Preface Generally acknowledged as India's greatest mathematician, Srinivasa Ramanujan is most often thought of as a number theorist, although he made substantial contributions to analysis and several other ar- eas of mathematics. For most number theorists, when Ramanujan's name is mentioned, the partition and tau functions immediately come to mind. His interest in these arithmetic functions was inextricably intertwined with his primary interests of theta functions and other q- series. In fact, most of Ramanujan's research in number theory arose out of g-series and theta functions. Theta functions are the fundamen- tal building blocks in the theory of elliptic functions, and Ramanujan independently developed his own theory of elliptic functions, which is quite unlike the classical theory. We do not formally define an elliptic function, but, roughly, elliptic functions are meromorphic functions with two linearly independent periods over the real numbers. The concept of double periodicity is not used in this book, and, to the best of our knowledge, Ramanujan never utilized this idea. The purpose of this book is to provide an introduction to this large expanse of Ramanujan's work in number theory. Needless to say, we shall be able to cover only a very small fraction of Ramanu- jan's work on theta functions and g-series and their connections with number theory. However, after developing only a few facts about IX
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