Ramanujan tau function

The Ramanujan tau function, studied by Ramanujan (1916), is the function ${\displaystyle \tau :\mathbb {N} \rightarrow \mathbb {Z} }$ defined by the following identity:

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

where q = exp(2πiz) with Im z > 0, ${\displaystyle \phi }$ is the Euler function, η is the Dedekind eta function, and the function Δ(z) is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form (some authors, notably Apostol, write ${\displaystyle \Delta /(2\pi )^{12}}$ instead of ${\displaystyle \Delta }$). It appears in connection to an "error term" involved in counting the number of ways of expressing an integer as a sum of 24 squares. A formula due to Ian G. Macdonald was given in Dyson (1972).