# 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 non-primitive, 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.

Title Dirichlet character DirichletCharacter 2013-03-22 13:22:31 2013-03-22 13:22:31 bbukh (348) bbukh (348) 10 bbukh (348) Definition msc 11A25 CharacterOfAFiniteGroup trivial character primitive character conductor induced character