Continued Fractions and Modular Functions

It is widely recognized that the work of Ramanujan deeply influenced the direction of modern number theory. This influence resonates clearly in the “Ramanujan conjectures.” Here I will explore another part of his work whose position within number theory seems to be less well understood, even though it is more elementary, namely that related to continued fractions. I will concentrate on the special values of continued fractions that represent modular functions, especially the Rogers-Ramanujan continued fraction. These give analogues of the simple continued fraction expansions of units in real quadratic fields. My primary motivation is to furnish a coherent treatment of this topic, around which an air of mystery seems to linger. Another is to provide an inviting and non-standard introduction to the classical theory of modular functions. This is largely an expository paper; most of the ideas I discuss are well known. Yet it is hoped that the elaboration given here combines these ideas in a novel way. Although this paper is not intended to be comprehensive, its later sections contain more material than is likely needed to gain a clear impression of the main themes, which the first six sections should provide. These will take the general reader through a proof of the first main result, Theorem 1, introducing the needed concepts along the way. The sections that follow these assume a somewhat greater background in number theory.