TITLE: Does the Gamma function preserve integers?
QUESTION [7 upvotes]: Does the Gamma function $\Gamma: \mathbb{C} \to \mathbb{C}$ preserve the Kummer ring $\mathbb{Z}[\exp(2\pi\imath/m)]$? And if not, then what about the Gaussian integers $\mathbb{Z}[\imath]$ or the Eisenstein integers $\mathbb{Z}[\exp(2\pi\imath/3)]$?
Is it possible to characterize holomorphic function which do preserve these lattices?

REPLY [13 votes]: Note that
$$\frac{\Gamma(z) \Gamma(1-z)}{\Gamma(2z) \Gamma(1 - 2z)} = 2 \cos(\pi z),$$
and the RHS is transcendental for any non-rational algebraic number $z$ (by the Gelfond–Schneider theorem). So $\Gamma$ certainly won't preserve any number field $K$. It's most likely true that $\Gamma(z)$ is transcendental for algebraic $z \notin \mathbf{Z}$, but I'm not sure if that's known.

REPLY [7 votes]: I don't know about characterization, but there are lots of such functions.  In fact, for any map $g$ of the lattice $L$ into itself, there are continuum-many entire functions $f$ such that $f(z) = g(z)$ for $z \in L$.  This is because you can
get entire functions that take prescribed values on any subset of $\mathbb C$ without limit points.