Editing Dirichlet character (section)

== Relation to group characters ==

The word "[[Character (mathematics)|character]]" is used several ways in mathematics. In this section it refers to a [[homomorphism]] from a group <math>G</math> (written multiplicatively) to the multiplicative group of the field of complex numbers:
:<math>\eta: G\rightarrow \mathbb{C}^\times,\;\;\eta(gh)=\eta(g)\eta(h),\;\;\eta(g^{-1})=\eta(g)^{-1}.</math>

The set of characters is denoted <math>\widehat{G}.</math> If the product of two characters is defined by pointwise multiplication <math>\eta\theta(a)=\eta(a)\theta(a),</math> the identity by the trivial character <math>\eta_0(a)=1</math> and the inverse by complex inversion  <math>\eta^{-1}(a)=\eta(a)^{-1}</math> then <math>\widehat{G}</math> becomes an abelian group.<ref>See [[Multiplicative character]]</ref>

If <math>A</math> is a [[abelian group#Finite abelian groups|finite abelian group]] then<ref name="IR">Ireland and Rosen p. 253-254</ref> there is an [[isomorphism]] <math>A\cong\widehat{A}</math>, and the orthogonality relations:<ref>See [[Character group#Orthogonality of characters]]</ref>

:<math>\sum_{a\in A} \eta(a)=

\begin{cases}
|A|&\text{ if  } \eta=\eta_0\\
0&\text{ if  } \eta\ne\eta_0
\end{cases}
</math> &nbsp; &nbsp; and &nbsp; &nbsp;  <math>\sum_{\eta\in\widehat{A}}\eta(a)=
\begin{cases}
|A|&\text{ if  } a=1\\
0&\text{ if  } a\ne 1.
\end{cases}
</math>

The elements of the finite abelian group <math>(\mathbb{Z}/m\mathbb{Z})^\times</math> are the residue classes <math>[a]=\{x:x\equiv a\pmod m\}</math> where <math>(a,m)=1.</math>

A group character <math>\rho:(\mathbb{Z}/m\mathbb{Z})^\times\rightarrow \mathbb{C}^\times</math> can be extended to a Dirichlet character <math>\chi:\mathbb{Z}\rightarrow \mathbb{C}</math> by defining
:<math>
\chi(a)=
\begin{cases}
0 &\text{if } [a]\not\in(\mathbb{Z}/m\mathbb{Z})^\times&\text{i.e. }(a,m)> 1\\
\rho([a])&\text{if } [a]\in(\mathbb{Z}/m\mathbb{Z})^\times&\text{i.e. }(a,m)= 1,
\end{cases}</math>

and conversely, a Dirichlet character mod <math>m</math> defines a group character on <math>(\mathbb{Z}/m\mathbb{Z})^\times.</math>

Paraphrasing Davenport,<ref>Davenport p. 27</ref> Dirichlet characters can be regarded as a particular case of  Abelian group characters. But this article follows Dirichlet in giving a direct and constructive account of them. This is partly for historical reasons, in that Dirichlet's work preceded by several decades the development of group theory, and partly for a mathematical reason, namely that the group in question has a simple and interesting structure which is obscured if one treats it as one treats the general Abelian group.