Note on Fractional Sums with Fixed GCD

We investigate fractional sums of arithmetic functions over products of two or three integers, with emphasis on fixed greatest common divisors and multiplicative weights. Let $f$ be an arithmetic function satisfying $f(n)\ll n^{\alpha }$ for some $0\le \alpha <1$. For $r\ge 2$, let ${\tau }_r(n)$ denote the number of representations of $n$ as a product of $r$ positive integers, and more generally, ${\tau }_r^{(d)}(n)$ the number of representations with $\gcd$ factors equal to $d$. We establish asymptotic formulas for the fractional sums $$S_{f,r}^{(d)}(x)=\sum _{n\le x}{\tau }_r^{(d)}(n)f\text{\hspace{0.17em}⁣}\left(\lfloor \frac{x}{n}\rfloor \right),$$ in the cases $r=2$ and $r=3$.