Is prime-counting function #P-complete? Recall $\pi(n)$ the number of primes $\le n$ is the prime-counting function.  By "PRIMES in P", computing $\pi(n)$ is in #P.  Is the problem #P-complete?  Or, perhaps, there is a complexity reason to believe that this problem is not #P-complete?  

P.S.  I realize this is a bit naive since somebody must have studied the problem and proved/disproved/conjectured this, but I can't seem to find the answer in the literature.  See here if you are curious why I care.
Some heuristic evidence: to the best of our knowledge  $\pi(n)$ looks like a simple function corrected by random fluctuations.  Thus I’d expect a poly-time machine with a $\pi(n)$ oracle to be no stronger than such a machine with a random oracle, and w.r.t. a random oracle $X$ adding a separate random oracle $Y$ to $\mathsf{P}$ gives $\#\mathsf{P}^X \not\subset \mathsf{P}^{XY}$ with probability 1 (here $Y$ corresponds to $\pi(n)$ and $X$ is an independent random oracle).