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 $te$ 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 $\mathbb{Q}$.

arXiv comment12 pages, no figures. Inclusion of a new theorem (Theor.3). Submitted to Expos. Math. (Feb/07/2014)