Congruence properties of the partition function and associated functions

Abstract

The subject of this paper is the study of an unpublished manuscript by the late Srinivasa Ramanujan, the Indian mathematician. The manuscript covers forty-three pages of foolscap, and it is now in the possession of Prof. G. N. Watson. It is entitled‘Properties of p(n) and τ(n) defined by the relationsand was sent to the late Prof. G. H. Hardy by Ramanujan a few months before the latter's death in 1920. The work in the manuscript is concerned with the congruence properties of p(n) and τ(n), and Hardy extracted from it (Ramanujan (16)) proofs of the theorems