Factorization Theorems for Hadamard Products and Higher-Order Derivatives of Lambert Series Generating Functions

TL;DR

This paper develops new factorization theorems for Lambert series, focusing on Hadamard products and higher derivatives, linking multiplicative functions to partition theory and deriving novel identities and series related to prime numbers and the Riemann zeta function.

Contribution

It introduces new analogs of Lambert series factorization theorems for Hadamard products and derivatives, expanding the theoretical framework and applications in number theory.

Findings

Derived new series for the Riemann zeta function

Established identities for the number of distinct prime divisors

Connected Lambert series expansions to partition theory

Abstract

We first summarize joint work on several preliminary canonical Lambert series factorization theorems. Within this article we establish new analogs to these original factorization theorems which characterize two specific primary cases of the expansions of Lambert series generating functions: factorizations for Hadamard products of Lambert series and for higher-order derivatives of Lambert series. The series coefficients corresponding to these two generating function cases are important enough to require the special due attention we give to their expansions within the article, and moreover, are significant in that they connect the characteristic expansions of Lambert series over special multiplicative functions to the explicitly additive nature of the theory of partitions. Applications of our new results provide new exotic sums involving multiplicative functions, new summation-based interpretations of the coefficients of the integer-order $j^{th}$ derivatives of Lambert series generating functions, several new series for the Riemann zeta function, and an exact identity for the number of distinct primes dividing $n$.