Editing Dirichlet character (section)

=== Powers of 2 ===

<math>(\mathbb{Z}/2\mathbb{Z})^\times</math> is the trivial group with one element. <math>(\mathbb{Z}/4\mathbb{Z})^\times</math> is cyclic of order 2. For 8, 16, and higher powers of 2, there is no primitive root; the powers of 5 are the units <math>\equiv 1\pmod{4}</math> and their negatives are the units <math>\equiv 3\pmod{4}.</math><ref>Landau pp. 107-108</ref>
For example
:<math>5^1\equiv 5,\;5^2\equiv5^0\equiv 1\pmod{8}</math>
:<math>5^1\equiv 5,\;5^2\equiv 9,\;5^3\equiv 13,\;5^4\equiv5^0\equiv 1\pmod{16}</math>
:<math>5^1\equiv 5,\;5^2\equiv 25,\;5^3\equiv 29,\;5^4\equiv 17,\;5^5\equiv 21,\;5^6\equiv 9,\;5^7\equiv 13,\;5^8\equiv5^0\equiv 1\pmod{32}.</math>
Let <math>q=2^k, \;\;k\ge3</math>; then <math>(\mathbb{Z}/q\mathbb{Z})^\times</math> is the direct product of a cyclic group of order 2 (generated by −1) and a cyclic group of order <math>\frac{\phi(q)}{2}</math> (generated by 5).
For odd numbers <math>a</math> define the functions <math>\nu_0</math> and <math>\nu_q</math> by
:<math>a\equiv(-1)^{\nu_0(a)}5^{\nu_q(a)}\pmod{q},</math>
:<math>0\le\nu_0<2,\;\;0\le\nu_q<\frac{\phi(q)}{2}.</math>
For odd <math>a</math> and <math>b, \;\;a\equiv b\pmod{q}</math> if and only if <math>\nu_0(a)=\nu_0(b)</math> and <math>\nu_q(a)=\nu_q(b).</math>
For odd <math>a</math> the value of <math> \chi(a)</math> is determined by the values of <math> \chi(-1)</math> and <math>\chi(5).</math>

Let <math>\omega_q = \zeta_{\frac{\phi(q)}{2}}</math> be a primitive <math>\frac{\phi(q)}{2}</math>-th root of unity. The possible values of <math> \chi((-1)^{\nu_0(a)}5^{\nu_q(a)})</math> are
<math> \pm\omega_q, \pm\omega_q^2, ... \pm\omega_q^{\frac{\phi(q)}{2}}=\pm1.</math> These distinct values give rise to  <math>\phi(q)</math> Dirichlet characters mod <math>q.</math> For odd <math>r </math> define <math>\chi_{q,r}(a)</math> by
:<math>
\chi_{q,r}(a)=
\begin{cases}
0 &\text{if } a\text{ is even}\\
(-1)^{\nu_0(r)\nu_0(a)}\omega_q^{\nu_q(r)\nu_q(a)}&\text{if } a \text{ is odd}.
\end{cases}</math>
Then for odd <math>r</math> and <math>s</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> showing that <math>\widehat{(\mathbb{Z}/2^{k}\mathbb{Z})^\times}\cong (\mathbb{Z}/2^{k}\mathbb{Z})^\times.</math>

==== Examples ''m'' = 2, 4, 8, 16 ====

The only character mod 2 is the principal character <math>\chi_{2,1}</math>.

−1 is a primitive root mod 4 (<math>\phi(4)=2</math>)

:<math>
\begin{array}{|||}
    a    & 1 & 3  \\
\hline
\nu_0(a) & 0 & 1  \\
\end{array}
</math>

The  nonzero values of the characters mod 4 are
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
             & 1 & 3    \\
\hline
\chi_{4,1} & 1 & 1        \\
\chi_{4,3} & 1 & -1  \\

\end{array}
</math>

−1 is and 5 generate the units mod 8 (<math>\phi(8)=4</math>)

:<math>
\begin{array}{|||}
    a    & 1 & 3 & 5 & 7  \\
\hline
\nu_0(a) & 0 & 1 & 0 & 1 \\
\nu_8(a) & 0 & 1 & 1 & 0  \\
\end{array}
</math>.

The  nonzero values of the characters mod 8 are
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
             & 1 & 3  &  5  & 7     \\
\hline
\chi_{8,1} & 1 & 1  &  1  & 1      \\
\chi_{8,3} & 1 & 1  & -1  & -1 \\
\chi_{8,5} & 1 & -1 & -1  & 1    \\
\chi_{8,7} & 1 & -1 & 1   & -1     \\
\end{array}
</math>

−1 and 5 generate the units mod 16 (<math>\phi(16)=8</math>)
:<math>
\begin{array}{|||}
    a    & 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 \\
\hline
\nu_0(a) & 0 & 1 & 0 & 1 & 0 & 1  & 0  & 1  \\
\nu_{16}(a) & 0 & 3 & 1 & 2 & 2 & 1  & 3  & 0  \\
\end{array}
</math>.

The  nonzero values of the characters mod 16 are

:<math>
\begin{array}{|||}
                   & 1 & 3  & 5  & 7  & 9  & 11  & 13 & 15 \\
\hline
\chi_{16,1}    & 1 & 1  & 1  & 1  & 1  & 1   & 1  & 1  \\
\chi_{16,3}    & 1 & -i & -i & 1  & -1 & i   & i  & -1   \\
\chi_{16,5}    & 1 & -i & i  & -1 & -1 & i   & -i & 1  \\
\chi_{16,7}    & 1 & 1  & -1 & -1 & 1  & 1   & -1 & -1  \\
\chi_{16,9}    & 1 & -1 & -1 & 1  & 1  & -1  & -1 & 1  \\
\chi_{16,11}   & 1 & i  & i  & 1  & -1 & -i  & -i & -1  \\
\chi_{16,13}   & 1 & i  & -i & -1 & -1 & -i  & i  & 1  \\
\chi_{16,15}   & 1 & -1 & 1  & -1 & 1  & -1  & 1  & -1  \\

\end{array}
</math>.