Ramanujan's Modular Equations of "Large” Prime Degree

Abstract

Many areas of analysis and number theory today bear the imprint of Ramanujan's ingenuity. Because of the vast variety of bis ideas, with few exceptions, Ramanujan did not devote long stretches of his short life to the study of single topics. One topic that Ramanujan studied rather completely is hypergeometric series, as two entire chapters of his published notebooks [20] are devoted to his discoveries about these series. But perhaps the subject examined most folly by Ramanujan is modular equations. Over two chapters in his notebooks, as well as two chapters of necessary background material on theta functions, are devoted to Ramanujan's findings about modular equations. The extent of Ramanujan's work in this area is so vast, that he likely found more modular equations than all of his predecessors combined. Some modular equations are stated without proof in [19], [21, pp. 23-39], and Hardy [10, Chapter 12] has very briefly described a very small portion of Ramanujan's work, but otherwise non of Ramanujan's work on modular equations has ever been published.