§27.8 Dirichlet Characters

If $k$ $(>1)$ is a given integer, then a function $\chi\left(n\right)$ 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 $\displaystyle\chi\left(1\right)$ $\displaystyle=1,$ ⓘ Symbols: $\chi\left(\NVar{n}\right)$: Dirichlet character and $k$: positive integer Permalink: http://dlmf.nist.gov/27.8.E1 Encodings: TeX, pMML, png See also: Annotations for §27.8 and Ch.27 27.8.2 $\displaystyle\chi\left(mn\right)$ $\displaystyle=\chi\left(m\right)\chi\left(n\right),$ $m,n=1,2,\dots$, ⓘ Symbols: $\chi\left(\NVar{n}\right)$: Dirichlet character, $k$: positive integer, $m$: positive integer and $n$: positive integer Permalink: http://dlmf.nist.gov/27.8.E2 Encodings: TeX, pMML, png See also: Annotations for §27.8 and Ch.27 27.8.3 $\displaystyle\chi\left(n+k\right)$ $\displaystyle=\chi\left(n\right),$ $n=1,2,\dots$, ⓘ Symbols: $\chi\left(\NVar{n}\right)$: 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 and Ch.27 27.8.4 $\displaystyle\chi\left(n\right)$ $\displaystyle=0,$ $\left(n,k\right)>1$.

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

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

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

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

A Dirichlet character $\chi\pmod{k}$ 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 $\chi\left(a\right)\neq 1$. If $k$ is prime, then every nonprincipal character $\chi\pmod{k}$ is primitive. A divisor $d$ of $k$ is called an induced modulus for $\chi$ if

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

Every Dirichlet character $\chi$ (mod $k$) is a product

 27.8.8 $\chi\left(n\right)=\chi_{0}\left(n\right)\chi_{1}\left(n\right),$ ⓘ Symbols: $\chi\left(\NVar{n}\right)$: 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 and Ch.27

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