Dirichlet character
A Dirichlet character^{} modulo $m$ is a group homomorphism^{} from ${\left(\frac{\mathbb{Z}}{m\mathbb{Z}}\right)}^{*}$ to ${\u2102}^{*}$. Dirichlet characters are usually denoted by the Greek letter $\chi $. The function^{}
$$\gamma (n)=\{\begin{array}{cc}\chi (nmodm),\hfill & \text{if}\mathrm{gcd}(n,m)=1,\hfill \\ 0,\hfill & \text{if}\mathrm{gcd}(n,m)1.\hfill \end{array}$$ 
is also referred to as a Dirichlet character. The Dirichlet characters modulo $m$ form a group if one defines $\chi {\chi}^{\prime}$ to be the function which takes $a\in {\left(\frac{\mathbb{Z}}{m\mathbb{Z}}\right)}^{*}$ to $\chi (a){\chi}^{\prime}(a)$. It turns out that this resulting group is isomorphic to ${\left(\frac{\mathbb{Z}}{m\mathbb{Z}}\right)}^{*}$. The trivial character is given by $\chi (a)=1$ for all $a\in {\left(\frac{\mathbb{Z}}{m\mathbb{Z}}\right)}^{*}$, and it acts as the identity element^{} for the group. A character^{} $\chi $ modulo $m$ is said to be induced by a character ${\chi}^{\prime}$ modulo ${m}^{\prime}$ if ${m}^{\prime}\mid m$ and $\chi (n)={\chi}^{\prime}(nmod{m}^{\prime})$. A character which is not induced by any other character is called primitive. If $\chi $ is nonprimitive, the $\mathrm{gcd}$ of all such ${m}^{\prime}$ is called the conductor of $\chi $.
Examples:

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Legendre symbol^{} $\left(\frac{n}{p}\right)$ is a Dirichlet character modulo $p$ for any odd prime $p$. More generally, Jacobi symbol^{} $\left(\frac{n}{m}\right)$ is a Dirichlet character modulo $m$.

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The character modulo $4$ given by $\chi (1)=1$ and $\chi (3)=1$ is a primitive character modulo $4$. The only other character modulo $4$ is the trivial character.
Title  Dirichlet character 

Canonical name  DirichletCharacter 
Date of creation  20130322 13:22:31 
Last modified on  20130322 13:22:31 
Owner  bbukh (348) 
Last modified by  bbukh (348) 
Numerical id  10 
Author  bbukh (348) 
Entry type  Definition 
Classification  msc 11A25 
Related topic  CharacterOfAFiniteGroup 
Defines  trivial character 
Defines  primitive character 
Defines  conductor 
Defines  induced character 