 
Summary: Character and Object
Becky Morris
December 28, 2011
1 Introduction
The concept of function permeates most branches of modern mathematics: from
analysis, algebra and logic to probability and mathematical physics. However, the
settheoretic conception of function as a particular mathematical object (specif
ically, as a set of ordered pairs satisfying conditions of totality and single val
uedness) was a long time in development. Indeed, historians generally cite the
eighteenth century as the time during which the concept of function first made an
explicit appearance. Moreover, from the time it was first introduced to the present
day, the concept of function underwent significant development and changed con
siderably (see e.g. [Mon72], [You76], [Kle89], [Luz98a], and [Luz98b]). Most
of the historical literature focuses primarily on the development of the function
concept within mathematical analysis, but the concept also underwent significant
changes within number theory. In this thesis, we will explore a facet of the devel
opment of the function concept in number theory by looking at the evolution of
particular types of functions, and the associated philosophical consequences.
In particular, we shall examine the history of certain functions called Dirich
let characters and Dirichlet Lfunctions used in the proof of a result first estab
