§27.8 Dirichlet Characters

Notes:: See Apostol (1976, pp. 137–142, 167–172).
Defines:: $\mathop{\chi\/}\nolimits\!\left(n\right)$ : Dirichlet character
Keywords:: Dirichlet characters, characters, number-theoretic functions, number theory
Referenced by:: §24.16(ii), §25.11(x), §25.15(i), §25.15(i), §27.3
Permalink:: http://dlmf.nist.gov/27.8

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 $\mathop{\chi\/}\nolimits\!\left(1\right)=1,$

Symbols:: $\mathop{\chi\/}\nolimits\!\left(n\right)$ : Dirichlet character and $k$ : positive integer
Permalink:: http://dlmf.nist.gov/27.8.E1
Encodings:: TeX, pMML, png

27.8.2 $\mathop{\chi\/}\nolimits\!\left(mn\right)=\mathop{\chi\/}\nolimits\!\left(m\right)\mathop{\chi\/}\nolimits\!\left(n\right),$ $m,n=1,2,\dots$ ,

Symbols:: $\mathop{\chi\/}\nolimits\!\left(n\right)$ : Dirichlet character, $k$ : positive integer, $m$ : positive integer and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.8.E2
Encodings:: TeX, pMML, png

27.8.3 $\mathop{\chi\/}\nolimits\!\left(n+k\right)=\mathop{\chi\/}\nolimits\!\left(n\right),$ $n=1,2,\dots$ ,

Symbols:: $\mathop{\chi\/}\nolimits\!\left(n\right)$ : Dirichlet character, $k$ : positive integer and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.8.E3
Encodings:: TeX, pMML, png

27.8.4 $\mathop{\chi\/}\nolimits\!\left(n\right)=0,$ $\left(n,k\right)>1$ .

Symbols:: $\mathop{\chi\/}\nolimits\!\left(n\right)$ : Dirichlet character, $k$ : positive integer and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.8.E4
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{\chi\/}\nolimits\!\left(n\right)$ : Dirichlet character, $k$ : positive integer and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.8.E5
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\chi\/}\nolimits\!\left(n\right)$ : Dirichlet character, $\mathop{\phi _{{k}}\/}\nolimits\!\left(n\right)$ : sum of powers of integers relatively prime to $n$ , $\conj{z}$ : complex conjugate, $k$ : positive integer, $m$ : positive integer and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.8.E6
Encodings:: TeX, pMML, png

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 $d<k$ ), 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 $d$)},$ $\left(a,k\right)=1$ .

Symbols:: $\mathop{\chi\/}\nolimits\!\left(n\right)$ : Dirichlet character, $d$ : positive integer and $k$ : positive integer
Permalink:: http://dlmf.nist.gov/27.8.E7
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{\chi\/}\nolimits\!\left(n\right)$ : Dirichlet character, $k$ : positive integer and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.8.E8
Encodings:: TeX, pMML, png

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.