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

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 n/3). 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.

Schläfli modular equations for generalized Weber functions

“…In fact Ramanujan was the first to calculate 'signature three' functions in his lost notebook. Ramanujan's various claims on signature three invariants were established by Berndt, Chan, Kang, and Zhang [4]. Further values were established by Chan, Gee, and Tan [7], and by Chan, Liaw and Tan [8].…”

Questionable claims found in Ramanujan’s lost notebook

“…Furthermore, some formulas in the lost notebook contain errors that are a bit more serious than mere misprints, but which we have been able to correct. [11], [3,Chapter 9], respectively. Most of the values in each table are missing, but it is clear that Ramanujan could have completed the tables; he was just running out of time, and there were other theorems to record and establish before he died.…”

Cubic singular moduli, Ramanujan’s class invariants λnand the explicit Shimura Reciprocity Law

“…Using cubic Russell-type modular equations, Kronecker's Limit Formulas and other techniques, Berndt, Chan, S.-Y. Kang and L.-C. Zhang [7] provided proofs of all these values except for n = 73, 97, 193, 217, and 241. In Section 4, we modify our method in Sections 2 and 3 and determine rigorously these remaining values of λ n .…”