TITLE: Generating function for the characteristic function of primes
QUESTION [6 upvotes]: What do we know about the generating function of $\chi(n)$ (A010051)
$$
  f(x) = \sum_{n=0}^\infty \chi(n)x^n = \sum_{p\text{ prime}} x^p
$$
for $\chi(n)$ the characteristic function of the primes:
$$
  \chi(n) = 
     \begin{cases}
        1, & \text{if $n$ is prime}\\
        0, & \text{otherwise}
     \end{cases}
$$
Are there some references I could take a look at?

REPLY [6 votes]: This is essentially the inverse Mellin transform of the Prime zeta function. Perhaps you'd like to look in that direction.