I was reading about the divisor function on Wikipedia, and I stumbled upon the formula $$\sum_{n\geq 1}\frac{\sigma_a(n)}{n^s}=\zeta(s)\zeta(s-a).$$ Here $\sigma_a(n)=\sum_{d|n}d^a$ for an integer $a\geq 0$. Let us fix $s=1$. The sum still makes sense but does not converge anymore to the right hand side. However my question is not about the sum $\sum_{n\geq 1}\frac{\sigma_a(n)}{n}$. I was wondering if the generating function associated to the sequence $\frac{\sigma_a(n)}{n}$ is known:
Question: If $x$ is a formal variable, how can we express the power series $$ \sum_{n\geq 1}\frac{\sigma_a(n)}{n}x^n,$$ e.g. is it a rational function, an infinite product, or anything else related to $\zeta$, or...?
Generally, when you are dealing with multiplicative functions (which $\sigma_a(n)/n$ is) you get Lambert Series (in fact, I was amused to see that when I was looking for the link, I saw that the wikipedia article had exactly this example).