Editing Dirichlet character (section)

== Classification of characters ==

=== Conductor; Primitive and induced characters ===

Any character mod a prime power is also a character mod every larger power. For example, mod 16<ref>This section follows Davenport pp. 35-36,</ref>

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

\chi_{16,9}    & 1 & -1 & -1 & 1  & 1  & -1  & -1 & 1  \\

\chi_{16,15}   & 1 & -1 & 1  & -1 & 1  & -1  & 1  & -1  \\

\end{array}
</math>

<math>\chi_{16,3}</math> has period 16, but <math>\chi_{16,9}</math> has period 8 and <math>\chi_{16,15}</math> has period 4: &nbsp; <math>\chi_{16,9}=\chi_{8,5}</math> and &nbsp;<math>\chi_{16,15}=\chi_{8,7}=\chi_{4,3}.</math>

We say that a character <math>\chi</math> of modulus <math>q</math> has a '''quasiperiod of <math>d</math>''' if <math>\chi(m)=\chi(n)</math> for all <math>m</math>, <math>n</math> coprime to <math>q</math> satisfying <math>m\equiv n</math> mod <math>d</math>.<ref>{{cite web |last1=Platt |first1=Dave |title=Dirichlet characters Def. 11.10. |url=https://people.maths.bris.ac.uk/~madjp/Teaching/lecture_dc.pdf |access-date=April 5, 2024}}</ref> For example, <math>\chi_{2,1}</math>, the only Dirichlet character of modulus <math>2</math>, has a quasiperiod of <math>1</math>, but ''not'' a period of <math>1</math> (it has a period of <math>2</math>, though). The smallest positive integer for which <math>\chi</math> is quasiperiodic is  the '''conductor''' of <math>\chi</math>.<ref>{{cite web |title=Conductor of a Dirichlet character (reviewed) |url=http://www.lmfdb.org/knowledge/show/character.dirichlet.conductor |website=LMFDB |access-date=April 5, 2024}}</ref> So, for instance, <math>\chi_{2,1}</math> has a conductor of <math>1</math>.

The conductor of <math>\chi_{16,3}</math> is 16, the conductor of <math>\chi_{16,9}</math> is 8 and that of  <math>\chi_{16,15}</math> and <math>\chi_{8,7}</math> is 4. If the modulus and conductor are equal the character is '''primitive''', otherwise '''imprimitive'''. An imprimitive character is '''induced''' by the character for the smallest modulus: <math>\chi_{16,9}</math> is induced from <math>\chi_{8,5}</math> and <math>\chi_{16,15}</math> and <math>\chi_{8,7}</math> are induced from <math>\chi_{4,3}</math>.

A related phenomenon can happen with a character mod the product of primes; its ''nonzero values'' may be periodic with a smaller period.

For example, mod 15,
:<math>
\begin{array}{|||}
                 & 1 & 2  &3 & 4  &5&6 & 7   & 8 &9&10   & 11&12  & 13 & 14 &15 \\
\hline

\chi_{15,8}    & 1 &  i &0 & -1 &0&0 & -i  & -i &0&0  & -1  &0& i   & 1 &0 \\
\chi_{15,11}   & 1 & -1 &0 & 1  &0&0 & 1   & -1 &0&0 & -1  &0& 1  & -1  &0\\
\chi_{15,13}   & 1 & -i &0 & -1 &0&0 & -i  & i  &0&0 & 1 &0 & i   & -1  &0\\

\end{array}
</math>.

The nonzero values of <math>\chi_{15,8}</math> have period 15, but those of <math>\chi_{15,11}</math> have period 3 and those of <math>\chi_{15,13}</math> have period 5. This is easier to see by juxtaposing them with characters mod 3 and 5:

:<math>
\begin{array}{|||}
                 & 1  & 2  & 3 & 4  & 5  & 6 & 7   & 8  & 9  & 10 & 11 & 12  & 13 & 14 &15\\
\hline
\chi_{15,11}   & 1  & -1 & 0 & 1  & 0  & 0 & 1   & -1 & 0  & 0  & -1 & 0  & 1  & -1  &0\\
\chi_{3,2}     & 1  & -1 & 0 & 1  & -1 & 0 & 1   & -1 & 0  & 1  & -1 & 0  & 1  & -1  &0\\
\hline
\chi_{15,13}   & 1  & -i & 0 & -1 &  0 & 0 & -i  & i  & 0  & 0  & 1  & 0  & i  & -1  &0\\
\chi_{5,3}     & 1  & -i & i & -1 &  0 & 1 & -i  & i  & -1 & 0  & 1  & -i & i  & -1  &0\\
\end{array}
</math>.

If a character mod <math>m=qr,\;\; (q,r)=1, \;\;q>1,\;\; r>1</math>  is defined as
:<math> \chi_{m,\_}(a)=
\begin{cases}
0&\text{ if  }\gcd(a,m)>1\\
\chi_{q,\_}(a)&\text{ if  }\gcd(a,m)=1
\end{cases}
</math>, &nbsp; or equivalently as <math> \chi_{m,\_}= \chi_{q,\_} \chi_{r,1},</math>
its nonzero values are determined by the character mod <math>q</math> and have period <math>q</math>.

