Editing Dirichlet character (section)

=== Products of prime powers ===
Let <math>m=p_1^{m_1}p_2^{m_2} \cdots p_k^{m_k} = q_1q_2 \cdots q_k</math> where <math> p_1<p_2< \dots < p_k</math> be the factorization of <math>m</math> into prime powers. The group of units mod <math>m</math> is isomorphic to the direct product of the groups mod the <math>q_i</math>:<ref>See [[Multiplicative group of integers modulo n#General composite numbers|group of units]] for details</ref>
:<math>(\mathbb{Z}/m\mathbb{Z})^\times \cong(\mathbb{Z}/q_1\mathbb{Z})^\times \times(\mathbb{Z}/q_2\mathbb{Z})^\times \times \dots \times(\mathbb{Z}/q_k\mathbb{Z})^\times .</math>
This means that 1) there is a one-to-one correspondence between <math>a\in (\mathbb{Z}/m\mathbb{Z})^\times</math> and <math>k</math>-tuples <math>(a_1, a_2,\dots, a_k)</math> where <math>a_i\in(\mathbb{Z}/q_i\mathbb{Z})^\times </math> 
and 2) multiplication mod <math>m</math> corresponds to coordinate-wise multiplication of <math>k</math>-tuples:  
:<math>ab\equiv c\pmod{m}</math> corresponds to 
:<math>(a_1,a_2,\dots,a_k)\times(b_1,b_2,\dots,b_k)=(c_1,c_2,\dots,c_k)</math> where <math>c_i\equiv a_ib_i\pmod{q_i}.</math>

The [[Chinese remainder theorem]] (CRT) implies that the <math>a_i</math> are simply <math>a_i\equiv a\pmod{q_i}.</math>

There are subgroups <math> G_i<(\mathbb{Z}/m\mathbb{Z})^\times</math> such that <ref>To construct the <math>G_i, </math> for each <math> a\in (\mathbb{Z}/q_i\mathbb{Z})^\times </math> use the CRT to find <math>a_i\in (\mathbb{Z}/m\mathbb{Z})^\times</math> where
:<math>a_i\equiv 
\begin{cases}
a &\mod q_i\\
1&\mod q_j, j\ne i.
\end{cases}
</math>
</ref>
:<math>G_i\cong(\mathbb{Z}/q_i\mathbb{Z})^\times </math> and
:<math>G_i\equiv
\begin{cases}
(\mathbb{Z}/q_i\mathbb{Z})^\times &\mod q_i\\
\{1\}&\mod q_j, j\ne i.
\end{cases}
</math>

Then <math>(\mathbb{Z}/m\mathbb{Z})^\times \cong G_1\times G_2\times...\times G_k</math>
and every <math>a\in (\mathbb{Z}/m\mathbb{Z})^\times</math> corresponds to a <math>k</math>-tuple  <math>(a_1, a_2,...a_k)</math> where <math>a_i\in G_i </math> and <math>a_i\equiv a\pmod{q_i}. </math>  
Every <math>a\in (\mathbb{Z}/m\mathbb{Z})^\times</math> can be uniquely factored as <math>a =a_1a_2...a_k.</math>
<ref>Assume <math>a</math> corresponds to <math>(a_1,a_2, ...)</math>. By construction <math>a_1</math> corresponds to <math>(a_1,1,1,...)</math>, <math>a_2</math> to <math>(1,a_2,1,...)</math> etc. whose coordinate-wise product is <math>(a_1,a_2, ...).</math></ref>
<ref>For example let <math>m=40, q_1=8, q_2=5.</math> Then <math>G_1=\{1,11,21,31\}</math> and <math>G_2=\{1,9,17,33\}.</math> The factorization of the elements of <math>(\mathbb{Z}/40\mathbb{Z})^\times</math> is 

:<math>
\begin{array}{|c|c|c|c|c|c|c|}
     & 1  &  9   &  17  & 33  \\
\hline
  1  & 1  &  9   &  17  & 33  \\
  11 & 11 &  19  &  27  & 3   \\
  21 & 21 &  29  &  37  & 13  \\
  31 & 31 &  39  &   7  & 23  \\
\end{array}
</math>
</ref>

If <math>\chi_{m,\_}</math> is a character mod <math>m,</math> on the subgroup <math>G_i</math> it must be identical to some <math>\chi_{q_i,\_}</math> mod <math>q_i</math> Then
:<math>\chi_{m,\_}(a)=\chi_{m,\_}(a_1a_2...)=\chi_{m,\_}(a_1)\chi_{m,\_}(a_2)...=\chi_{q_1,\_}(a_1)\chi_{q_2,\_}(a_2)...,</math>
showing that every character mod <math> m</math> is the product of characters mod the <math>q_i</math>. 

