On arithmetic sums involving divisor functions in two variables

TL;DR

This paper extends an identity related to divisor functions in two variables, enabling the evaluation of sums that count representations of positive integers involving radicals, thus connecting divisor sums with number representations.

Contribution

It proves an analogue of a known identity and applies it to evaluate complex divisor sums in two variables, linking them to integer representations involving radicals.

Findings

Derived new identities for divisor sums in two variables.

Evaluated sums that count representations of integers with radicals.

Established connections between divisor functions and number representations.

Abstract

We prove the analogue of an identity of Huard, Ou, Spearman and Williams and apply it to evaluate a variety of sums involving divisor functions in two variables. It turns out that these sums count representations of positive integers involving radicals.