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].
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