Overpartitions and functions from multiplicative number theory

TL;DR

This paper explores the relationship between overpartitions and multiplicative number theory functions using generalized Lambert series, revealing new connections and identities in partition theory and number theory.

Contribution

It introduces a novel approach to connect overpartition counts with multiplicative number theory functions via generalized Lambert series.

Findings

Established new identities linking overpartitions and multiplicative functions

Provided a framework for analyzing linear combinations of overpartition statistics

Connected overpartition enumeration with classical number theoretic functions

Abstract

Let $\alpha$ and $\beta$ be two nonnegative integers such that $\beta < \alpha$. For an arbitrary sequence $\{a_n\}_{n\geqslant 1}$ of complex numbers, we consider the generalized Lambert series in order to investigate linear combinations of the form $\sum_{k\geqslant 1} S(\alpha k-\beta,n) a_k$, where $S(k,n)$ is the total number of non-overlined parts equal to $k$ in all the overpartitions of $n$. The general nature of the numbers $a_n$ allows us to provide connections between overpartitions and functions from multiplicative number theory.