# Primitive Dirichlet Characters

Let $\chi$ be a Dirichlet characters of modulus $q$. Thus $\chi$ is a periodic function of period $q$. It is possible, that for values of $a$ restricted by condition $\gcd(a,q)=1$, the function $\chi$ may have d period less than $q$.

Definition (Primitive characthers) We say that a Dirichlet character $\chi$  of modulus $q$ is imprimitive if $\chi(a)$ restricted by $\gcd(a,q)=1$ has a period which is less than $q$, Otherwise we say that $\chi$ is primitive.

For maximal clarity, let us write out exactly what we mean $\chi(a)$ restricted $(a,q)=1$ has a period $n\in\mathbf{Z}^+$. This means: For any integers $a,b$ with $\gcd(a,q)=\gcd(b,q)=1$ and $a=b\ \text{mod}\ n$ we have $\chi(a)=\chi(b)$.

Definition (Conductor) Given a Dirichlet character $\chi$ of modulus $q$, the conductor of $\chi$ is defined to be the smallest positive integer $q'$ such that $\chi(a)$ restricted by $\gcd(a,q)=1$ has a period $q'$.

Lemma If $\chi$ is a Dirichlet character of modulus $q$ and if $q_1\in\mathbf{Z}^+$ is a period of $\chi(a)$ restricted by $\gcd(a,q)=1$, then $c(\chi)\mid q_1$. Hence in particular ,we have $c(\chi)\mid q$.

Lemma For each Dirichlet character $\chi$ of modulus $q$, there is a unique Dirichlet character of $\chi'$ of modulus $c(\chi)$ such that

$\displaystyle \chi(n)=\begin{cases}\chi'(n) & \text{if}\ \gcd(a,q)=1\\ 0 & \text{if} \gcd(a,q)>1. \end{cases}$

This Dirichlet character $\chi'$ is primitive