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 \(4\).