§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 $\displaystyle\mathop{\chi\/}\nolimits\!\left(1\right)$ $\displaystyle=1,$ Symbols: $\mathop{\chi\/}\nolimits\!\left(\NVar{n}\right)$: Dirichlet character and $k$: positive integer Permalink: http://dlmf.nist.gov/27.8.E1 Encodings: TeX, pMML, png See also: Annotations for 27.8 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 $\displaystyle\mathop{\chi\/}\nolimits\!\left(n+k\right)$ $\displaystyle=\mathop{\chi\/}\nolimits\!\left(n\right),$ $n=1,2,\dots$, Symbols: $\mathop{\chi\/}\nolimits\!\left(\NVar{n}\right)$: Dirichlet character, $k$: positive integer and $n$: positive integer Permalink: http://dlmf.nist.gov/27.8.E3 Encodings: TeX, pMML, png See also: Annotations for 27.8 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 $\mathop{\chi_{1}\/}\nolimits\!\left(n\right)=\begin{cases}1,&\left(n,k\right)=% 1,\\ 0,&\left(n,k\right)>1.\end{cases}$

For any character $\mathop{\chi\/}\nolimits\pmod{k}$, $\mathop{\chi\/}\nolimits\!\left(n\right)\neq 0$ 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 $\mathop{\chi_{1}\/}\nolimits,\dots,\mathop{\chi_{\mathop{\phi\/}\nolimits\!% \left(k\right)}\/}\nolimits$. If $\mathop{\chi\/}\nolimits$ is a character (mod $k$), so is its complex conjugate $\overline{\mathop{\chi\/}\nolimits}$. If $\left(n,k\right)=1$, then the characters satisfy the orthogonality relation

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

A Dirichlet character $\mathop{\chi\/}\nolimits\pmod{k}$ is called primitive (mod $k$) if for every proper divisor $d$ of $k$ (that is, a divisor $d), there exists an integer $a\equiv 1\pmod{d}$, with $\left(a,k\right)=1$ and $\mathop{\chi\/}\nolimits\!\left(a\right)\neq 1$. If $k$ is prime, then every nonprincipal character $\mathop{\chi\/}\nolimits\pmod{k}$ 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 $\mathop{\chi\/}\nolimits\!\left(n\right)=\mathop{\chi_{0}\/}\nolimits\!\left(n% \right)\mathop{\chi_{1}\/}\nolimits\!\left(n\right),$ Symbols: $\mathop{\chi\/}\nolimits\!\left(\NVar{n}\right)$: Dirichlet character, $k$: positive integer and $n$: positive integer Permalink: http://dlmf.nist.gov/27.8.E8 Encodings: TeX, pMML, png See also: Annotations for 27.8

where $\mathop{\chi_{0}\/}\nolimits$ is a character (mod $d$) for some induced modulus $d$ for $\mathop{\chi\/}\nolimits$, and $\mathop{\chi_{1}\/}\nolimits$ 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.