Entry 124

In Entry 123, the $24$th power of the golden ratio $\phi$ and Gauss' constant $G$ was discussed. We will use it again and connect it to the Dedekind eta function $\eta(\tau)$ and Watson's triple integral  $$\begin{align}I_1 &= \frac{1}{\pi^3}\int_0^\pi \int_0^\pi \int_0^\pi \frac{dx\, dy\, dz}{1-\cos x\cos y\cos z}\\ &=\frac{\Gamma^4(\frac{1}{4})}{4\pi^3}\\ &= 2\,G^2 = 4\,\eta(i)^4 = 1.393203\dots\end{align}$$ where $$I_1 =\frac{\,25\phi^6}{\sqrt{\phi^{24}-4}}\sum_{n=0}^\infty \frac{(6n)!}{(3n)!\,n!^3} \left(\frac{-\phi^{16}}{4(\phi^{24}-4)}\right)^{3n}=1.393203\dots$$ Notice that the $24$th power of the golden ratio is off by $4$.