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
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