Mike HirschhornAffiliation: 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 $$ \sum_{n\ge1}\tau(n)q^n=q\prod_{n\ge1}(1q^n)^{24}. $$
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, $$ \prod_{n\ge1}(1+a^{1}q^{2n1})(1+aq^{2n1})(1q^{2n})=\sum_{\infty}^\infty a^nq^{n^2}. $$
I will present one or more of these proofs. WARNING: This page contains MATHJAX
Last update made Thu Feb 11 08:27:48 PST 2016.
