Note on a paper by Bordellès, Dai, Heyman, Pan and Shparlinski, 2

Denote by $[t]$ the integral part of $t$. Under some simple hypothesis on the growth of the arithmetic function $f$, we prove asymptotic formulas for $$ S_f(x):= \sum _{n\le x} f\left (\left [\frac {x}{n}\right ]\right ) $$ as $x\to \infty $. The improve