primitive Dirichlet characters

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1: 27.8 Dirichlet Characters

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27.8.6

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $\varphi \left(n\right)$: Euler’s totient, $\overline{z}$: complex conjugate, $\equiv$: modular equivalence, $k$: positive integer, $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E6

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Annotations for §27.8 and Ch.27

►A Dirichlet character $\chi (modk)$ is called primitive (mod $k$) if for every proper divisor $d$ of $k$ (that is, a divisor ), there exists an integer $a\equiv 1(modd)$, with $\left(a,k\right)=1$ and $\chi \left(a\right)\ne 1$. If $k$ is prime, then every nonprincipal character $\chi (modk)$ is primitive. …

2: 24.16 Generalizations

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§24.16(ii) Character Analogs

►Let $\chi$ be a primitive Dirichlet character $modf$ (see §27.8). …

3: 27.10 Periodic Number-Theoretic Functions

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27.10.11 $${|G(1,\chi )|}^2=k.$$

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $G(n,\chi )$: Gauss sum, $k$: positive integer and $\chi$: Dirichlet character

Permalink:

http://dlmf.nist.gov/27.10.E11

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Annotations for §27.10 and Ch.27

… ►The finite Fourier expansion of a primitive Dirichlet character $\chi (modk)$ has the form …

4: 25.15 Dirichlet $L$-functions

… ►When $\chi$ is a primitive character (mod $k$) the $L$-functions satisfy the functional equation: … ►Since $L(s,\chi )\ne 0$ if $\mathrm{\Re }s>1$, (25.15.5) shows that for a primitive character $\chi$ the only zeros of $L(s,\chi )$ for (the so-called trivial zeros) are as follows: …