Eisenstein series in Ramanujan's lost notebook

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

Research output: Contribution to journalArticle

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 (US)
Pages (from-to)81-114
Number of pages34
JournalRamanujan Journal
Volume4
Issue number1
DOIs
StatePublished - 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

Fingerprint Dive into the research topics of 'Eisenstein series in Ramanujan's lost notebook'. Together they form a unique fingerprint.

  • Cite this