What is known about meromorphic functions agreeing with $\pi(n)$?

Question

Let $f$ be a meromorphic function in some region containing the positive real axis such that $f(n) = \pi(n)$ for all but finitely many positive integers $n$, where $\pi(n)$ is the number of primes less than or equal to $n$. Can anything interesting be said about $f$, perhaps of the nature "$f$ must be at least this pathological..."?