**Student Mathematical Library**

Volume: 34;
2006;
187 pp;
Softcover

MSC: Primary 11;
Secondary 33

Print ISBN: 978-0-8218-4178-5

Product Code: STML/34

List Price: $38.00

Individual Price: $30.40

**Electronic ISBN: 978-1-4704-2145-8
Product Code: STML/34.E**

List Price: $38.00

Individual Price: $30.40

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#### Supplemental Materials

# Number Theory in the Spirit of Ramanujan

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*Bruce C. Berndt*

Ramanujan is recognized as one of the great
number theorists of the twentieth century. Here now is the first book
to provide an introduction to his work in number theory. Most of
Ramanujan's work in number theory arose out of \(q\)-series and
theta functions. This book provides an introduction to these two
important subjects and to some of the topics in number theory that are
inextricably intertwined with them, including the theory of
partitions, sums of squares and triangular numbers, and the Ramanujan
tau function. The majority of the results discussed here are
originally due to Ramanujan or were rediscovered by him. Ramanujan did
not leave us proofs of the thousands of theorems he recorded in his
notebooks, and so it cannot be claimed that many of the proofs given
in this book are those found by Ramanujan. However, they are all in
the spirit of his mathematics.

The subjects examined in this book have a rich history dating back
to Euler and Jacobi, and they continue to be focal points of
contemporary mathematical research. Therefore, at the end of each of
the seven chapters, Berndt discusses the results established in the
chapter and places them in both historical and contemporary contexts.
The book is suitable for advanced undergraduates and beginning
graduate students interested in number theory.

#### Table of Contents

# Table of Contents

## Number Theory in the Spirit of Ramanujan

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface ix10 free
- Chapter 1. Introduction 122 free
- Chapter 2. Congruences for p(n) and r(n) 2748
- Chapter 3. Sums of Squares and Sums of Triangular Numbers 5576
- §3.1. Lambert Series 5576
- §3.2. Sums of Two Squares 5677
- §3.3. Sums of Four Squares 5980
- §3.4. Sums of Six Squares 6384
- §3.5. Sums of Eight Squares 6788
- §3.6. Sums of Triangular Numbers 7192
- §3.7. Representations of Integers by x[(sup)2] + 2y[(sup)2], x[(sup)2] + 3y[(sup)2], and x[(sup)2] + xy + y[(sup)2] 7293
- §3.8. Notes 79100

- Chapter 4. Eisenstein Series 85106
- Chapter 5. The Connection Between Hypergeometric Functions and Theta Functions 109130
- Chapter 6. Applications of the Primary Theorem of Chapter 5 133154
- Chapter 7. The Rogers–Ramanujan Continued Fraction 153174
- Bibliography 171192
- Index 185206
- Back Cover Back Cover1210

#### Readership

Undergraduate and graduate students interested in number theory, including \(q\)-series and theta functions.

#### Reviews

... undergraduates will find no better place to meet the mind behind the towering reputation.

-- D. V. Feldman, University of New Hampshire for CHOICE Reviews

This is a delightful little book on selected topics in number theory. ...I highly recommend this book to all mathematicians. It is a great resource both to learn from and to teach from. Even the experts will enjoy his new perspective on these old questions.

-- Journal of Approximation Theory

This slender volume is extremely well-written and contains a wealth of material. It is a lucid and accessible introduction to a rich and fascinating area of mathematics, written by the world's leading expert. For anyone with a knowledge of calculus wanting to learn about the mathematical work of Ramanujan, this book is the best place to start.

-- Shaun Cooper, Massey University - New Zealand Newsletter

...“Number theory in the spirit of Ramanujan” is a gem that deserves a place in every mathematician's library.

-- Zentralblatt MATH