Ramanujan’s Cubic Analogue of the Classical Ramanujan–Weber Class Invariants

At the top of page 212 in his lost notebook [244], Ramanujan defines the function $\lambda_n$ by

$$\lambda_n=\frac{e^{\pi/2\sqrt{n/3}}}{3\sqrt3}\{(1+e^{-\pi\sqrt{n/3}}) (1-e^{-2\pi\sqrt{n/3}})(1-e^{-4\pi\sqrt{n/3}})\cdots\}^6,$$

((9.1.1))

and then devotes the remainder of the page to stating several elegant values of $\lambda_n$, for n ≡ 1 (mod 8). The quantity $\lambda_n$ can be thought of as an analogue in Ramanujan’s cubic theory of elliptic functions [57, Chapter 33] of the classical Ramanujan–Weber class invariant Gn, which is defined by

$$G_n := 2^{-1/4}q^{-1/24}(-q;q^2)_\infty,$$

((9.1.2))

where $q= \exp(-\pi \sqrt n)$ and n is any positive rational number.