Ramanujan’s Notebooks: Part IV

458 Pages·1994·5.9 MB·English

by  Bruce C. Berndt (auth.)

Ramanujan's Notebooks
Part IV
Bruce C. Berndt
Ramanujan's Notebooks
Part IV
Springer Science+Business Media, LLC
Bruce C. Bemdt
Department of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL 61801
USA
With 4 Illustrations
AMS Subject Classifications: 11-00, 11-03, 01A60, 01A75, 33E05, 11AXX, 33-00, 33-03
Library of Congress Cataloging-in-Publication Data
(Revised for voI. 4)
Ramanujan Aiyangar, Srinivasa, 1887-1920.
Ramanujan's notebooks.
Inc1udes bibliographies and indexes.
1. Mathematics. 1. Berndt, Bruce C., 1939-
II. Title.
QA3.R33 1985 510 84-20201
ISBN 978-1-4612-6932-8 ISBN 978-1-4612-0879-2 (eBook)
DOI 10.1007/978-1-4612-0879-2
Printed on acid-free paper.
© 1994 Springer Science+Business Media New York
Originally published by Springer-Verlag Berlin Heidelberg New York in 1994
Softcover reprint ofthe hardcover Ist edition 1994
Ali rights reserved. This work may not be translated or copied in whole or in part without the
written permission ofthe publisher Springer Science+Business Media, LLC, except for brief
excerpts in connection with reviews or scholarly analysis.
U se in connection with any form of information storage and retrieval, electronic adaptation,
computer software, or by similar or dissimilar methodology now known or hereafter developed
is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if
the former are not especially identified, is not to be taken as a sign that such names, as
understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely
byanyone.
Production coordinated by Brian Howe and managed by Bill Imbornoni; manufacturing
supervised by Jacqui Ashri.
Typeset by Techset Composition Ltd., England, UK.
9 8 7 6 543 2 1
ISBN 978-1-4612-6932-8
Dedicated to
the People of India
Passport photo of Srinivasa Ramanujan.
Reprinted with courtesy of Professor S. Chandrasekhar, F.R.S.
Preface
During the years 1903-1914, Ramanujan recorded most of his mathematical
discoveries without proofs in notebooks. Although many of his results were
already found in the literature, most were not. Almost a decade after
Ramanujan's death in 1920, G. N. Watson and B. M. Wilson began to edit
Ramanujan's notebooks, but they never completed the task. A photostat
edition, with no editing, was published by the Tata Institute of Fundamental
Research in Bombay in 1957.
This book is the fourth of five volumes devoted to the editing of
Ramanujan's notebooks. Part I, published in 1985, contains an account of
Chapters 1-9 in the second notebook as well as a description of Ramanujan's
quarterly reports. Part II, published in 1989, comprises accounts of Chapters
10-15 in the second notebook. Part III, published in 1991, provides an
account of Chapters 16-21 in the second notebook. This is the first of two
volumes devoted to proving the results found in the unorganized portions
of the second notebook and in the third notebook. We also shall prove those
results in the first notebook that are not found in the second or third
notebooks. For those results that are known, we provide references in the
literature where proofs may be found. Otherwise, we give complete proofs.
Urbana, Illinois Bruce C. Berndt
February, 1993
Contents
Preface
IX
In trodu ction 1
CHAPTER 22
Elementary Results 7
CHAPTER 23
Number Theory 51
CHAPTER 24
Ramanujan's Theory of Prime Numbers 111
CHAPTER 25
Theta-Functions and Modular Equations 138
CHAPTER 26
Inversion Formulas for the Lemniscate and Allied Functions 245
CHAPTER 27
q-Series 261
CHAPTER 28
Integrals 287
CHAPTER 29
Special Functions 334
xii Contents
CHAPTER 30
Partial Fraction Expansions 355
CHAPTER 31
Elementary and Miscellaneous Analysis 382
Location in Notebook 2 of the Material in the 16 Chapters of
Notebook 1 401
References 433
Index 447
Introduction
If you have built castles in the air, your work need not be lost; that is where
they should be. Now put the foundations under them.
H. D. Thoreau-Walden
Ramanujan built many castles. Although some may have been lost, most
were preserved. Since his death in 1920, many mathematicians have been
constructing the foundations for these magnificent structures. We continue
this task in the present volume.
In the first three volumes of this series (Berndt [2], [4], [6]), we gave
references or proofs for all of the results claimed by Ramanujan in the 21
chapters of his second notebook. At the end of this notebook, after the 21
