# primitive

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## 7 matching pages

##### 1: 27.8 Dirichlet Characters
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. …
##### 2: 27.2 Functions
and if $\phi\left(n\right)$ is the smallest positive integer $f$ such that $a^{f}\equiv 1\pmod{n}$, then $a$ is a primitive root mod $n$. …
27.2.12 $\mu\left(n\right)=\begin{cases}1,&n=1,\\ (-1)^{\nu\left(n\right)},&a_{1}=a_{2}=\dots=a_{\nu\left(n\right)}=1,\\ 0,&\mbox{otherwise}.\end{cases}$
27.2.13 $\lambda\left(n\right)=\begin{cases}1,&n=1,\\ (-1)^{a_{1}+\dots+a_{\nu\left(n\right)}},&n>1.\end{cases}$
##### 3: 27.10 Periodic Number-Theoretic Functions
This is the sum of the $n$th powers of the primitive $k$th roots of unity. … For a primitive character $\chi\pmod{k}$, $G\left(n,\chi\right)$ is separable for every $n$, and Conversely, if $G\left(n,\chi\right)$ is separable for every $n$, then $\chi$ is primitive (mod $k$). The finite Fourier expansion of a primitive Dirichlet character $\chi\pmod{k}$ has the form …
##### 4: 27.21 Tables
8 gives examples of primitive roots of all primes $\leq 9973$; Table 24. …
##### 5: 29.17 Other Solutions
They are algebraic functions of $\operatorname{sn}\left(z,k\right)$, $\operatorname{cn}\left(z,k\right)$, and $\operatorname{dn}\left(z,k\right)$, and have primitive period $8K$. …
##### 6: 25.15 Dirichlet $L$-functions
where $\chi_{0}$ is a primitive character (mod $d$) for some positive divisor $d$ of $k$27.8). When $\chi$ is a primitive character (mod $k$) the $L$-functions satisfy the functional equation: … Since $L\left(s,\chi\right)\neq 0$ if $\Re s>1$, (25.15.5) shows that for a primitive character $\chi$ the only zeros of $L\left(s,\chi\right)$ for $\Re s<0$ (the so-called trivial zeros) are as follows: …
##### 7: 24.16 Generalizations
###### §24.16(ii) Character Analogs
Let $\chi$ be a primitive Dirichlet character $\mod f$ (see §27.8). …