Haihong Fan, Wenguang Zhai
For any real number $x,$ $[x]$ denotes the integer part of $x.$ $\mathcal{F}_{1}, \mathcal{F}_{2}$ denote two multiplicative function classes which are small in numerical sense. In this paper, we study the summation $\sum_{n\leq x} f([x/n])$ for $f\in \mathcal{F}_{1}$. As specific cases, we take $d^{(e)}(n), \beta(n), a(n), \mu_{2}(n)$ denoting the number of exponential divisors of $n$, the number of square-full divisors of $n,$ the number of non-isomorphic Abelian groups of order $n,$ and the characteristic function of the square-free integers, respectively. In the case of $\mu_{2}(n),$ we improved the result of Liu, Wu and Yang. The sums shaped like $\Sigma_{n\leq x} f([x/n]+f([x/n]))$ for $f\in \mathcal{F}_{2}$ are also researched.
