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.
1
http://dx.doi.org/10.1090/stml/034/01
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