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