Is there a simpler form to $f(x)=\sum_{r=1}^{\infty}\left(\sum_{d|r}d\mu(d)\right)x^r/r$ where $\mu(d)$ is the Mobius function and $\sum_{d|r}$ is a sum over the divisors $d$ of $r$?
I searched the literature and didn't find that this sum has been calculated before. I also tried solving it using the Lambert series (I found that $\sum_{n=1}^{\infty} f(x^n)=-\log(1-x)$). I also tried plotting the function, which converges in the range $[0,1)$, but was unable to fit it to any relatively simple function. I am not sure I know what else I can try (I am not a mathematician).