I've started reading Shakarchi's Complex Analysis, and I thought about something interesting.
If I haven't mistaken, for any subsequence $A\subset \mathbb{Z}^+$, $\sum_{n\in A} z^n$ has radius of convergence $1/(\limsup |a_n|^{1/n})=1$.
So we can take $f(z)=\sum_p z^p$ for prime $p$, which has radius of convergence 1. This looks like a really fascinating function to me. Does anyone know a prior study done on this subject?