Year: 2026
Type: preprint
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 $r=3$.
Source: arXiv (Cornell University)
Author: Karras, Meselem
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7129017264
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