chapters of organized material, there are exactly 100 pages of unorganized
results. In the Tata Institute's publication (Ramanujan [22]) of the second
notebook, three pages of further results precede the 21 chapters. Between
1920 and the publication of [22] in 1957, these three pages evidently were
shifted from the unorganized part to the beginning, for in G. N. Watson's
copy of the second notebook, the three pages appear among the unorganized
pages at the end of the second notebook. Ramanujan's third notebook,
published by the Tata Institute in the same volume as the second notebook,
contains only 33 pages of unorganized material. In this volume and the next
(Berndt [9]), we provide proofs or give references for all of the results found
in the unorganized pages of the second and third notebooks.
The first notebook, published as volume 1 by the Tata Institute, is a
preliminary version of the second. However, the first notebook contains
several results not found in the second. In Parts IV and V, we also provide
proofs for these theorems as well.
Since the results in the aforementioned 136 pages of miscellaneous
material in the second and third notebooks were not organized by Ramanu-
2 Introduction
jan into chapters, we have taken the liberty of doing so. Generally, within
each chapter, we have recorded the results in chronological order as they
appear in the unorganized pages. This volume contains ten of the fifteen
chapters into which we have organized Ramanujan's theorems. Part V will
contain chapters on continued fractions, Ramanujan's theory of elliptic
functions to alternative bases, class invariants and singular moduli, asympto
tic analysis and approximations, and infinite series.
In this volume, we also provide an account of the 16 chapters of organized
material in the first notebook. More precisely, for each result in the first
notebook that can be found in the second, we indicate where in the second
notebook the corresponding result is located. Furthermore, we provide
proofs for those theorems in the 16 chapters of Notebook 1 that cannot be
found in the second notebook. Most of the results in these 16 chapters that
Ramanujan failed to record in Notebook 2 are either wrong or relatively
easy to prove. However, some are more interesting and more challenging to
prove. The miscellaneous material in the first notebook contains substanti
ally more results not found in the second notebook; we prove these in Part V.
Brief descriptions of the contents of the ten chapters in this volume will
now be given.
The first chapter, Chapter 22, is devoted to elementary results. Most
require only high school algebra to prove. Despite the modest title, Chapter
22 contains many very interesting results. Several were submitted by Rama
nujan as problems in the Journal of the Indian Mathematical Society. Many
entries in the chapter pertain to polynomial equations or systems of equa
tions. In particular, we mentioned Entries 4, 5, and 32 on certain systems of
equations wherein the solutions are represented as infinite nested radicals.
Although Ramanujan is known to most mathematicians as a number
theorist, Chapters 1-21 in the second notebook contain little number theory,
although much of this material, for example, the chapters on theta-functions,
is related to number theory. Ramanujan's interest in number theory appears
to have commenced only one or two years before he wrote G. H. Hardy in
January, 1913. Most of the discoveries in number theory that Ramanujan
made before departing for England in 1914 are found in the 136 pages of
miscellaneous material in the second and third notebooks. Chapter 23
contains results in number theory, except that discoveries in the theory of
prime numbers are reserved for the following chapter. While in England
under the influence of Hardy, Ramanujan worked primarily on number
theory. As we shall see in Chapters 23 and 24, many of the principal ideas
in Ramanujan's papers in the theory of numbers published in England had
their geneses in India. Thus, most of Chapters 23 and 24 concerns previously
published material. Fortunately, in the unorganized pages of the second and
third notebooks, Ramanujan provides sketches of some of his methods. In
particular, Ramanujan indicates his proof of the asymptotic formula for the
number of integers less than or equal to x that can be represented as a sum
of two squares. He also sketches a general method which he undoubtedly