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

Abstract

In his lost notebook, Ramanujan defined a parameter λn by a certain quotient of Dedekind eta-functions at the argument q = exp(-π√n/3). He then recorded a table of several values of λn. 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.