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.
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3$\begingroup$ 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. $\endgroup$– reunsDec 3, 2022 at 23:06
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5$\begingroup$ 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 ... $\endgroup$– Mark BennetDec 3, 2022 at 23:15
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1$\begingroup$ Also, the function can't be multiplicative if the number used wasn't $24.$ $\endgroup$– mathlanderDec 3, 2022 at 23:20
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1$\begingroup$ Cf. this $\endgroup$– J. W. TannerDec 3, 2022 at 23:39
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1 Answer
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$24$ 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.
