Ramanujan's tau function

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

where ${\displaystyle q=\exp(2\pi iz)}$ with ${\displaystyle \Im z>0}$ and ${\displaystyle \eta }$ is the Dedekind eta function and the function ${\displaystyle \Delta (z)}$ is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form. 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).

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