Proof of conjecture regarding the level of Rose's generalized sum-of-divisor functions

TL;DR

This paper proves Rose's conjecture that certain generalized sum-of-divisor functions are quasi-modular forms for the congruence subgroup (n), confirming their modular properties.

Contribution

The paper provides a proof confirming that these generalized sum-of-divisor functions are quasi-modular forms for (n), advancing understanding of their modular nature.

Findings

Confirmed the quasi-modular form status for (n)

Validated Rose's conjecture on generalized sum-of-divisor functions

Enhanced the theory of modular forms associated with divisor functions

Abstract

In a recent paper, Rose proves that certain generalized sum-of-divisor functions are quasi-modular forms for some congruence subgroup and conjectures that these forms are quasi-modular for $\Gamma_1(n)$. Here, we prove this conjecture.