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{{Short description|Complex-valued arithmetic function}}
In [[analytic number theory]] and related branches of mathematics, a complex-valued [[arithmetic function]]  <math>\chi:\mathbb{Z}\rightarrow\mathbb{C}</math> is a '''Dirichlet character of modulus <math>m</math>''' (where <math>m</math> is a positive integer) if for all integers <math>a</math> and <math>b</math>:<ref>This is the standard definition; e.g. Davenport p.27; Landau p. 109; Ireland and Rosen p. 253</ref>
# <math>\chi(ab) = \chi(a)\chi(b);</math> that is, <math>\chi</math> is [[Completely multiplicative function|completely multiplicative]].
# <math>
\chi(a)
\begin{cases}
=0 &\text{if } \gcd(a,m)>1\\
\ne 0&\text{if }\gcd(a,m)=1.
\end{cases}</math>  (gcd is the [[greatest common divisor]])
# <math>\chi(a + m) = \chi(a)</math>; that is, <math>\chi</math> is periodic with period <math>m</math>.
The simplest possible character, called the '''principal character''', usually denoted <math>\chi_0</math>, (see [[#Notation|Notation]] below) exists for all moduli:<ref>Note the special case of modulus 1: the unique character mod 1 is the constant 1; all other characters are 0 at 0</ref>
:<math>
\chi_0(a)=
\begin{cases}
0 &\text{if } \gcd(a,m)>1\\
1 &\text{if } \gcd(a,m)=1.
\end{cases}</math>
The German mathematician [[Peter Gustav Lejeune Dirichlet]]—for whom the character is named—introduced these functions in his 1837 paper on [[Dirichlet's theorem on arithmetic progressions|primes in arithmetic progressions]].<ref>Davenport p. 1</ref><ref>An English translation is in External Links</ref>

== Notation ==

<math>\phi(n)</math> is [[Euler's totient function]].<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Totient Function |url=https://mathworld.wolfram.com/TotientFunction.html |access-date=2025-02-09 |website=mathworld.wolfram.com |language=en}}</ref>

<math>\zeta_n</math> is a complex primitive [[Root of unity|n-th root of unity]]:
:<math>
\zeta_n^n=1,</math> but <math>\zeta_n\ne 1, \zeta_n^2\ne 1, ... \zeta_n^{n-1}\ne 1.</math>

<math>(\mathbb{Z}/m\mathbb{Z})^\times</math> is the [[Multiplicative group of integers modulo n|group of units mod <math>m</math>]]. It has order <math>\phi(m).</math>

<math>\widehat{(\mathbb{Z}/m\mathbb{Z})^\times}</math> is the group of Dirichlet characters mod <math>m</math>.

<math>p, p_k,</math> etc. are  [[prime number]]s.

<math>(m,n)</math> is a standard<ref>Used in Davenport, Landau, Ireland and Rosen</ref> abbreviation<ref><math>(rs,m)=1</math> is equivalent to <math>\gcd(r,m)=\gcd(s,m)=1</math></ref> for <math>\gcd(m,n)</math>

<math>\chi(a), \chi'(a), \chi_r(a),</math> etc. are Dirichlet characters. (the lowercase [[Chi (letter)|Greek letter chi]] for "character")

There is no standard notation for Dirichlet characters that includes the modulus. In many contexts (such as in the proof of Dirichlet's theorem) the modulus is fixed. In other contexts, such as this article, characters of different moduli appear. Where appropriate this article employs a variation of [https://lmfdb.org/knowledge/show/character.dirichlet.conrey Conrey labeling] (introduced by [[Brian Conrey]] and used by the [https://www.lmfdb.org/ LMFDB]).

In this labeling characters for modulus <math>m</math> are denoted <math>\chi_{m, t}(a)</math> where the index <math>t</math> is described in the section [[#The group of characters|the group of characters]] below. In this labeling, <math>\chi_{m,\_}(a)</math> denotes an unspecified character and
<math>\chi_{m,1}(a)</math> denotes the principal character mod <math>m</math>.

