**AMS Chelsea Publishing**

Volume: 136;
1991;
254 pp;
Hardcover

MSC: Primary 11;
Secondary 01

**Print ISBN: 978-0-8218-2023-0
Product Code: CHEL/136.H**

List Price: $44.00

AMS Member Price: $39.60

MAA Member Price: $39.60

**Electronic ISBN: 978-1-4704-6711-1
Product Code: CHEL/136.E**

List Price: $44.00

AMS Member Price: $39.60

MAA Member Price: $39.60

# Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work

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*G. H. Hardy*

AMS Chelsea Publishing: An Imprint of the American Mathematical Society

Ramanujan occupies a unique place in analytic number theory.
His formulas, identities, and calculations are still amazing
three-quarters of a century after his death. Many of his discoveries
seem to have appeared as if from the ether. His mentor and primary
collaborator was the famous G. H. Hardy. Here, Hardy collects twelve
of his own lectures on topics stemming from Ramanujan's life and work.
The topics include partitions, hypergeometric series, Ramanujan's
\(\tau\)-function and round numbers.

Hardy was the first to recognize the brilliance of Ramanujan's
ideas. As one of the great mathematicians of the time, it is
fascinating to read Hardy's accounts of their importance and
influence. The book concludes with a chapter by chapter overview
written by Bruce C. Berndt. In this overview, Berndt gives references
to current literature, developments since Hardy's original lectures,
and background information on Ramanujan's research, including his
unpublished papers.

#### Readership

Graduate students and research mathematicians interested in number theory.

#### Reviews & Endorsements

From the fact that practically all topics of analytic number theory are mentioned, briefly or extensively, in this book in connection with one or the other of Ramanujan's ideas, theorems, conjectures, we realize the far-reaching influence which his work has had on present-day mathematics … the book is not only an homage to Ramanujan's genius; it is a survey of many branches of modern arithmetic and analysis and, altogether, a book which makes fascinating reading.

-- Hans Rademacher, Mathematical Reviews

#### Table of Contents

# Table of Contents

## Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work

- RAMANUJAN 22
- PREFACE 66
- PREFACE TO THE FOURTH PRINTING 88
- CONTENTS 1010
- I THE INDIAN MATHEMATICIAN RAMANUJAN 1212
- II RAMANUJAN AND THE THEORYOF PRIME NUMBERS 3333
- III ROUND NUMBERS 5959
- IV SOME MORE PROBLEMS OF THE ANALYTIC THEORY OF NUMBERS 6969
- V A LATTICE-POINT PROBLEM 7878
- VI RAMANUJAN’S WORK ON PARTITIONS 9494
- VII HYPERGEOMETRIC SERIES 112112
- VIII ASYMPTOTIC THEORY OF PARTITIONS 124124
- IX THE REPRESENTATION OF NUMBERS AS SUMS OF SQUARES 143143
- X RAMANUJAN’S FUNCTION r(n) 172172
- XI DEFINITE INTEGRALS 197197
- XII ELLIPTIC AND MODULAR FUNCTIONS 223223
- BIBLIOGRAPHY 242242
- COMMENTARY ON RAMANUJAN BY G. H. HARDY 248248