For <math>(t,m)=1</math> define<ref>See [https://lmfdb.org/knowledge/show/character.dirichlet.conrey Conrey labeling].</ref>
:<math> \chi_{m,t}=\chi_{q_1,t}\chi_{q_2,t}...</math>
Then for <math>(rs,m)=1</math> and all <math>a</math> and <math>b</math><ref>Because these formulas are true for each factor.</ref>
:<math>\chi_{m,r}(a)\chi_{m,r}(b)=\chi_{m,r}(ab),</math> showing that <math>\chi_{m,r}</math> is a character and
:<math>\chi_{m,r}(a)\chi_{m,s}(a)=\chi_{m,rs}(a),</math> showing an isomorphism <math>\widehat{(\mathbb{Z}/m\mathbb{Z})^\times}\cong(\mathbb{Z}/m\mathbb{Z})^\times.</math>


==== Examples ''m'' = 15, 24, 40 ====

<math>(\mathbb{Z}/15\mathbb{Z})^\times\cong(\mathbb{Z}/3\mathbb{Z})^\times\times(\mathbb{Z}/5\mathbb{Z})^\times.</math>

The factorization of the characters mod 15 is
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
             & \chi_{5,1} &  \chi_{5,2} &  \chi_{5,3} &  \chi_{5,4}   \\
\hline
\chi_{3,1} & \chi_{15,1} & \chi_{15,7} & \chi_{15,13} & \chi_{15,4}  \\
\chi_{3,2} & \chi_{15,11} & \chi_{15,2} & \chi_{15,8} & \chi_{15,14} \\
\end{array} </math>
The nonzero values of the characters mod 15 are

:<math>
\begin{array}{|||}
                 & 1 & 2   & 4   & 7   & 8    & 11  & 13 & 14 \\
\hline
\chi_{15,1}    & 1 & 1   & 1   & 1   & 1    & 1   & 1   & 1  \\
\chi_{15,2}    & 1 & -i  & -1  & i   & i   & -1  & -i  & 1   \\
\chi_{15,4}    & 1 & -1  & 1   & -1  & -1   & 1   & -1  & 1  \\
\chi_{15,7}    & 1 &  i  & -1  & i   & -i   & 1   & -i  & -1  \\
\chi_{15,8}    & 1 &  i  & -1  & -i  & -i   & -1  & i   & 1  \\
\chi_{15,11}   & 1 & -1  & 1   & 1   &  -1  & -1  & 1  & -1  \\
\chi_{15,13}   & 1 & -i   & -1  & -i  & i   & 1  & i   & -1  \\
\chi_{15,14}   & 1 & 1   & 1   & -1   & 1    & -1  & -1   & -1  \\

\end{array}
</math>.

<math>(\mathbb{Z}/24\mathbb{Z})^\times\cong(\mathbb{Z}/8\mathbb{Z})^\times\times(\mathbb{Z}/3\mathbb{Z})^\times.</math>
The factorization of the characters mod 24 is
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
             & \chi_{8,1} &  \chi_{8,3} &  \chi_{8,5} &  \chi_{8,7}   \\
\hline
\chi_{3,1} & \chi_{24,1} & \chi_{24,19} & \chi_{24,13} & \chi_{24,7}  \\
\chi_{3,2} & \chi_{24,17} & \chi_{24,11} & \chi_{24,5} & \chi_{24,23} \\
\end{array} </math>

The nonzero values of the characters mod 24 are

:<math>
\begin{array}{|||}
                 & 1 & 5   & 7   & 11   & 13    & 17  & 19 & 23 \\
\hline
\chi_{24,1}    & 1 & 1   & 1   & 1   & 1    & 1   & 1   & 1  \\
\chi_{24,5}    & 1 & 1  & 1  & 1   & -1   & -1  & -1  & -1   \\
\chi_{24,7}    & 1 & 1  & -1   & -1  & 1   & 1   & -1  & -1  \\
\chi_{24,11}    & 1 &  1  & -1  & -1   & -1   & -1   & 1  & 1  \\
\chi_{24,13}    & 1 &  -1  & 1  & -1  & -1   & 1  & -1   & 1  \\
\chi_{24,17}   & 1 & -1  & 1   & -1   &  1  & -1  & 1  & -1  \\
\chi_{24,19}   & 1 & -1   & -1  & 1  & -1   & 1  & 1   & -1  \\
\chi_{24,23}   & 1 & -1   & -1   & 1   & 1    & -1  & -1   & 1  \\

\end{array}
</math>.

