Editing Dirichlet character (section)

=== Powers of odd primes ===

If <math>q=p^k</math> is an odd number <math>(\mathbb{Z}/q\mathbb{Z})^\times</math> is cyclic of order <math>\phi(q)</math>; a generator is called a [[Primitive root modulo n|primitive root]] mod <math>q</math>.<ref>There is a primitive root mod <math>p</math> which is a primitive root mod <math>p^2</math> and all higher powers of <math>p</math>. See, e.g., Landau p. 106</ref>
Let <math>g_q</math> be a primitive root and  for <math>(a,q)=1</math> define the function <math>\nu_q(a)</math> (the '''index''' of <math>a</math>) by
:<math>a\equiv g_q^{\nu_q(a)}\pmod {q},</math>
:<math>0\le\nu_q<\phi(q).</math>
For <math>(ab,q)=1,\;\;a \equiv b\pmod{q}</math> if and only if <math>\nu_q(a)=\nu_q(b).</math> Since
:<math>\chi(a)=\chi(g_q^{\nu_q(a)})=\chi(g_q)^{\nu_q(a)},</math> &nbsp; <math>\chi</math> is determined by its value at <math>g_q.</math>

Let <math>\omega_q= \zeta_{\phi(q)}</math>  be a primitive <math>\phi(q)</math>-th root of unity. From property 7) above the possible values of <math> \chi(g_q)</math> are
<math> \omega_q, \omega_q^2, ... \omega_q^{\phi(q)}=1.</math> These distinct values give rise to  <math>\phi(q)</math> Dirichlet characters mod <math>q.</math> For <math>(r,q)=1</math> define <math>\chi_{q,r}(a)</math> as
:<math>
\chi_{q,r}(a)=
\begin{cases}
0 &\text{if } \gcd(a,q)>1\\
\omega_q^{\nu_q(r)\nu_q(a)}&\text{if } \gcd(a,q)=1.
\end{cases}</math>

Then for <math>(rs,q)=1</math> and all <math>a</math> and <math>b</math>
:<math>\chi_{q,r}(a)\chi_{q,r}(b)=\chi_{q,r}(ab),</math> showing that <math>\chi_{q,r}</math> is a character and
:<math>\chi_{q,r}(a)\chi_{q,s}(a)=\chi_{q,rs}(a),</math> which gives an explicit isomorphism <math>\widehat{(\mathbb{Z}/p^k\mathbb{Z})^\times}\cong(\mathbb{Z}/p^k\mathbb{Z})^\times.</math>

==== Examples ''m'' = 3, 5, 7, 9 ====
2 is a primitive root mod 3. &nbsp; (<math>\phi(3)=2</math>)
:<math>2^1\equiv 2,\;2^2\equiv2^0\equiv 1\pmod{3},</math>
so the values of <math>\nu_3</math> are
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
a & 1 & 2  \\
\hline
\nu_3(a) & 0 &  1\\
\end{array}
</math>.
The nonzero values of the characters mod 3 are
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
             & 1 & 2  \\
\hline
\chi_{3,1} & 1 & 1  \\
\chi_{3,2} & 1 & -1  \\
\end{array}
</math>

2 is a primitive root mod 5. &nbsp; (<math>\phi(5)=4</math>)
:<math>2^1\equiv 2,\;2^2\equiv 4,\;2^3\equiv 3,\;2^4\equiv2^0\equiv 1\pmod{5},</math>
so the values of <math>\nu_5</math> are
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
a & 1 & 2 & 3 & 4  \\
\hline
\nu_5(a) & 0 & 1 & 3 & 2 \\
\end{array}
</math>.
The nonzero values of the characters mod 5 are
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
             & 1 & 2  & 3    &  4  \\
\hline
\chi_{5,1} & 1 & 1   & 1   &  1 \\
\chi_{5,2} & 1 & i   & -i  & -1\\
\chi_{5,3} & 1 & -i  & i   & -1\\
\chi_{5,4} & 1 & -1  & -1  &  1\\
\end{array}
</math>

3 is a primitive root mod 7. &nbsp; (<math>\phi(7)=6</math>)
:<math>3^1\equiv 3,\;3^2\equiv 2,\;3^3\equiv 6,\;3^4\equiv 4,\;3^5\equiv 5,\;3^6\equiv3^0\equiv 1\pmod{7},</math>
so the values of <math>\nu_7</math> are
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
a        & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
\nu_7(a) & 0 & 2 & 1 & 4 & 5 & 3 \\
\end{array}
</math>.
The nonzero values of the characters mod 7 are (<math>\omega=\zeta_6, \;\;\omega^3=-1</math>)
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
             & 1 & 2         & 3         & 4        & 5          & 6   \\
\hline
\chi_{7,1} & 1 & 1         & 1         & 1        & 1          & 1   \\
\chi_{7,2} & 1 & -\omega   & \omega^2  & \omega^2 & -\omega    & 1   \\
\chi_{7,3} & 1 & \omega^2  & \omega    & -\omega  & -\omega^2  & -1  \\
\chi_{7,4} & 1 & \omega^2  & -\omega   & -\omega  & \omega^2   & 1   \\
\chi_{7,5} & 1 & -\omega   & -\omega^2 & \omega^2 & \omega     & -1  \\
\chi_{7,6} & 1 & 1         & -1        & 1        & -1         & -1  \\
\end{array}
</math>.

2 is a primitive root mod 9. &nbsp; (<math>\phi(9)=6</math>)
:<math>2^1\equiv 2,\;2^2\equiv 4,\;2^3\equiv 8,\;2^4\equiv 7,\;2^5\equiv 5,\;2^6\equiv2^0\equiv 1\pmod{9},</math>
so the values of <math>\nu_9</math> are
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
a & 1 & 2 &4 & 5&7&8 \\
\hline
\nu_9(a) & 0 & 1 & 2 & 5&4&3 \\
\end{array}
</math>.
The nonzero values of the characters mod 9 are (<math>\omega=\zeta_6, \;\;\omega^3=-1</math>)
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
               & 1 & 2         & 4         & 5         &7          & 8 \\
\hline
\chi_{9,1} & 1 & 1         & 1         & 1         & 1         & 1 \\
\chi_{9,2} & 1 & \omega    & \omega^2  & -\omega^2 & -\omega   & -1 \\
\chi_{9,4} & 1 & \omega^2  & -\omega   & -\omega   & \omega^2  & 1  \\
\chi_{9,5} & 1 & -\omega^2 & -\omega   & \omega    & \omega^2  & -1  \\
\chi_{9,7} & 1 & -\omega   &  \omega^2 & \omega^2  & -\omega   & 1  \\
\chi_{9,8} & 1 & -1        & 1         & -1        & 1         & -1  \\
\end{array}
</math>.