New Factor Pairs for Factorizations of Lambert Series Generating Functions

TL;DR

This paper introduces new variants of Lambert series factorizations, expanding the toolkit for deriving identities involving key arithmetic functions and demonstrating their applications to classical number theoretic functions.

Contribution

It presents novel generalized factorization theorems for Lambert series, extending previous results and enabling new identities for important arithmetic functions.

Findings

New identities involving the Euler partition function

Generalized sum-of-divisors function identities

Applications to Möbius, Euler totient, and Jordan totient functions

Abstract

We prove several new variants of the Lambert series factorization theorem established in the first article "Generating special arithmetic functions by Lambert series factorizations" by Merca and Schmidt (2017). Several characteristic examples of our new results are presented in the article to motivate the formulations of the generalized factorization theorems. Applications of these new factorization results include new identities involving the Euler partition function and the generalized sum-of-divisors functions, the M\"obius function, Euler's totient function, the Liouville lambda function, von Mangoldt's lambda function, and the Jordan totient function.