Quasi-modularity of generalized sum-of-divisors functions

TL;DR

This paper establishes that generalized sum-of-divisors generating functions can be viewed as quasi-modular forms within a Jacobi form framework, extending classical divisor sum theory.

Contribution

It introduces a new framework linking divisor sum generating functions to Jacobi forms and proves their quasi-modularity.

Findings

Generalized divisor sum generating functions are quasi-modular forms.

The framework connects divisor sums with Jacobi forms.

Provides a new perspective on classical divisor sum theory.

Abstract

In 1919, P. A. MacMahon studied generating functions for generalized divisor sums. In this paper, we provide a framework in which to view these generating functions in terms of Jacobi forms, and prove that they are quasi-modular forms.