== Relation to group characters ==

The word "[[Character (mathematics)|character]]" is used several ways in mathematics. In this section it refers to a [[homomorphism]] from a group <math>G</math> (written multiplicatively) to the multiplicative group of the field of complex numbers:
:<math>\eta: G\rightarrow \mathbb{C}^\times,\;\;\eta(gh)=\eta(g)\eta(h),\;\;\eta(g^{-1})=\eta(g)^{-1}.</math>

The set of characters is denoted <math>\widehat{G}.</math> If the product of two characters is defined by pointwise multiplication <math>\eta\theta(a)=\eta(a)\theta(a),</math> the identity by the trivial character <math>\eta_0(a)=1</math> and the inverse by complex inversion  <math>\eta^{-1}(a)=\eta(a)^{-1}</math> then <math>\widehat{G}</math> becomes an abelian group.<ref>See [[Multiplicative character]]</ref>

If <math>A</math> is a [[abelian group#Finite abelian groups|finite abelian group]] then<ref name="IR">Ireland and Rosen p. 253-254</ref> there is an [[isomorphism]] <math>A\cong\widehat{A}</math>, and the orthogonality relations:<ref>See [[Character group#Orthogonality of characters]]</ref>

:<math>\sum_{a\in A} \eta(a)=

\begin{cases}
|A|&\text{ if  } \eta=\eta_0\\
0&\text{ if  } \eta\ne\eta_0
\end{cases}
</math> &nbsp; &nbsp; and &nbsp; &nbsp;  <math>\sum_{\eta\in\widehat{A}}\eta(a)=
\begin{cases}
|A|&\text{ if  } a=1\\
0&\text{ if  } a\ne 1.
\end{cases}
</math>

The elements of the finite abelian group <math>(\mathbb{Z}/m\mathbb{Z})^\times</math> are the residue classes <math>[a]=\{x:x\equiv a\pmod m\}</math> where <math>(a,m)=1.</math>

A group character <math>\rho:(\mathbb{Z}/m\mathbb{Z})^\times\rightarrow \mathbb{C}^\times</math> can be extended to a Dirichlet character <math>\chi:\mathbb{Z}\rightarrow \mathbb{C}</math> by defining
:<math>
\chi(a)=
\begin{cases}
0 &\text{if } [a]\not\in(\mathbb{Z}/m\mathbb{Z})^\times&\text{i.e. }(a,m)> 1\\
\rho([a])&\text{if } [a]\in(\mathbb{Z}/m\mathbb{Z})^\times&\text{i.e. }(a,m)= 1,
\end{cases}</math>

and conversely, a Dirichlet character mod <math>m</math> defines a group character on <math>(\mathbb{Z}/m\mathbb{Z})^\times.</math>

Paraphrasing Davenport,<ref>Davenport p. 27</ref> Dirichlet characters can be regarded as a particular case of  Abelian group characters. But this article follows Dirichlet in giving a direct and constructive account of them. This is partly for historical reasons, in that Dirichlet's work preceded by several decades the development of group theory, and partly for a mathematical reason, namely that the group in question has a simple and interesting structure which is obscured if one treats it as one treats the general Abelian group.

== Elementary facts ==

4) Since  <math>\gcd(1,m)=1,</math> property 2) says <math>\chi(1)\ne 0</math> so it can be canceled from both sides of  <math>\chi(1)\chi(1)=\chi(1\times 1) =\chi(1)</math>:

:<math>\chi(1)=1.</math><ref>These properties are derived in all introductions to the subject, e.g. Davenport p. 27, Landau p. 109.</ref>

5) Property 3) is equivalent to

:if <math>a \equiv b \pmod{m}</math> &nbsp; then <math>\chi(a) =\chi(b).</math>

6) Property 1) implies that, for any positive integer <math>n</math>
:<math>\chi(a^n)=\chi(a)^n.</math>

7) [[Euler's theorem]] states that if <math>(a,m)=1</math> then <math>a^{\phi(m)}\equiv 1 \pmod{m}.</math> Therefore,
:<math>\chi(a)^{\phi(m)}=\chi(a^{\phi(m)})=\chi(1)=1.</math>

That is, the nonzero values of <math>\chi(a)</math> are <math>\phi(m)</math>-th [[root of unity|roots of unity]]:

:<math>
\chi(a)=
\begin{cases}
0 &\text{if } \gcd(a,m)>1\\
\zeta_{\phi(m)}^r&\text{if } \gcd(a,m)=1
\end{cases}</math>

for some integer <math>r</math> which depends on <math>\chi, \zeta,</math> and <math>a</math>. This implies there are only a finite number of characters for a given modulus.

