Ramanujan tau function

In mathematics, the Ramanujan tau function, studied by Srinivasa Ramanujan, is the function ${\displaystyle \tau :\mathbb {N} \to \mathbb {Z} }$ defined by

${\displaystyle \sum _{n=1}^{\infty }\tau (n)q^{n}=q\prod _{n=1}^{\infty }(1-q^{n})^{24}=q\phi (q)^{24}=\eta (z)^{24}=\Delta (z),}$

where ${\displaystyle \phi }$ is the Euler function, ${\displaystyle \eta }$ is the Dedekind eta function, ${\displaystyle \Delta (z)}$ is the modular discriminant, and ${\displaystyle q=e^{2\pi iz}}$ with ${\displaystyle \mathrm {Im} (z)>0}$.