Power Series with digits of $\pi$

Question

Sorry if this has already been asked, but I haven't found a post. Can anything be said about the function $$f(z)=\sum_{n=0}^\infty a_n z^n$$ where the $a_n$'s are the digits of $\pi= 3.14159...$, so $$a_0=3,a_1=1,a_2=4,a_3=1,...$$ and so on. Since the coefficients are pretty random, the behaviour of the function is hard to predict. Bounding $|f(z)|\leq9\sum_{n=0}^\infty |z|^n$, it is seen that the radius of convergence is $1$, but what exactly happens at the boundary? Is it even extendible to outside of the unit-disc or are there infinitely many singularities?