8) If <math>\chi</math> and <math>\chi'</math> are two characters for the same modulus so is their product <math>\chi\chi',</math> defined by pointwise multiplication:
:<math>\chi\chi'(a) = \chi(a)\chi'(a)</math> &nbsp; (<math>\chi\chi'</math> obviously satisfies 1-3).<ref>In general, the product of a character mod <math>m</math> and a character mod <math>n</math> is a character mod <math>\operatorname{lcm}(m,n)</math></ref>

The principal character is an identity:
:<math>
\chi\chi_0(a)=\chi(a)\chi_0(a)=
\begin{cases}
0 \times 0 &=\chi(a)&\text{if } \gcd(a,m)>1\\
\chi(a)\times 1&=\chi(a) &\text{if } \gcd(a,m)=1.
\end{cases}</math>

9) Let <math>a^{-1}</math> denote the inverse of <math>a</math>  in <math>(\mathbb{Z}/m\mathbb{Z})^\times</math>.
Then
:<math>\chi(a)\chi(a^{-1})=\chi(aa^{-1})=\chi(1)=1,
</math> so <math>\chi(a^{-1})=\chi(a)^{-1},</math> which extends 6) to all integers.

The [[complex conjugate]] of a root of unity is also its inverse (see [[root of unity#Elementary properties|here]] for details), so for <math>(a,m)=1</math>
:<math>\overline{\chi}(a)=\chi(a)^{-1}=\chi(a^{-1}). </math> &nbsp; (<math>\overline\chi</math> also obviously satisfies 1-3).

Thus for all integers <math>a</math>
:<math>
\chi(a)\overline{\chi}(a)=
\begin{cases}
0 &\text{if } \gcd(a,m)>1\\
1 &\text{if } \gcd(a,m)=1
\end{cases}; </math> &nbsp; in other words  <math>\chi\overline{\chi}=\chi_0</math>.&nbsp;

10) The multiplication and identity defined in 8) and the inversion defined in 9) turn the set of Dirichlet characters for a given modulus into a [[abelian group#Finite abelian groups|finite abelian group]].

== The group of characters ==

There are three different cases because the groups <math>(\mathbb{Z}/m\mathbb{Z})^\times</math>  have different structures depending on whether <math>m</math> is a power of 2, a power of an odd prime, or the product of prime powers.<ref>Except for the use of the modified Conrie labeling, this section follows  Davenport pp. 1-3, 27-30</ref>

=== 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>.

=== 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>.

=== 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>.

=== Summary ===
Let <math>m=p_1^{k_1}p_2^{k_2}\cdots = q_1q_2 \cdots</math>, <math>p_1<p_2< \dots </math> be the factorization of <math>m</math> and assume <math>(rs,m)=1.</math>

There are <math>\phi(m)</math> Dirichlet characters mod <math>m.</math> They are denoted by <math>\chi_{m,r},</math> where <math>\chi_{m,r}=\chi_{m,s}</math> is equivalent to <math>r\equiv s\pmod{m}.</math>
The identity <math>\chi_{m,r}(a)\chi_{m,s}(a)=\chi_{m,rs}(a)\;</math> is an isomorphism <math>\widehat{(\mathbb{Z}/m\mathbb{Z})^\times}\cong(\mathbb{Z}/m\mathbb{Z})^\times.</math><ref>This is true for all finite abelian groups: <math>A\cong\hat{A}</math>; See Ireland & Rosen pp. 253-254</ref>

Each character mod <math>m</math> has a unique factorization as the product of characters mod the prime powers dividing <math>m</math>:
:<math>\chi_{m,r}=\chi_{q_1,r}\chi_{q_2,r}...</math>

If <math>m=m_1m_2, (m_1,m_2)=1</math> the product <math>\chi_{m_1,r}\chi_{m_2,s}</math> is a character <math>\chi_{m,t}</math> where <math>t</math> is given by <math>t\equiv r\pmod{m_1}</math> and <math>t\equiv s\pmod{m_2}.</math>

Also,<ref>because the formulas for <math>\chi</math> mod prime powers are symmetric in <math>r</math> and <math>s</math> and the formula for products preserves this symmetry. See Davenport, p. 29.</ref><ref>This is the same thing as saying that the n-th column and the n-th row in the tables of nonzero values are the same.</ref>
<math> \chi_{m,r}(s)=\chi_{m,s}(r)</math>

