Hyperbolic Summation for Fractional Sums

Let $f(n)$ be an arithmetic function with $f(n)\ll n^{\alpha }$ for some $\alpha \in [0,1)$ and let $\lfloor \mathrm{.}\rfloor$ denote the integer part function. In this paper, we evaluate asymptotically the sums $$\sum _{n_1{n}_2\le x}f\left(\lfloor \frac{x}{n_1{n}_2}\rfloor \right),$$ we use the estimation of three-dimensional exponential sums due to Robert and Sargos.