Some transcendence results from a harmless irrationality theorem

Abstract

The arithmetic nature of values of some functions of a single variable, particularly, $\sin{z}$, $\cos{z}$, $\sinh{z}$, $\cosh{z}$, $e^z$, and $\ln{z}$, is a relevant topic in number theory. For instance, all those functions return transcendental values for all non-zero algebraic values of $z$ ($z \ne 1$ in the case of $\ln{z}$). On the other hand, not even an irrationality proof is known for some numbers like $\,e^e$, $\,\pi^e$, $\,\pi^\pi$, $\,\ln{\pi}$, $\,\pi + e\,$ and $\,\pi \, e$, though it is well-known that at least one of the last two numbers is irrational. In this note, I first derive a more general form of this last result, showing that at least one of the sum and product of any two transcendental numbers is transcendental. I then use this to show that, given any complex number $\,t \ne 0, 1/e$, at least two of the numbers $\,\ln{t}$, $\,t + e\,$ and $\,t \, e\,$ are transcendental. I also show that $\,\cosh{z}$, $\sinh{z}\,$ and $\,\tanh{z}\,$ return transcendental values for all $\,z = r \, \ln{t}$, $\,r \in \mathbb{Q}$, $r \ne 0$. Finally, I use a recent algebraic independence result by Nesterenko to show that, for all integer $\,n > 0$, $\,\ln{\pi}\,$ and $\,\sqrt{n} \, \pi\,$ are linearly independent over $\mathbb{Q}$.