论涉及小算术函数的一些和

IF 0.8 3区数学 Q2 MATHEMATICS

Acta Mathematica Sinica-English Series Pub Date : 2024-07-18 DOI:10.1007/s10114-024-2129-y

Wen Guang Zhai

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。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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On Some Sums Involving Small Arithmetic Functions

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) ≪ n^ε, then the error term E_f(x) in the asymptotic formula of S_f(x) has the form O(x^1/2+ε). In this paper, we shall study the mean square of E_f(x) and establish some new results of E_f(x) for some special functions.

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来源期刊

Acta Mathematica Sinica-English Series 数学-数学

CiteScore

1.00

自引率

0.00%

发文量

138

审稿时长

14.5 months

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.