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I'm trying to prove the algebraic independence of $\pi, e^\pi$ and $ \Gamma(1/4)$ while using Nesterenko's Theorem ($\{q, P(q), Q(q), R(q)\}$ contains at least three algebraically independent numbers over the rationals, if $|q| < 1$, where $E_2(z) = P(e^{2\pi iz}), E_4(z) = Q(e^{2\pi iz})$ and $E_6(z) = R(e^{2\pi iz})$ and $E_{2k}$ denotes the classical Eisenstein series) as a black box.

I know it follows from $$P(e^{-2\pi}) = \frac{3}{\pi}, Q(e^{-2\pi}) = \frac{3\Gamma(1/4)^8}{(2\pi)^6}, R(e^{-2\pi}) = 0$$

but I have no idea how to prove these equations. Could anyone point me to a source or explain how it works?

Thanks!

morris
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  • The values of these functions are given by their link with elliptic integrals. For $P(e^{-2\pi})$ see this thread. For $Q$ use the formula $$Q(q^2)=(2K(k)/\pi)^4(1-k^2+k^4)$$ where $k$ is elliptic modulus corresponding to nome $q$ and $K(k) =\int_0^{\pi/2}\frac{dx}{\sqrt{1-k^2\sin^2x}}$. For $q=e^{-\pi}$ we have $k=1/\sqrt{2}$. – Paramanand Singh Aug 15 '23 at 01:20
  • For $R$ it is best to note that $E_6(-1/z)=z^6E_6(z)$ and put $z=i$ to get $E_6(i)=0$. – Paramanand Singh Aug 15 '23 at 01:21

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