<math>(\mathbb{Z}/40\mathbb{Z})^\times\cong(\mathbb{Z}/8\mathbb{Z})^\times\times(\mathbb{Z}/5\mathbb{Z})^\times.</math>
The factorization of the characters mod 40 is
:<math>
\begin{array}{|c|c|c|c|c|c|c|}
             & \chi_{8,1}   &  \chi_{8,3}  &  \chi_{8,5}  &  \chi_{8,7}   \\
\hline
\chi_{5,1} & \chi_{40,1}  & \chi_{40,11} & \chi_{40,21} & \chi_{40,31}  \\
\chi_{5,2} & \chi_{40,17} & \chi_{40,27} & \chi_{40,37} & \chi_{40,7} \\
\chi_{5,3} & \chi_{40,33} & \chi_{40,3}  & \chi_{40,13} & \chi_{40,23}  \\
\chi_{5,4} & \chi_{40,9}  & \chi_{40,19} & \chi_{40,29} & \chi_{40,39} \\
\end{array} </math>

The nonzero values of the characters mod 40 are

:<math>
\begin{array}{|||}
                 & 1 & 3  & 7   & 9   & 11   & 13  & 17 & 19 & 21 & 23  & 27  & 29 & 31 & 33  & 37 & 39 \\
\hline
\chi_{40,1}    & 1 & 1  & 1   & 1   & 1    & 1   & 1  & 1  & 1  & 1   & 1   & 1  & 1  & 1   & 1  & 1  \\
\chi_{40,3}    & 1 & i  & i   & -1  & 1    & -i  & -i & -1 & -1 & -i  & -i  & 1  & -1 & i   & i  & 1  \\
\chi_{40,7}    & 1 & i  & -i  & -1  & -1   & -i  & i  & 1  & 1  & i   & -i  & -1 & -1 & -i  & i  & 1  \\
\chi_{40,9}    & 1 & -1 & -1  & 1   & 1    & -1  & -1 & 1  & 1  & -1  & -1  &  1 & 1  & -1  & -1 & 1  \\
\chi_{40,11}   & 1 & 1  & -1  & 1   & 1    & -1  & 1  & 1  & -1 & -1  & 1   & -1 & -1 & 1   & -1 & -1 \\
\chi_{40,13}   & 1 & -i & -i  & -1  & -1   & -i  & -i & 1  & -1 & i   & i   & 1  &  1 & i   & i  & -1 \\
\chi_{40,17}   & 1 & -i & i   & -1  & 1    & -i  & i  & -1 & 1  & -i  & i   & -1 &  1 & -i  & i  & -1 \\
\chi_{40,19}   & 1 & -1 & 1   & 1   & 1    & 1   & -1 & 1  & -1 & 1   & -1  & -1 & -1 & -1  & 1 & -1  \\
\chi_{40,21}   & 1 & -1 & 1   & 1   & -1   & -1  & 1  & -1 & -1 & 1   & -1  & -1 &  1 & 1   & -1 & 1  \\
\chi_{40,23}   & 1 & -i & i   & -1  & -1   & i   & -i & 1  & 1  & -i  & i   & -1 & -1 & i   & -i & 1  \\
\chi_{40,27}   & 1 & -i & -i  & -1  & 1    & i   & i  & -1 & -1 & i   & i   & 1  & -1 & -i  & -i & 1  \\
\chi_{40,29}   & 1 & 1  & -1  & 1   & -1   & 1   & -1 & -1 & -1 & -1  & 1   & -1 &  1 & -1  & 1  & 1  \\
\chi_{40,31}   & 1 & -1 & -1  & 1   & -1   & 1   & 1  & -1 & 1  & -1  & -1  & 1  & -1 & 1   & 1  & -1 \\
\chi_{40,33}   & 1 & i  & -i  & -1  & 1    & i   & -i & -1 &  1 & i   & -i  & -1 & 1  & i   & -i & -1 \\
\chi_{40,37}   & 1 & i  & i   & -1  & -1   & i   & i  & 1  & -1 & -i  & -i  &  1 &  1 & -i  & -i & -1 \\
\chi_{40,39}   & 1 & 1  & 1   & 1   & -1   & -1  & -1 & -1 & 1  & 1   & 1   & 1  & -1 & -1  & -1 & -1 \\
\end{array}
</math>.