# Mike Hirschhorn

Affiliation: University of New South Wales, Australia

Email: m.hirschhorn@unsw.edu.au

Title Of Talk: Ramanujan's tau function

Abstract: Ramanujan's tau function is defined by $$\sum_{n\ge1}\tau(n)q^n=q\prod_{n\ge1}(1-q^n)^{24}.$$

The tau function has many fascinating properties. One of these is that for prime $p$, \begin{eqnarray} \tau(pn)=\tau(p)\tau(n)-p^{11}\tau\left (\frac{n}{p}\right ), \tag{1} \end{eqnarray} where it is understood that $\displaystyle \tau\left (\frac{n}{p}\right )=0$ if $p\nmid n$.

I have recently managed to give proofs of (1) for $p=2,\ 3,\ 5$ and $7$ which require nothing more than Jacobi's triple product identity, $$\prod_{n\ge1}(1+a^{-1}q^{2n-1})(1+aq^{2n-1})(1-q^{2n})=\sum_{-\infty}^\infty a^nq^{n^2}.$$

I will present one or more of these proofs.