On some sums involving small arithmetic functions

Zhai Wenguang

On some sums involving small arithmetic functions

Number Theory 2026-07-20 v1

Authors: Zhai Wenguang

Abstract

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.

Cite

@article{arxiv.2607.17474,
  title  = {On some sums involving small arithmetic functions},
  author = {Zhai Wenguang},
  journal= {arXiv preprint arXiv:2607.17474},
  year   = {2026}
}