Hyperbolic summation for fractional sums

Abstract

Let $\tau (n)$ denote the number of positive divisors of an integer $n\geq 1$ and let $\lfloor \cdot \rfloor $ denote the integer part function. We evaluate asymptotically the sums $$ \sum _{n\leq x}f (\lfloor x/n\rfloor )\tau (n), $$ where $f$ is an arit