Entry 189

Ramanujan's \(g_m\)-function has a consistent form for fundamental discriminants \(d=4m\) with class number \(2\) and \(4\), where \(m=2p\) for primes \(p\equiv 3\,\text{mod}\,4\). For example, for \(p=3,11\) and \(p=7,23,71\) then

$$\begin{align}g_{6}&= \sqrt[6]{1+\sqrt2}\\ g_{22}&= \sqrt{1+\sqrt2}\\ g_{14}&= \sqrt{\frac{-1+\sqrt2}4}+  \sqrt{\frac{4+(-1+\sqrt2)}4}\\ g_{46}&= \sqrt{\frac{1+\sqrt2}4}+  \sqrt{\frac{4+(1+\sqrt2)}4}\\ g_{142}&= \sqrt{\frac{(1+\sqrt2)^3}4}+  \sqrt{\frac{4+(1+\sqrt2)^3}4}\end{align}$$

It seems \(\sqrt{p}\) is not needed but only \(\sqrt{2}\), or the fundamental unit \(U_2 = 1+\sqrt{2}\) also known as the silver ratio. For the modular lambda function \(\lambda(\sqrt{-m})\) for these same \(m\), see also Entry 134 where

$$\frac1{\lambda(\sqrt{-14})} = \Big(n+\sqrt{n^2-1}\Big)^2 \Big(\sqrt{n^2-2}+\sqrt{n^2-1}\Big)^2 ,\quad n=2(1+\sqrt2)$$