A certain quotient of eta-functions found in Ramanujan’s lost notebook

Bruce C. Berndt, Heng Huat Chan, Soon-Yi Kang and Liang-Cheng Zhang

Vol. 202 (2002), No. 2, 267–304

DOI: 10.2140/pjm.2002.202.267

Abstract

In his lost notebook, Ramanujan defined a parameter λ_n by a certain quotient of Dedekind eta-functions at the argument q = exp(−π). He then recorded a table of several values of λ_n. To prove these values (and others), we develop several methods, which include modular equations, the modular j-invariant, Kronecker’s limit formula, Ramanujan’s “cubic theory” of elliptic functions, and an empirical process.

Milestones

Received: 11 February 2000

Revised: 19 June 2000

Published: 1 February 2002