TITLE: Does there exist a meromorphic function all of whose Taylor coefficients are prime?
QUESTION [15 upvotes]: More precisely, does there exist an unbounded sequence $a_0, a_1, ... \in \mathbb{N}$ of primes such that the function
$\displaystyle O(z) = \sum_{n \ge 0} a_n z^n$
is meromorphic on $\mathbb{C}$?  
[A previous version of the question also asked about the exponential generating function of $(a_n)$.  However, such a function can trivially be entire. - GJK]

REPLY [19 votes]: Borel proved the following much stronger result: if a power series with integer coefficients
represents a function f(z) that is meromorphic in a disk of radius >1, then f(z) extends to a rational function on all of C. I found this result without a reference on page 3 of www.mathematik.uni-bielefeld.de/~anugadre/Adeles.pdf.