Note on Fractional Sums with Fixed GCD
Meselem Karras
TL;DR
This paper studies fractional sums of multiplicative arithmetic functions over products of integers with fixed gcd constraints. It develops asymptotic formulas for $S_{f,r}(x)$ and its fixed-gcd variants $S_{f,r}^{(d)}(x)$, extending prior work of Karras–Li–Stucky to the case of prescribed gcds and to the case $r=3$, while incorporating polynomial weights through the general sum $F_{f}^{a,b,(d)}(x)$. The author derives explicit main-term constants and error terms, using Möbius inversion to relate restricted sums to unrestricted ones and employing divisor-sum estimates and analytic number theory tools. The results yield precise asymptotics for $r=2$ and $r=3$, including $S_{f,3}^{*}(x)$ with a quadratic-log main term and a detailed constant structure, as well as upper bounds for weighted and coprimality-restricted sums, thereby broadening the scope of fractional-sum analyses under gcd constraints.
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 $τ_r(n)$ 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 $r=3$.