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Number Theory in the Spirit of Ramanujan

American Mathematical Society
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One of the richest areas of mathematical research today lies in the exploration and extension of the insights of Srinivasa Ramanujan, the self-taught genius who died tragically young and left a legacy of unpublished results that are still being sorted out. Now that Berndt has completed the five volumes of Ramanujan's Notebooks, and Andrews and Berndt have begun to publish the volumes of Ramanujan's Lost Notebook, the process of explaining and setting into context all of Ramanujan's unpublished results is nearing completion. These volumes, however, have left a very conspicuous void in the literature: an introduction to this body of work, a well-paced development of Ramanujan's methods and tools that begins with minimal prerequisites and takes the reader up to the point at which Ramanujan's prolific and varied output can be appreciated. The book under review does an excellent job of filling this void.

It is a bit of a stretch to say that this book is appropriate for undergraduates. The primary audience will be graduate students and even professional mathematicians who seek an accessible introduction to Ramanujan's body of work. But there really are very minimal prerequisites. The methods of this book are almost entirely formal manipulation of infinite series and products. The first three chapters could form the foundation of a very nice senior seminar. The first chapter introduces and proves the fundamental identities on which so much of Ramanujan's work relies: the q-binomial theorem, Jacobi's triple product identity, Euler's pentagonal number theorem, Ramanujan's 1ψ1 summation, and the quintuple product identity. In the next two chapters, these are used to explore the partition function, p(n), its congruence identities, the congruence identities for Ramanujan's τ function, and results on the number of representations of an integer as a sum of squares and as a sum of triangular numbers.

After chapter 3, the book begins to prepare the reader for the heart of Ramanujan's contributions, his work on modular functions. We are introduced to the identities that connect Eisenstein series, hypergeometric series, theta functions, and elliptic integrals. Chapter 5 leads up to and proves the identity that Berndt refers to as the Main Event:

\[ \left(\sum_{n=-\infty}^\infty q^{n^2}\right)^2 = {}_2F_1(1/2.1/2;1,x),\]


\[ q = \exp\left(-\pi\frac{{\phantom{F}}_2F_1(1/2,1/2;1,1-x)}{{\phantom{F}}_2F_1(1/2,1/2;1,x}\right) \qquad \text{ and } \qquad {\phantom{F}}_2F_1(1/2,1/2;1,x) = \sum_{n=0}^\infty \binom{2n}{n}^2\left(\frac{x}{16}\right)^n.\] 

This is followed by a chapter and a half of illustrations of the power of this identity.

The last chapter is an introduction to Ramanujan's work on continued fractions. One cannot talk about Ramanujan's mathematics without including continued fractions, but to include full proofs of even the simplest results would have required a far longer and more difficult book. For this reason, this chapter is primarily expository.

My only regret is that chapters 4 through 6 never explain the context of Eisenstein series, elliptic integrals, and modular functions, what these identities mean and why they are so interesting. I recognize that within the scope of this book, that would have been too much to ask. But the reader who knows nothing about elliptic integrals would profit by reading these chapters in conjunction with McKean and Moll's Elliptic Curves.

David M. Bressoud is DeWitt Wallace Professor Mathematics at Macalester College in St. Paul, Minnesota.

Date Received: 
Monday, August 21, 2006
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Bruce C. Berndt
Student Mathematical Library 34
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David M. Bressoud
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Monday, January 17, 2011