Note on Fractional Sums with Fixed GCD

Abstract: 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 $f(n) \ll n^α$ for some $0 \le α< 1$. For $r \ge 2$, let $τr(n)$ denote the number of representations of $n$ as a product of $r$ positive integers, and more generally, $τ_r^{(d)}(n)$ 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} τ_r^{(d)}(n) f!\left(\left\lfloor \frac{x}{n}\right\rfloor \right), ] in the cases $r=2$ and $r=3$.