Introduction

This book constitutes the fifth and final volume of our attempts to establish all the results claimed by the great Indian mathematician Srinivasa Ramanujan in hisNotebooksfirst published in a photostat edition by the Tata Institute of Fundamental Research in 1957 [9]. Although each of the five volumes contains many deep results, perhaps the average depth in this volume is greater than in the first four. As will be seen in the following paragraphs, several mathematicians made important contributions to the completion of this volume. However, I particularly extend my deepest gratitude to Heng Huat Chan and Liang—Cheng Zhang without whose contributions this volume would have been woefully deficient. This volume, however, should not be regarded as the closing chapter on Ramanujan’s notebooks. Instead, it is just the first milestone on our journey to understanding Ramanujan’s ideas. Many of the proofs given here and in other volumes certainly do not reflect Ramanujan’s motivation, insights, proofs, and wisdom. It is our fervent wish that these volumes will serve as springboards for further investigations by mathematicians intrigued by Ramanujan’s remarkable ideas. As in the other four volumes, for each correct claim, we either provide a proof or cite references in the literature where proofs can be found. We emphasize that Ramanujan made extremely few errors, and that most “mistakes” are either minor misprints, or, in fact, they are errors made by the author arising from misinterpretations of Ramanujan’s claims, which are occasionally fuzzy.