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.
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2I do not think it is known whether $\zeta(3)$ is transcendental. It is known to be irrational. – André Nicolas May 21 '14 at 17:36
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This superb answer by Achille Hui shows the transcendence of $\sum10^{-(n^2+n)/2}$ and provides a pointer to a survey paper about the use of Jacobi theta functions to prove that a large class of numbers are transcendental. – MJD May 21 '14 at 20:12
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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.
Gerry Myerson
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