Note on Fractional Sums with Fixed GCD

TL;DR

This paper derives asymptotic formulas for fractional sums involving arithmetic functions over products of two or three integers with fixed gcd, expanding understanding of multiplicative structures in number theory.

Contribution

It introduces new asymptotic formulas for fractional sums of arithmetic functions with fixed gcd constraints for the cases r=2 and r=3.

Findings

Established asymptotic formulas for r=2 and r=3 cases.

Analyzed sums involving gcd-restricted divisor functions.

Extended fractional sum analysis to multiplicative weights.

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 $f(n) \ll n^\alpha$ 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 \[ S_{f,r}^{(d)}(x) = \sum_{n \le x} \tau_r^{(d)}(n) f\!\left(\left\lfloor \frac{x}{n}\right\rfloor \right), \] in the cases $r=2$ and $r=3$.