Factorization Theorems for Generalized Lambert Series and Applications
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 La(α, β, q) := Σn ≥ 1 an qα n-β / (1-qα n-β) for integers α, β defined such that α ≥ 1 and 0 ≤ β < α. 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.