Is there an analytic continuation of the function, $P(n)$, that gives the $n$-th prime number?

Question

Is there a continuation of $P(n)$ to an analytic function, where $P(n)$ denotes the $n$-th prime number?

I do realize there is a continuation for $\pi(n)$, the prime counter.

Answer 1

Yes. In fact, any sequence of complex numbers is the sequence of values $f(n)$ of some entire function $f$ at the natural numbers $n$. This is a corollary of a theorem of Mittag-Leffler about existence of meromorphic functions with poles at exactly the natural numbers and with prescribed principal parts there. Taking such a function with only simple poles and multiplying it by $\sin(2\pi z)$, you get an entire function as desired.