I got the following series in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$(which converges uniformly in any compact subset), and I am not sure whether there is a closed form for it. Let $d(k)$ be the number-of-divisors function, eg. $d(p)=1$ for any prime and $d(10)=|\{1,2,5,10\}|=4$. $$f(z)=\sum_{k=1}^{\infty}\frac{d(k)}{k}z^k.$$
By "closed form" I can accept even known special functions.