Explicit congruences for the partition function modulo every prime

Abstract.

Let p(n) denote the number of unrestricted partitions of the positive integer n, and let m be a prime $\geq 13$. We prove, for k = 1, explicit congruences of the form

$\sum_{n=0}^{\infty} p(m^kn+\delta_{m,k})q^{24n+r_{m,k}} \equiv \eta^{r_{m,k}}(24z)\phi_{m,k}(24z) ({\rm mod}\, m)$ where $r_{m,k}, \delta_{m,k}$ are integers depending on m and k and $\phi_{m,k}(z)$ are explicitly computable level one holomorphic modular forms of small weight. We also give theoretical and numerical support that the congruences also hold for k > 1. Our main idea is a level reduction result for the modular forms which originated from Atkin-Lehner. From our result, we deduce periodicity properties for the partition function with short periods which improve upon recent results of K. Ono.