Relations in the Set of Divisors of an Integer $n$

TL;DR

This paper extends a method to analyze the relationships among divisors of an integer, establishing bounds on the number of divisor triples satisfying a sum relation, thus contributing to number theory's understanding of divisor structures.

Contribution

It generalizes a previous method to study divisor sets and provides a new bound on the count of divisor triples with a sum relation, improving understanding of divisor interactions.

Findings

Bound on the number of divisor triples with sum relation: rac{}{} au(n)^{2-}

Established a specific numerical value for =0.045072

Extended previous divisor analysis methods to new bounds

Abstract

Let $\mathcal{D}_{n} \subset \mathbb{N}$ be the set of the $\tau(n)$ divisors of $n$. We generalize a method developed by Erd\H os, Tenenbaum and de la Bret\`eche for the study of the set $\mathcal{D}_{n}$. In particular, using these ideas, we establish that $$ |\{(d_{1},d_{2},d_{3}) \in \mathcal{D}_{n}^3 : d_{1}+d_{2}=d_{3}\}| \le \tau(n)^{2-\delta} $$ with $\delta=0.045072$.