Eisenstein series in Ramanujan's lost notebook

Bruce C. Berndt, Heng Huat Chan, Jaebum Sohn, Seung Hwan Son

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15 Citations (Scopus)

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 languageEnglish
Pages (from-to)81-114
Number of pages34
JournalRamanujan Journal
Volume4
Issue number1
DOIs
Publication statusPublished - 2000 Mar

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All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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