Factorization Theorems for Generalized Lambert Series and Applications

TL;DR

This paper develops new factorization theorems for generalized Lambert series, expanding their applicability to divisor sums, convolutions, and partition functions, with potential implications in number theory.

Contribution

It introduces generalized Lambert series factorization theorems for a broader class of series involving parameters α and β, extending prior results.

Findings

New factorization formulas for generalized Lambert series.

Applications to divisor sums and multiplicative function convolutions.

Connections established between convolutions and partition functions.

Abstract

We prove new variants of the Lambert series factorization theorems studied by Merca and Schmidt (2017) which correspond to a more general class of Lambert series expansions of the form $L_a(\alpha, \beta, q) := \sum_{n \geq 1} a_n q^{\alpha n-\beta} / (1-q^{\alpha n-\beta})$ for integers $\alpha, \beta$ defined such that $\alpha \geq 1$ and $0 \leq \beta < \alpha$. Applications of the new results in the article are given to restricted divisor sums over several classical special arithmetic functions which define the cases of well-known, so-termed "ordinary" Lambert series expansions cited in the introduction. We prove several new forms of factorization theorems for Lambert series over a convolution of two arithmetic functions which similarly lead to new applications relating convolutions of special multiplicative functions to partition functions and $n$-fold convolutions of one of the special functions.