27 Functions of Number Theory Multiplicative Number Theory 27.7 Lambert Series as Generating Functions 27.9 Quadratic Characters

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

27.8.1 $\mathop{\chi\/}\nolimits\!\left(1\right)=1,$

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

An example is the principal character (mod ):

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, : positive integer and : 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 ), 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 ), 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 , $\conj{z}$ : complex conjugate, : positive integer, : positive integer and : 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 ) if for every proper divisor of (that is, a divisor ), 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 is prime, then every nonprincipal character $\mathop{\chi\/}\nolimits\;\;(\mathop{{\rm mod}}k)$ is primitive. A divisor of 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, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/27.8.E7
Encodings:: TeX, pMML, png

Every Dirichlet character $\mathop{\chi\/}\nolimits$ (mod ) 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, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/27.8.E8
Encodings:: TeX, pMML, png

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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 27.7 Lambert Series as Generating Functions 27.9 Quadratic Characters