Euler products in ramanujan's lost notebook

Bruce C. Berndt, Byungchan Kim, Kenneth S. Williams

Research output: Contribution to journalArticlepeer-review

Abstract

In his famous paper, «On certain arithmetical functions», Ramanujan offers for the first time the Euler product of the Dirichlet series in which the coefficients are given by Ramanujan's tau-function. In his lost notebook, Ramanujan records further Euler products for L-series attached to modular forms, and, typically, does not record proofs for these claims. In this semi-expository article, for the Euler products appearing in his lost notebook, we provide or sketch proofs using elementary methods, binary quadratic forms, and modular forms.

Original languageEnglish (US)
Pages (from-to)1313-1349
Number of pages37
JournalInternational Journal of Number Theory
Volume9
Issue number5
DOIs
StatePublished - Aug 2013

Keywords

  • Dedekind eta-function
  • Dirichlet series with Euler products
  • Eisenstein series
  • Ramanujan tau-function
  • Ramanujan's lost notebook
  • binary quadratic forms

ASJC Scopus subject areas

  • Algebra and Number Theory

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