Eisenstein series in Ramanujan's lost notebook

In his lost notebook, Ramanujan stated without proofs several beautifuliden tities for the three classsical Eisenstein series (in Ramanujan's notation) P(q), Q(q), and R(q). The identities are given in terms of certain quotien ts of Dedekind eta-functions called Hauptmoduls. These identities were firs t proved by S. Raghavan and S.S. Rangachari, but their proofs used the theo ry of modular forms, with which Ramanujan was likely unfamiliar. In this pa per we prove all these identities by using classical methods which would ha ve been well known to Ramanujan. In fact, all our proofs use only results f rom Ramanujan's notebooks.