# §27.8 Dirichlet Characters

If $k$ $(>1)$ is a given integer, then a function $\mathop{\chi\/}\nolimits\!\left(n\right)$ is called a Dirichlet character (mod $k$) if it is completely multiplicative, periodic with period $k$, and vanishes when $\left(n,k\right)>1$. In other words, Dirichlet characters (mod $k$) satisfy the four conditions:

 27.8.1 $\displaystyle\mathop{\chi\/}\nolimits\!\left(1\right)$ $\displaystyle=1,$ 27.8.2 $\displaystyle\mathop{\chi\/}\nolimits\!\left(mn\right)$ $\displaystyle=\mathop{\chi\/}\nolimits\!\left(m\right)\mathop{\chi\/}\nolimits% \!\left(n\right),$ $m,n=1,2,\dots$, 27.8.3 $\displaystyle\mathop{\chi\/}\nolimits\!\left(n+k\right)$ $\displaystyle=\mathop{\chi\/}\nolimits\!\left(n\right),$ $n=1,2,\dots$, 27.8.4 $\displaystyle\mathop{\chi\/}\nolimits\!\left(n\right)$ $\displaystyle=0,$ $\left(n,k\right)>1$.

An example is the principal character (mod $k$):

 27.8.5 $\mathop{\chi\/}\nolimits_{1}(n)=\begin{cases}1,&\left(n,k\right)=1,\\ 0,&\left(n,k\right)>1.\end{cases}$

For any character $\mathop{\chi\/}\nolimits\;\;(\mathop{{\rm mod}}k)$, $\mathop{\chi\/}\nolimits\!\left(n\right)\neq 0$ if and only if $\left(n,k\right)=1$, in which case the Euler–Fermat theorem (27.2.8) implies $\left(\mathop{\chi\/}\nolimits\!\left(n\right)\right)^{\mathop{\phi\/}% \nolimits\!\left(k\right)}=1$. There are exactly $\mathop{\phi\/}\nolimits\!\left(k\right)$ different characters (mod $k$), which can be labeled as $\mathop{\chi\/}\nolimits_{1},\dots,\mathop{\chi\/}\nolimits_{\mathop{\phi\/}% \nolimits\!\left(k\right)}$. If $\mathop{\chi\/}\nolimits$ is a character (mod $k$), so is its complex conjugate $\conj{\mathop{\chi\/}\nolimits}$. If $\left(n,k\right)=1$, then the characters satisfy the orthogonality relation

 27.8.6 $\sum_{r=1}^{\mathop{\phi\/}\nolimits\!\left(k\right)}\mathop{\chi\/}\nolimits_% {r}(m)\conj{\mathop{\chi\/}\nolimits}_{r}(n)=\begin{cases}\mathop{\phi\/}% \nolimits\!\left(k\right),&m\equiv n\;\;(\mathop{{\rm mod}}k),\\ 0,&\mbox{otherwise}.\end{cases}$

A Dirichlet character $\mathop{\chi\/}\nolimits\;\;(\mathop{{\rm mod}}k)$ is called primitive (mod $k$) if for every proper divisor $d$ of $k$ (that is, a divisor $d), there exists an integer $a\equiv 1\;\;(\mathop{{\rm mod}}d)$, with $\left(a,k\right)=1$ and $\mathop{\chi\/}\nolimits\!\left(a\right)\neq 1$. If $k$ is prime, then every nonprincipal character $\mathop{\chi\/}\nolimits\;\;(\mathop{{\rm mod}}k)$ is primitive. A divisor $d$ of $k$ is called an induced modulus for $\mathop{\chi\/}\nolimits$ if

 27.8.7 $\mathop{\chi\/}\nolimits\!\left(a\right)=1\text{ for all a\equiv 1 (mod d)},$ $\left(a,k\right)=1$.

Every Dirichlet character $\mathop{\chi\/}\nolimits$ (mod $k$) is a product

 27.8.8 $\mathop{\chi\/}\nolimits\!\left(n\right)=\mathop{\chi\/}\nolimits_{0}(n)% \mathop{\chi\/}\nolimits_{1}(n),$

where $\mathop{\chi\/}\nolimits_{0}$ is a character (mod $d$) for some induced modulus $d$ for $\mathop{\chi\/}\nolimits$, and $\mathop{\chi\/}\nolimits_{1}$ is the principal character (mod $k$). A character is real if all its values are real. If $k$ is odd, then the real characters (mod $k$) are the principal character and the quadratic characters described in the next section.