Entry 87

Define \(d_k = \eta(k\tau)\) with the Dedekind eta function \(\eta(\tau)\). We have the nice $$\left(\frac{\sqrt2\,d_1\,d_4^2}{d_2^3}\right)^8+\left(\frac{d_1^2\,d_4}{d_2^3}\right)^8 = 1$$ Equivalently $$\left(\frac{d_2^3}{d_1\,d_4^2}\right)^8-8=\left(\frac{d_1}{d_4}\right)^8+8$$ Focusing on the first term, note that the three similar Monster functions

$$j_{4C} =\left(\frac{d_2^3}{d_1\,d_4^2}\right)^8,\quad  j_{8E} =\left(\frac{d_4^3}{d_2\,d_8^2}\right)^4,\quad  j_{16B} =\left(\frac{d_8^3}{d_4\,d_{16}^2}\right)^2$$

being the McKay-Thompson series of class \(4C, 8E, 16B\), respectively, are necessary to the \(9\) dependencies found by Conway, Norton, and Atkins such that the moonshine functions span a linear space of \(172-9=163\) dimensions (discussed in previous entries). In fact, scaled and flipped over, it is an important function,  $$\lambda(\tau)=\left(\frac{\sqrt2\,d_{1/2}\,d_2^2}{d_1^3}\right)^8 = \left(\frac{\sqrt{2}\,\eta(\tfrac{\tau}{2})\,\eta^2(2\tau)}{\eta^3(\tau)}\right)^8$$ known as the modular lambda function \(\lambda(\tau)\).