The smallest period of the nonzero values is the '''conductor''' of the character. For example, the conductor of <math>\chi_{15,8}</math> is 15, the conductor of <math>\chi_{15,11}</math> is 3, and that of <math>\chi_{15,13}</math> is 5.

As in the prime-power case, if the conductor equals the modulus the character is '''primitive''', otherwise '''imprimitive'''. If imprimitive it is '''induced''' from the character with the smaller modulus. For example, <math>\chi_{15,11}</math> is induced from <math>\chi_{3,2}</math> and <math>\chi_{15,13}</math> is induced from <math>\chi_{5,3}</math>

The principal character is not primitive.<ref>Davenport classifies it as neither primitive nor imprimitive; the LMFDB induces it from <math>\chi_{1,1}.</math></ref>

The character <math>\chi_{m,r}=\chi_{q_1,r}\chi_{q_2,r}...</math> is primitive if and only if each of the factors is primitive.<ref name="twop">Note that if <math>m</math> is two times an odd number, <math>m=2r</math>, all characters mod <math> m </math> are imprimitive because <math>\chi_{m,\_}=\chi_{r,\_}\chi_{2,1}</math></ref>

Primitive characters often simplify (or make possible) formulas in the theories of [[Dirichlet L-function|L-functions]]<ref>For example the functional equation of <math>L(s,\chi)</math> is only valid for primitive <math>\chi</math>. See  Davenport, p. 85</ref> and [[modular form]]s.

=== Parity ===

<math>\chi(a)</math> is '''even''' if <math>\chi(-1)=1</math>  and is '''odd''' if <math>\chi(-1)=-1.</math>

This distinction appears in the [[Dirichlet L-function#Functional equation|functional equation]] of the [[Dirichlet L-function]].

=== Order ===

The '''order''' of a character is its [[Order (group theory)|order as an element of the group]] <math>\widehat{(\mathbb{Z}/m\mathbb{Z})^\times}</math>, i.e. the smallest positive integer <math>n</math> such that <math>\chi^n= \chi_0.</math> Because of the isomorphism <math>\widehat{(\mathbb{Z}/m\mathbb{Z})^\times}\cong(\mathbb{Z}/m\mathbb{Z})^\times</math> the order of <math>\chi_{m,r}</math>  is the same as the order of <math>r</math> in  <math>(\mathbb{Z}/m\mathbb{Z})^\times. </math> The principal character has order 1; other [[#Real characters|real characters]] have order 2, and imaginary characters have order 3 or greater. By [[Lagrange's theorem (group theory)|Lagrange's theorem]] the order of a character divides the order of <math>\widehat{(\mathbb{Z}/m\mathbb{Z})^\times}</math> which is <math>\phi(m)</math>

=== Real characters ===

<math>\chi(a)</math> is '''real''' or '''quadratic''' if all of its values are real (they must be <math>0,\;\pm1</math>); otherwise it is '''complex''' or '''imaginary.'''

<math>\chi</math> is real if and only if  <math>\chi^2=\chi_0</math>; <math>\chi_{m,k} </math> is real if and only if <math>k^2\equiv1\pmod{m}</math>; in particular, <math>\chi_{m,-1} </math> is real and non-principal.<ref>In fact, for prime modulus <math>p\;\;\chi_{p,-1}</math>  is the [[Legendre symbol]]: <math>\chi_{p,-1}(a)=\left(\frac{a}{p}\right).\;</math> Sketch of proof: <math>\nu_p(-1)=\frac{p-1}{2},\;\;\omega^{\nu_p(-1)}=-1, \;\;\nu_p(a)</math> is even (odd) if a is a quadratic residue (nonresidue)</ref>

Dirichlet's original proof that <math>L(1,\chi)\ne0</math> (which was only valid for prime moduli) took two different forms depending on whether <math>\chi</math> was real or not. His later proof, valid for all moduli, was based on his [[class number formula]].<ref>Davenport, chs. 1, 4.</ref><ref>Ireland and Rosen's proof, valid for all moduli, also has these two cases. pp. 259 ff</ref>

Real characters are [[Kronecker symbol]]s;<ref>Davenport p. 40</ref> for example, the principal character can be written<ref>The notation <math>\chi_{m,1}=\left(\frac{m^2}{\bullet}\right)</math> is a shorter way of writing <math>\chi_{m,1}(a)=\left(\frac{m^2}{a}\right)</math></ref>
<math>\chi_{m,1}=\left(\frac{m^2}{\bullet}\right)</math>.

