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Dirichlet character
A Dirichlet character modulo $m$ is a group homomorphism from $\left(\frac{\mathbb{Z}}{m\mathbb{Z}}\right)^{*}$ to $\mathbb{C^{*}}$. Dirichlet characters are usually denoted by the Greek letter $\chi$. The function
$\gamma(n)=\begin{cases}\chi(n\bmod m),&\text{if }\gcd(n,m)=1,\\ 0,&\text{if }\gcd(n,m)>1.\end{cases}$ 
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 $\chi(n)=\chi^{{\prime}}(n\bmod m^{{\prime}})$. A character which is not induced by any other character is called primitive. If $\chi$ is nonprimitive, the $\gcd$ of all such $m^{{\prime}}$ is called the conductor of $\chi$.
Examples:

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

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.
Mathematics Subject Classification
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