Editing Dirichlet character (section)

=== 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.