Ordinary generating functions related to prime numbers?

Question

I know that typically to study prime numbers with polynomials or generating function you would use a Dirichlet series, but is their anything interesting which can be said about $$\sum \pi(n)x^n$$ or $$\sum \mathbb{1}_{n\in\mathbb{P}}x^n$$ Some quick reasearch leads to figuring out this is part of analytic number theory, so my first question is what are the prerequisites to try these problems myself? And some more searching leads me to believe this is uninteresting because all results I find related to this have dirichlet series, or other more complicated functions than "simple" generating functions. So my second question, is this even a worthwhile question, or do we need to use series of the form $\sum \frac{a_n}{n^s}$ if $a_n$ is related to primes?