First of all, I must clarify that I do not really know how much it is known about this subject, but, as it seems to be just a little bit/nothing (at least because of the little information I could find about it over the Internet), I will post this question anyway. Excuse me if you find it slightly "open".
Do we know any Ordinary Generating Function
$$f(x) = \sum_{n=0}^\infty a_n x^n$$
with some type of closed form for which $a_n$ contains or involves the Prime Counting Function $\pi(n)$?
To clarify, some examples of what I am looking for are:
$$f_1(x) = \sum_{n=0}^\infty \pi(n) x^n$$
$$f_2(x) = \sum_{n=0}^\infty \frac{\pi(n)}{n!} x^n$$
$$f_3(x) = \sum_{n=0}^\infty \frac{1}{\pi(n)} x^n$$
Etc.
Motivation: I thought about this while reading some conclusions about the famous formula
$$\int_2^\infty \frac{\pi(t)}{t^{s+1}-t} dt =\frac{\log \zeta(s)}{s}$$
For $\text{Re}(s)>1$.
And I asked myself whether we knew about any similar result on Ordinary Generating Functions.
Personally, it seems out of reach finding such a closed form. However, has anyone ever discovered any? It would be great if I could be referred to any paper about this kind of series, since I have not found any.
Thank you.
Let $\displaystyle \Psi(x) = \sum_{p^k} \frac{e^{-x p^k}}{k}$ the sum being over prime powers. Then for $\Re(s) > 1$ $$\int_0^\infty \Psi(x) x^{s-1}dx = \sum_{p^k} \frac{1}{k}\int_0^\infty e^{-x p^k} x^{s-1}dx=\sum_{p^k} \frac{1}{k} p^{-sk} \Gamma(s) = \Gamma(s) \log \zeta(s)$$ Thus by inverse Mellin/Laplace/Fourier transform, for $\sigma > 1$ $$\Psi(x) = \frac{1}{2i\pi} \int_{\sigma-i\infty}^{\sigma+i\infty} \frac{x^{-s}}{s}\Gamma(s)\log \zeta(s) ds$$ And the prime number theorem is that $\displaystyle\Psi(x) \sim \sum_{n=2}^\infty \frac{e^{-nx}}{\log n}$ as $x \to 0$.