SPIRI T OF R A M A N U J A N xvn The first chapter of this book is devoted to basic facts about g-series and theta functions, including the g-binomial theorem, the Jacobi triple product identity, the pentagonal number theorem, Ra- manujan's 1^1 summation theorem, and the quintuple product iden- tity. Many of the theorems proved in Chapter 1 can be found in Chapter 16 of Ramanujan's second notebook [193], [34]. Chapter 2 focuses on congruences for the partition function p(n) and Ramanujan's tau function r(n). Much of this material is taken from Ramanujan's handwritten manuscript on p(n) and r{ri), which was first published in 1988 along with Ramanujan's lost notebook [194]. Adding details to many of Ramanujan's proofs and discussing Ramanujan's theorems in light of the literature written after Ra- manujan's death, the present author and K. Ono [50] published an expanded version of this manuscript. In his notebooks [193], Ramanujan recorded a large number of entries on Lambert series. These identities for Lambert series were used by Ramanujan to establish theta function identities and formu- las for the number of representations of an integer as a sum of a certain numbers of squares or of triangular numbers. We introduce readers to Lambert series in Chapter 3 and establish many identi- ties leading to formulas for sums of squares and triangular numbers. A manuscript with no proofs on precisely this subject is another of those manuscripts published with Ramanujan's lost notebook [194], [19, Chapter 18]. His second notebook also contains a large number of such theorems. Eisenstein series permeate Ramanujan's notebooks [193] and lost notebook [194]. Much of our exposition on Eisenstein series in Chap- ter 4, however, is taken from Ramanujan's epic paper [186], [192, 136-162]. One of Ramanujan's approaches to congruences for p(n) is based on Eisenstein series, which we demonstrate at the close of Chapter 4. In Chapter 5, we introduce readers to hypergeometric functions and elliptic integrals. Our goal in this chapter is to prove one of the most fundamental theorems of elliptic functions relating hyper- geometric functions and elliptic integrals to theta functions. This theorem enables us to express theta functions and Eisenstein series
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