Abstract
In his lost notebook, Ramanujan stated without proofs several beautiful identities for the three classical Eisenstein series (in Ramanujan's notation) P(q), Q(q), and R(q). The identities are given in terms of certain quotients of Dedekind eta-functions called Hauptmoduls. These identities were first proved by S. Raghavan and S.S. Rangachari, but their proofs used the theory of modular forms, with which Ramanujan was likely unfamiliar. In this paper we prove all these identities by using classical methods which would have been well known to Ramanujan. In fact, all our proofs use only results from Ramanujan's notebooks.
Original language | English (US) |
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Pages (from-to) | 81-114 |
Number of pages | 34 |
Journal | Ramanujan Journal |
Volume | 4 |
Issue number | 1 |
DOIs | |
State | Published - Mar 2000 |
Keywords
- Dedekind eta function
- Differential equations for Eisenstein series
- Eisenstein series
- Modular equations
- Ramanujan's lost notebook
- Theta functions
ASJC Scopus subject areas
- Algebra and Number Theory