A remark on Fine's arithmetic functions

TL;DR

This paper revisits Fine's arithmetic functions, linking them to indefinite quadratic forms through Andrews' result, and derives new arithmetic theorems based on these connections.

Contribution

It introduces a novel connection between Fine's divisor functions and indefinite quadratic forms, leading to new arithmetic identities and theorems.

Findings

New identities relating divisor functions and quadratic forms

Arithmetic theorems derived from these identities

Enhanced understanding of Fine's functions in quadratic form context

Abstract

In this note, we take another look at some arithmetic identities of N.J. Fine associated with divisor functions. We connect these functions with indefinite quadratic forms using a result due to Andrews. As a consequence, arithmetic theorems are extracted.