Ramanujan tau

The Ramanujan tau function $\tau \colon \mathbb{N} \rightarrow \mathbb{Z}$ is defined by the formulas $$\displaystyle\sum_{n=1}^{\infty} \tau(n)q^n = q \prod_{n=1}^{\infty} (1-q^n)^{24} = \eta(z)^{24}=\Delta(z),$$ where $q=e^{2\pi i z}$ with $\mathrm{Re}(z)>0$, $\eta$ denotes the Dedekind eta function, and $\Delta$ denotes the discriminant modular form.

Properties

Theorem: $\tau(mn)=\tau(m)\tau(n)$ if $\gcd$$(m,n)=1$

Proof: █

Theorem: $\tau(p^{r+1})=\tau(p)\tau(p^r)-p^{11}\tau(p^{r-1})$ whenever $p$ is prime and $r>0$

Proof: █

Theorem: $|tau(p)| \leq 2p^{\frac{11}{2}}$ for all primes $p$

Proof: █