# Frank Garvan

## Extending Ramanujan’s Dyson rank function identity to all primes greater than 3

#### When/Where:

December 1, 2015, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

Let $$R(z,q)$$ be the two-variable generating function for Dyson’s rank function. In his lost notebook Ramanujan gives the 5-dissection of $$R(\zeta_p,q)$$
where $$\zeta_p$$  is a primitive $$p$$-th root of unity and $$p=5$$. This result is related to Dyson’s famous rank conjecture which was proved by Atkin
and Swinnerton-Dyer.
We show that there is an analogous result for the $$p$$-dissection of $$R(\zeta_p,q)$$ when $$p$$ is any prime greater than 3.
This extends previous work of Bringmann and Ono, and Ahlgren and Treneer.