Relationship between two types of partition functions

Question

After downvoting my previous thread, here is a more detailed explanation of my question. For $s\in \mathbb{C},\Re(s)>1 $, consider: $$\prod_{k=1}^{\infty}\prod_{n=2}^{\infty}\frac{1}{1-n^{-ks}}= \prod_{k=1}^{\infty} \sum_{m=1}^{\infty}\frac{\rho(m)}{m^{ks}}=\prod_{n=2}^{\infty}\sum_{j=0}^{\infty}\frac{p(j)}{n^{js}} $$ Where $\rho(m)$ is the multiplicative partition function of the integer $m$. And $p(j)$ is the additive partition function of the integer $j$. Based on the equation : $$\prod_{k=1}^{\infty} \sum_{m=1}^{\infty}\frac{\rho(m)}{m^{ks}}=\prod_{n=2}^{\infty}\sum_{j=0}^{\infty}\frac{p(j)}{n^{js}} $$ What arithmetical relationship(s) can be drawn between $\rho(m)$ and $p(j)$? For instance, how can we write $\rho(m)$ in terms of $p(j)$?