XXII

Introduction

such a reasonable approach would be to go through Chapter 1 sections 1-4.5

and sections 6 and 7. The above material is essentially background material

to acquaint the reader with the domains on which we will be doing anal-

ysis to obtain the combinatorial results. This reader should then continue

with Chapter 2, where the theory of the one dimensional theta function is

presented. This chapter should be read in its entirety. In Chapter 3 the

reader or instructor pressed for time could read quickly the first section for

the definitions and then go on to a careful study of sections 2 through 8.4.

Chapter 4 should be read in its entirety. The above could constitute a one

semester topics course for beginning graduate students.

The above suggestion leaves out Chapters 5, 6 and 7, which occupy much

of this book and are an important part of it, since they deal with the theory

of congruences for the Ramanujan partition function, the congruences for the

j-function, and the combinatorial interpretations of many of the identities

derived in Chapter 4. In a one-year course the material studied could include

Chapter 5 through section 10.8 and section

12,5

all of Chapters 6 and 7.

The professional mathematician who is interested in Riemann surface

theory should study Chapter 1, including sections 4.6 through 5.7, in order

to get a picture of where the theory can possibly go. If conversant with

the theory of the Riemann theta function, the reader can skip section 1 of

Chapter 2 and read the remainder of that chapter. The reader should then

proceed to the beginning of Chapter 3. Some of the introductory material

of this chapter can be skipped or read quickly to get acquainted with the

notation used; the choice of which of the remaining sections to read should

be guided by interests; we suggest that this include section 10. Chapter 4

should be read in its entirety and then Chapter 5 and Chapter 6, once again

guided by the interests of the reader. Chapter 7 should also be read in its

entirety.

The professional mathematician whose interests are in combinatorial

mathematics may wish to begin by looking at Chapter 7 and then proceed

backwards through the theory. Needless to say, Chapters 2, 3 and 4 will

have to be read at some point, and if interested in Ramanujan congruences,

Chapter 5 is a must. In any event section 12 of Chapter 5 should be reviewed.

There is lots of flexibility in the way the text can be studied and/or

approached. We trust the various readers will find their way through the

maze and enjoy the material they stop to study or, as we and others have

said, will enjoy this tour of Ramanujan's garden and the flowers they pick

there.

5 In a course where all nonstandard material is included, the instructor might want to spend

some time on the multiplicative properties of the 77-function. These could be taken from Knopp's

book [16].