AI 中文总结
研究涉及小算术函数的和$S_f(x)$，当$f(n)\ll n^\varepsilon$时其渐近公式误差项$E_f(x)$形式为$O(x^{1/2+\varepsilon})$，通过研究$E_f(x)$均方为某些特殊函数建立新结果。
AI 中文摘要
设$f$为任意算术函数，定义$S_f(x):=\sum_{n\leq x}f([x/n])$。若函数$f$较小，即$f(n)\ll n^\varepsilon$，则$S_f(x)$渐近公式中的误差项$E_f(x)$形式为$O(x^{1/2+\varepsilon})$。本文研究$E_f(x)$的均方，并为某些特殊函数建立$E_f(x)$的新结果。
英文摘要
Let $f$ be any arithmetic function and define $S_f(x):=\sum_{n\leq x}f([x/n])$. If the function $f$ is small, namely, $f(n)\ll n^\varepsilon,$ then the error term $E_f(x)$ in the asymptotic formula of $S_f(x)$ has the form $O(x^{1/2+\varepsilon}).$ 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.
Journal refActa Mathematica Sinica (English series) 40 (2024), 2497-2518
