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This is most likely open, since alredy $e^{\pi^2}$ is not known to be transcendental. As an added difficulty, I don't think that $\frac{\pi^2}{12 \log 2}$ is known to be transcendental either. There …

Actually Baker's theorem generalizes Lindemann–Weierstrass, so that alredy gives you an effective bound $$\bigg|\sum_i \beta_i e^{\alpha_i}\bigg| > Ce^{-(\log H)^k}$$ with $C$ an effectively computa …

The irrationality of $\log \pi$ is an open problem (see for example this recent paper). It is expected to be transcendental (page 34 of this slides by Michel Waldschmidt), and in fact this follows fr …

The claim that $\pi$ and $e$ are known to be algebraically independent is incorrect, see for example this MO question. The rationality of $\pi^n-e^n$ is a well-known open problem alredy for $n=1$, an …

I gather that the idea behind $\mathrm{Gal}(\pi)=\mathbb{Z}\backslash\{0\}$ (not $\mathbb{Z}$, $0$ is not a conjugate of $\pi$!) comes from Euler's formula: $$\prod_{n\in \mathbb{Z\backslash\{0\}}}\b …

For the most part "transalgebraic theory" seems an umbrella term for anything that relates (classic, differential, motivic, categorical) Galois theory with periods, trascendence results, special value …

As explained by Peter Humphries in the comments, the famous conjecture here is that the imaginary part of the non-trivial zeros of $\zeta(s)$ is transcendental (or equivalently, linearly independent o …