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 $\lfloor .\rfloor$ denote the integer part function. In this paper, we evaluate asymptotically the sums $$\sum_{n_{1}n_{2}\leq x}f \left( \left\lfloor \frac{x}{n_{1}n_{2}} \right\rfloor \right),$$ we use the estimation of three-dimensional exponential sums due to Robert and Sargos.