Ramanujan’s Tau Function

Abstract

Ramanujan’s tau function is defined by

$$ \sum _{n\ge 1}\tau (n)q^n=qE(q)^{24} $$

where $E(q)=\displaystyle \prod _{n\ge 1}(1-q^n)$. It is known that if p is prime,

$$ \tau (pn)=\tau (p)\tau (n)-p^{11}\tau \left( \frac{n}{p}\right) , $$

where it is understood that $\displaystyle \tau \left( \frac{n}{p}\right) =0$ if p does not divide n. We give proofs of this relation for $p=2,\,3,\,5,\,7$ and 13. which rely on nothing more than Jacobi’s triple product identity. I believe that the case $p=11$ is intrinsically more difficult, and I do not attempt it here.

This paper is dedicated to Krishna Alladi on the occasion of his 60th birthday