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

§27.8 Dirichlet Characters

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Defines:: $\chi\left(\NVar{n}\right)$ : Dirichlet character
Keywords:: Dirichlet, Dirichlet character, Dirichlet characters, characters, induced modulus, number theory, number-theoretic functions, orthogonality, orthogonality relation, primitive, principal, real
Notes:: See Apostol (1976, pp. 137–142, 167–172).
Referenced by:: §24.16(ii), §25.11(x), §25.15(i), §25.15(i), §27.3, Erratum (V1.0.19) for Notation
Permalink:: http://dlmf.nist.gov/27.8
See also:: Annotations for Ch.27

If is a given integer, then a function $\chi\left(n\right)$ 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 $\chi\left(1\right)=1,$

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Symbols:: $\chi\left(\NVar{n}\right)$ : Dirichlet character and : 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 $\chi\left(mn\right)=\chi\left(m\right)\chi\left(n\right),$ $m,n=1,2,\dots$ ,

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Symbols:: $\chi\left(\NVar{n}\right)$ : Dirichlet character, : positive integer, : positive integer and : 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 $\chi\left(n+k\right)=\chi\left(n\right),$ $n=1,2,\dots$ ,

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Symbols:: $\chi\left(\NVar{n}\right)$ : Dirichlet character, : positive integer and : 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 $\chi\left(n\right)=0,$ $\left(n,k\right)>1$ .

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Symbols:: $\chi\left(\NVar{n}\right)$ : Dirichlet character, $\left(\NVar{m},\NVar{n}\right)$ : greatest common divisor (gcd), : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/27.8.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §27.8 and Ch.27

An example is the principal character (mod ):

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

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Symbols:: $\chi\left(\NVar{n}\right)$ : Dirichlet character, $\left(\NVar{m},\NVar{n}\right)$ : greatest common divisor (gcd), : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/27.8.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §27.8 and Ch.27

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 ), which can be labeled as $\chi_{1},\dots,\chi_{\phi\left(k\right)}$ . If $\chi$ is a character (mod ), so is its complex conjugate $\overline{\chi}$ . If $\left(n,k\right)=1$ , then the characters satisfy the orthogonality relation

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

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Symbols:: $\chi\left(\NVar{n}\right)$ : Dirichlet character, $\phi\left(\NVar{n}\right)$ : Euler’s totient, $\overline{\NVar{z}}$ : complex conjugate, $\equiv$ : modular equivalence, : positive integer, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/27.8.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §27.8 and Ch.27

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

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

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Symbols:: $\chi\left(\NVar{n}\right)$ : Dirichlet character, $\left(\NVar{m},\NVar{n}\right)$ : greatest common divisor (gcd), $\equiv$ : modular equivalence, : positive integer and : positive integer
Permalink:: http://dlmf.nist.gov/27.8.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §27.8 and Ch.27

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

27.8.8 $\chi\left(n\right)=\chi_{0}\left(n\right)\chi_{1}\left(n\right),$

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Symbols:: $\chi\left(\NVar{n}\right)$ : Dirichlet character, : positive integer and : 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 ) for some induced modulus for $\chi$ , and $\chi_{1}$ 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.

27.7 Lambert Series as Generating Functions 27.9 Quadratic Characters

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