On a paper of Dressler and Van de Lune

Abstract

If $z\in {\mathbb {C}}$ and $1\le n$ is a natural number then

$$\begin{aligned} \sum _{d_1 d_2 =n} (1-z^{p_1})\cdots (1-z^{p_m}) z^{q_1 e_{1}+\cdots +q_i e_{i} }=1, \end{aligned}$$

where $d_1=p_1^{r_1}\dots p_m^{r_m }$, $d_2=q_1^{e_1}\dots q_i^{e_i }$ are the prime decompositions of $d_1, d_2$. This is one of the identities involving arithmetic functions that we prove using ideas from the paper of Dressler and van de Lune [3].