The sum of the first $n$-th roots of some given complex number $z$ in the unit open disk, $|z|<1$ $$\sum_{n=1}^N z^{1/n},$$ could be expressed as a polynomial series on $N$ or $z$, or it might exist a generating function or $L$-function with rational powers related.
Morever, some sort of $L$-function should have the form $$L(z,s):=\sum_{n=1}^\infty \frac{z^{1/n}}{n^s}.$$
Any ideas/references?