Prove of trancendence of $\ln(2)$.

Question

Where can I find some proofs for another transcendental numbers, like Hermite/Lindemann theorem proofs for $e/\pi$? For instance, prove that $\zeta(3)/\ln(2)$ is a transcendental number.

Answer 1

It is not clear whether the question concerns the quotient $\zeta(3)\div\log2$, or the individual numbers $\zeta(3)$ and $\log2$.

A theorem of Hermite and Lindemann says that if $\alpha$ is a nonzero number then at least one of the numbers $\alpha$ and $e^{\alpha}$ is transcendental (the source gives the hypothesis as, "if $\alpha$ is a nonzero algebraic number", but in that case the conclusion should simply be that $e^{\alpha}$ is transcendental). Applying this theorem to $\alpha=\log2$ shows that $\log2$ is transcendental.