Overpartitions and functions from multiplicative number theory

Abstract

Let $\alpha$ and $\beta$ be two nonnegative integers such that $\beta <\alpha$. For an arbitrary sequence $\{a_n{\}}_{n⩾1}$ of complex numbers, we consider the generalized Lambert series in order to investigate linear combinations of the form ${\sum }_{k⩾1}S(\alpha k-\beta ,n)a_k$, 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.

References in corpus (2)

• A Partition Identity Related to Stanley's Theorem
• The partition function $p(n)$ in terms of the classical Möbius function