Dirichlet character

A Dirichlet character modulo $m$ is a group homomorphism from ${\left(\frac{\mathbb{Z}}{m\mathbb{Z}}\right)}^{*}$ to ${ℂ}^{*}$. Dirichlet characters are usually denoted by the Greek letter $\chi$. The function

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 non-primitive, the $\gcd$ of all such $m^{\prime }$ is called the conductor of $\chi$.

Examples:

• 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 $m$.

• 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.