new results of <i>E</i><sub><i>f</i></sub>(<i>x</i>) for some special functions.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 10","pages":"2497 - 2518"},"PeriodicalIF":0.8000,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Some Sums Involving Small Arithmetic Functions\",\"authors\":\"Wen Guang Zhai\",\"doi\":\"10.1007/s10114-024-2129-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <i>f</i> be any arithmetic function and define <span>\\\\({S_{f}}(x):=\\\\sum\\\\nolimits_{{n \\\\le x}}f([x/n])\\\\)</span>. If the function <i>f</i> is small, namely, <i>f</i>(<i>n</i>) ≪ <i>n</i><sup><i>ε</i></sup>, then the error term <i>E</i><sub><i>f</i></sub>(<i>x</i>) in the asymptotic formula of <i>S</i><sub><i>f</i></sub>(<i>x</i>) has the form <i>O</i>(<i>x</i><sup>1/2+<i>ε</i></sup>). In this paper, we shall study the mean square of <i>E</i><sub><i>f</i></sub>(<i>x</i>) and establish some new results of <i>E</i><sub><i>f</i></sub>(<i>x</i>) for some special functions.</p></div>\",\"PeriodicalId\":50893,\"journal\":{\"name\":\"Acta Mathematica Sinica-English Series\",\"volume\":\"40 10\",\"pages\":\"2497 - 2518\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-07-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematica Sinica-English Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10114-024-2129-y\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Sinica-English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10114-024-2129-y","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
让 f 是任意算术函数，并定义 \({S_{f}}(x):=\sum\nolimits_{{n \le x}}f([x/n])\).如果函数 f 很小，即 f(n) ≪ nε，那么 Sf(x) 的渐近公式中的误差项 Ef(x) 的形式为 O(x1/2+ε)。本文将研究 Ef(x) 的均方值，并建立一些特殊函数 Ef(x) 的新结果。
Let f be any arithmetic function and define \({S_{f}}(x):=\sum\nolimits_{{n \le x}}f([x/n])\). If the function f is small, namely, f(n) ≪ nε, then the error term Ef(x) in the asymptotic formula of Sf(x) has the form O(x1/2+ε). In this paper, we shall study the mean square of Ef(x) and establish some new results of Ef(x) for some special functions.
期刊介绍：
Acta Mathematica Sinica, established by the Chinese Mathematical Society in 1936, is the first and the best mathematical journal in China. In 1985, Acta Mathematica Sinica is divided into English Series and Chinese Series. The English Series is a monthly journal, publishing significant research papers from all branches of pure and applied mathematics. It provides authoritative reviews of current developments in mathematical research. Contributions are invited from researchers from all over the world.
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