Asymptotic Result of A Generalization of A GCD-Sum

TL;DR

This paper investigates the asymptotic behavior of a generalized gcd-sum function, extending previous work and providing new asymptotic formulas using elementary methods.

Contribution

It introduces a broad generalization of the gcd-sum function and derives asymptotic results with elementary techniques, expanding understanding of gcd-related sums.

Findings

Established asymptotic formulas for the generalized gcd-sum

Extended previous results to broader classes of gcd-sums

Used elementary methods for asymptotic analysis

Abstract

In a paper in the American Mathematical Monthly, the corresponding author asks for an asymptotic of a gcd-sum function \begin{align}\sum_{ab\leq N}\tau(\gcd(a,b))\label{eqn:taugcdsum}\end{align} We extensively study generalizations of the sum in the aforementioned paper and establish asymptotic results using elementary methods only.