# What's so special about the number 24 in the definition of the Ramanujan tau function?

I'm learning about the Ramanujan tau function $$\tau \colon \mathbb{N} \to \mathbb{Z}$$ defined by $$\tau(1)q + \tau(2)q^2 + \tau(3)q^3 + \ldots = q \prod_{n = 1}^{\infty} \left(1 - q^n\right)^{24}.$$ However, I'm confused why the number $$24$$ is so special here. What if we used a different number instead of $$24?$$ All of the properties of the Ramanujan tau function I could find either had nothing to do with $$24$$ (such as the property that $$\tau$$ is multiplicative) or had some specific numbers in them.

• The RHS is a modular form and an eigenfunction of the Hecke operators. $24$ is the magical number that makes it work. For example with $48$ it is still a modular form but not an eigenform anymore. Commented Dec 3, 2022 at 23:06
• The fact that 24 is special is a mystery of Mathematics. It is distantly related to the fact that the group $S_6$ has a non-trivial outer automorphism and that the Leech Lattice exists. No-one has yer=t written a book on $24$ as a social mathematical number ... Commented Dec 3, 2022 at 23:15
• Also, the function can't be multiplicative if the number used wasn't $24.$ Commented Dec 3, 2022 at 23:20
• Cf. this Commented Dec 3, 2022 at 23:39

$$24$$ is the value such that $$\Delta$$ is a weight $$k$$ modular form with only one simple zero on the modular curve (at $$\infty$$).
Then, let $$f$$ be another weight $$k$$ cusp form. So $$\frac{f}{\Delta}$$ is a weight $$0$$ modular form, by the maximum modulus principle it has to be constant.
For each Hecke operator $$T_n \Delta$$ is a weight $$k$$ cusp form, thus equal to $$a_n \Delta$$ for some $$a_n$$, which has to be the $$n$$-th coefficient of $$\Delta$$.
Whence the coefficients of $$\Delta$$ are multiplicative.