XX
Introduction
many implications of these constructions, especially of the appearance of
constant functions. This material may not be new to the literature. We
present it in a unified way based on function theoretic foundations that
most of the time remove and in general isolate the mysteries in many of the
research monographs on the subject. This chapter is based almost entirely
on the properties of the classical 77-function, a very special case of the theory
described in the previous chapters. We need to know rather detailed infor-
mation on its multiplier system. The needed number theoretic arguments
are found in [16], for example. We would like to avoid dependence on these.
We have only partially succeeded in doing so and have hence not included
in this volume most of the results of this effort.
In Chapter 6 we begin by reviewing some concepts from covering space
theory and show how these ideas lead to beautiful identities among theta
constants and their interpretation as identities among infinite products.
Here the main tools are the uniformizations of the Riemann surfaces in
question. We then continue by showing how many of the ideas used in the
congruences related with the partition function and its generalizations carry
over to other modular forms. We treat in particular the j-function and the
congruences satisfied by the coefficients of its Laurent series expansion.
In Chapter 7 we show how statements about partitions are related to
other combinatorial quantities such as representations of positive integers
as sums of squares or of triangular numbers, and most importantly to the
divisors of an integer. In particular, we describe relations to the question of
primality of integers depending on statements about partitions. This sug-
gests that while primality is usually thought of as a subject in multiplicative
number theory, it can also be viewed as a part of additive number theory.
We give some examples to show what type of results can be expected in
this chapter in the expectation that these applications are the main interests
of some readers of this text. We emphasize that these results were not the
reason for writing this book. The list of examples is by no means exhaustive.
Let c(n) denote the classical a-function; that is, cr(n) is the sum of the
divisors of the positive integer n. We show that
whenever n is not of the form 3 m
2
+ m with m G Z. A companion related
result is
3=0
X 7
j=0
X 7
whenever n is not of the form m
2
hm with m G Z. We obtain a variant
of Jacobi's result on the number of ways a positive integer can be written
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