Generating Special Arithmetic Functions by Lambert Series Factorizations

TL;DR

This paper unifies matrix representations of Lambert series factorizations, explores properties and conjectures related to inverse matrices and partition functions, and advances understanding of special arithmetic functions.

Contribution

It provides a unified framework for Lambert series factorizations, connecting previous results and introducing new properties and conjectures.

Findings

Unified matrix representations for Lambert series factorizations

Proved properties of inverse matrix entries related to partition functions

Formulated and supported new conjectures in the area

Abstract

We summarize the known useful and interesting results and formulas we have discovered so far in this collaborative article summarizing results from two related articles by Merca and Schmidt arriving at related so-termed Lambert series factorization theorems. We unify the matrix representations that underlie two of our separate papers, and which commonly arise in identities involving partition functions and other functions generated by Lambert series. We provide a number of properties and conjectures related to the inverse matrix entries defined in Schmidt's article and the Euler partition function $p(n)$ which we prove through our new results unifying the expansions of the Lambert series factorization theorems within this article.