The real characters in the examples are:

==== Principal ====
If <math>m=p_1^{k_1}p_2^{k_2}...,\;p_1<p_2<\;...</math> the principal character is<ref>The product of primes ensures it is zero if <math>\gcd(m,\bullet) >1</math>; the squares ensure its only nonzero value is 1.</ref> <math>\chi_{m,1}=\left(\frac{p_1^2p_2^2...}{\bullet}\right).</math>

<math>\chi_{16,1}=\chi_{8,1}=\chi_{4,1}=\chi_{2,1}=\left(\frac{4}{\bullet}\right)</math> &nbsp;
<math>\chi_{9,1}=\chi_{3,1}=\left(\frac{9}{\bullet}\right)</math> &nbsp;
<math>\chi_{5,1}=\left(\frac{25}{\bullet}\right)</math> &nbsp;
<math>\chi_{7,1}=\left(\frac{49}{\bullet}\right)</math> &nbsp;
<math>\chi_{15,1}=\left(\frac{225}{\bullet}\right)</math> &nbsp;
<math>\chi_{24,1}=\left(\frac{36}{\bullet}\right)</math> &nbsp;
<math>\chi_{40,1}=\left(\frac{100}{\bullet}\right)</math> &nbsp;

==== Primitive ====

If the modulus is the absolute value of a  [[fundamental discriminant]] there is a real primitive character (there are two if the modulus is a multiple of 8); otherwise if there are any primitive characters<ref name="twop" /> they are imaginary.<ref>Davenport pp. 38-40</ref>

<math>\chi_{3,2}=\left(\frac{-3}{\bullet}\right)</math> &nbsp;
<math>\chi_{4,3}=\left(\frac{-4}{\bullet}\right)</math> &nbsp;
<math>\chi_{5,4}=\left(\frac{5}{\bullet}\right)</math> &nbsp;
<math>\chi_{7,6}=\left(\frac{-7}{\bullet}\right)</math> &nbsp;
<math>\chi_{8,3}=\left(\frac{-8}{\bullet}\right)</math> &nbsp;
<math>\chi_{8,5}=\left(\frac{8}{\bullet}\right)</math> &nbsp;
<math>\chi_{15,14}=\left(\frac{-15}{\bullet}\right)</math> &nbsp;
<math>\chi_{24,5}=\left(\frac{-24}{\bullet}\right)</math> &nbsp;
<math>\chi_{24,11}=\left(\frac{24}{\bullet}\right)</math> &nbsp;
<math>\chi_{40,19}=\left(\frac{-40}{\bullet}\right)</math> &nbsp;
<math>\chi_{40,29}=\left(\frac{40}{\bullet}\right)</math>

==== Imprimitive ====

<math>\chi_{8,7}=\chi_{4,3}=\left(\frac{-4}{\bullet}\right)</math> &nbsp;
<math>\chi_{9,8}=\chi_{3,2}=\left(\frac{-3}{\bullet}\right)</math> &nbsp;
<math>\chi_{15,4}=\chi_{5,4}\chi_{3,1}=\left(\frac{45}{\bullet}\right)</math> &nbsp;
<math>\chi_{15,11}=\chi_{3,2}\chi_{5,1}=\left(\frac{-75}{\bullet}\right)</math> &nbsp;
<math>\chi_{16,7}=\chi_{8,3}=\left(\frac{-8}{\bullet}\right)</math> &nbsp;
<math>\chi_{16,9}=\chi_{8,5}=\left(\frac{8}{\bullet}\right)</math> &nbsp;
<math>\chi_{16,15}=\chi_{4,3}=\left(\frac{-4}{\bullet}\right)</math> &nbsp;

<math>\chi_{24,7}=\chi_{8,7}\chi_{3,1}=\chi_{4,3}\chi_{3,1}=\left(\frac{-36}{\bullet}\right)</math> &nbsp;
<math>\chi_{24,13}=\chi_{8,5}\chi_{3,1}=\left(\frac{72}{\bullet}\right)</math> &nbsp;
<math>\chi_{24,17}=\chi_{3,2}\chi_{8,1}=\left(\frac{-12}{\bullet}\right)</math> &nbsp;
<math>\chi_{24,19}=\chi_{8,3}\chi_{3,1}=\left(\frac{-72}{\bullet}\right)</math> &nbsp;
<math>\chi_{24,23}=\chi_{8,7}\chi_{3,2}=\chi_{4,3}\chi_{3,2}=\left(\frac{12}{\bullet}\right)</math> &nbsp;

<math>\chi_{40,9}=\chi_{5,4}\chi_{8,1}=\left(\frac{20}{\bullet}\right)</math> &nbsp;
<math>\chi_{40,11}=\chi_{8,3}\chi_{5,1}=\left(\frac{-200}{\bullet}\right)</math> &nbsp;
<math>\chi_{40,21}=\chi_{8,5}\chi_{5,1}=\left(\frac{200}{\bullet}\right)</math> &nbsp;
<math>\chi_{40,31}=\chi_{8,7}\chi_{5,1}=\chi_{4,3}\chi_{5,1}=\left(\frac{-100}{\bullet}\right)</math> &nbsp;
<math>\chi_{40,39}=\chi_{8,7}\chi_{5,4}=\chi_{4,3}\chi_{5,4}=\left(\frac{-20}{\bullet}\right)</math> &nbsp;