X B. C. BERND T g-series and theta functions, we will be equipped to prove many in- teresting theorems. The arithmetic functions on which we focus are the partition function p(n), Ramanujan's tau function r(n), the num- ber of representations of a positive integer n as a sum of 2k squares denoted by r2fc(n), and other arithmetic functions closely allied to r2/e(n). Most of the material upon which we draw can be found in Ramanujan's published papers on p(n) and r(n), the later chapters in his second notebook, his lost notebook, and his handwritten man- uscript on p(n) and r(n) published with his lost notebook. We em- phasize that Ramanujan left behind few of his proofs, especially for his claims in his notebooks and lost notebook. Thus, for many of the theorems that we discuss, we do not know Ramanujan's proofs. This is particularly true for the theorems on sums of squares and similar arithmetic functions that we prove in Chapter 3. The requirements for reading and understanding the material in this book are relatively modest. An undergraduate course in elemen- tary number theory is advisable. For some of the analytic arguments, a solid undergraduate course in complex analysis is essential. How- ever, the occasions when deep analytical rigor is needed are few, and so readers who do not have a strong background in analysis can sim- ply verify formally the needed manipulations. Our intent here is not to give a rigorous course in analysis but to emphasize the most impor- tant ideas about g-series and theta functions and how they interplay with number theory. This book should be suitable for junior and senior undergraduates and beginning graduate students. Since many readers may not be familiar with Ramanujan's life, we begin with a short account of his life where readers learn about the notebooks and lost notebook in which he recorded his theorems over several years. We provide brief histories, first of the "ordinary" notebooks, and second of the lost notebook. After these biographical and historical narrations, we provide short summaries of the book's seven chapters. Ramanujan was born on December 22, 1887 in the home of his maternal grandmother in Erode, located in the southern Indian state of Tamil Nadu. After a few months, his mother, Komalatammal, re- turned with her son to her home in Kumbakonam approximately 160
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