Are there some general theorems about rationality/irrationality and abgebraicity/transcedentality of arithmetic-geometric mean? At least for some group of numbers (like natural numbers)?
Or even for particular AG means, like Gauss constant, is it known if it's irrational, transcendental? I've found no mention about this problem so far.
I know AG mean can be expressed in the form of complete elliptic integral of the first kind, but I don't think it helps.
I think this question may be important for number theory in general.
Edit
Gauss constant is transcendental, which we know from its expression through $\pi$ and $\Gamma(\frac{1}{4})$, however the general question still stands.