Chapter 1

Introduction

1.1. Notation and Arithmetical Functions

In this first section, we introduce notation that will be used through-

out the entire monograph. Secondly, we define the arithmetical func-

tions on which we will focus for most of this book, and which can be

studied by employing the theory of theta functions and g-series. A

brief introduction to theta functions and ^-series will be given in the

next section, to be followed in Section 1.3 by a few of the most useful

theorems about these functions.

Definition 1.1.1. Define

(1.1.1)

n - l

(a)0 := (a; q)o := 1, (a)n := (a; q)n := J J (1 -

aqk),

n 1,

k=0

CO

(1.1.2) (a)oo := (a; q)oo := ]J(l ~

aqk),

\q\ 1.

k=0

We call q the base, and if the identification of the base is clear, we

often omit q from the notation.

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http://dx.doi.org/10.1090/stml/034/01