== Orthogonality ==

The two orthogonality relations are<ref>See [[#Relation to group characters]] above.</ref>
:<math>\sum_{a\in(\mathbb{Z}/m\mathbb{Z})^\times} \chi(a)=
\begin{cases}
\phi(m)&\text{ if  }\;\chi=\chi_0\\
0&\text{ if  }\;\chi\ne\chi_0
\end{cases}
</math> &nbsp; &nbsp; and &nbsp; &nbsp;  <math>\sum_{\chi\in\widehat{(\mathbb{Z}/m\mathbb{Z})^\times}}\chi(a)=
\begin{cases}
\phi(m)&\text{ if  }\;a\equiv 1\pmod{m}\\
0&\text{ if  }\;a\not\equiv 1\pmod{m}.
\end{cases}
</math>

The relations can  be written in the symmetric form
:<math>\sum_{a\in(\mathbb{Z}/m\mathbb{Z})^\times} \chi_{m,r}(a)=
\begin{cases}
\phi(m)&\text{ if  }\;r\equiv 1\\
0&\text{ if  }\;r\not\equiv 1
\end{cases}
</math> &nbsp; &nbsp; and &nbsp; &nbsp;  <math>\sum_{r\in(\mathbb{Z}/m\mathbb{Z})^\times} \chi_{m,r}(a)=
\begin{cases}
\phi(m)&\text{ if  }\;a\equiv 1\\
0&\text{ if  }\;a\not\equiv 1.
\end{cases}
</math>

The first relation is easy to prove: If <math>\chi=\chi_0</math> there are <math>\phi(m)</math> non-zero summands each equal to 1. If <math>\chi\ne\chi_0</math>there is<ref>by the definition of <math>\chi_0</math></ref>  some <math>a^*,\; (a^*,m)=1,\;\chi(a^*)\ne1.</math> &nbsp;Then
:<math>\chi(a^*)\sum_{a\in(\mathbb{Z}/m\mathbb{Z})^\times} \chi(a)=\sum_{a}\chi(a^*) \chi(a)=\sum_{a} \chi(a^*a)=\sum_{a} \chi(a),
</math><ref name="permute">because multiplying every element in a group by a constant element merely permutes the elements. See [[Group (mathematics)]]</ref> &nbsp; implying
:<math>(\chi(a^*)-1)\sum_{a} \chi(a)=0.</math> &nbsp; Dividing by the first factor gives <math>\sum_{a} \chi(a)=0,</math> QED. The identity <math>\chi_{m,r}(s)=\chi_{m,s}(r)</math> for <math>(rs,m)=1</math> shows that the relations are equivalent to each other. 

The second relation can be proven directly in the same way, but requires a lemma<ref>Davenport p. 30 (paraphrase) To prove [the second relation] one has to use ideas that we have used in the construction [as in this article or Landau pp. 109-114], or appeal to the basis theorem for abelian groups [as in Ireland & Rosen pp. 253-254]</ref>
:Given <math>a \not\equiv 1\pmod{m},\;(a,m)=1,</math> there is a <math> \chi^*,\; \chi^*(a)\ne1.</math>

The second relation has an important corollary: if <math>(a,m)=1,</math> define the function
:<math>f_a(n)=\frac{1}{\phi(m)} \sum_{\chi} \bar{\chi}(a) \chi(n). </math> &nbsp; Then
:<math>f_a(n)
= \frac{1}{\phi(m)} \sum_{\chi} \chi(a^{-1}) \chi(n)
= \frac{1}{\phi(m)} \sum_{\chi} \chi(a^{-1}n)
= \begin{cases} 1, &  n \equiv a \pmod{m} \\ 0, & n\not\equiv a\pmod{m},\end{cases}</math>
That is <math>f_a=\mathbb{1}_{[a]}</math> the [[indicator function]] of the residue class <math>[a]=\{ x:\;x\equiv a \pmod{m}\}</math>. It is basic in the proof of Dirichlet's theorem.<ref>Davenport chs. 1, 4; Landau p. 114</ref><ref>Note that if <math>g:(\mathbb{Z}/m\mathbb{Z})^\times\rightarrow\mathbb{C} </math> is any function
<math>g(n)=\sum_{a\in(\mathbb{Z}/m\mathbb{Z})^\times} g(a)f_a(n)</math>; see [[Fourier transform on finite groups#Fourier transform for finite abelian groups]]</ref>

== 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;

== Applications ==

=== L-functions ===
{{Main|Dirichlet L-function}}

The Dirichlet L-series for a character <math>\chi</math> is

:<math>L(s,\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}.</math>

This series only converges for <math>\mathfrak{R}(s) >1</math>; it can be analytically  continued to a meromorphic function.

Dirichlet introduced the <math>L</math>-function along with the characters in his 1837 paper.

=== Modular forms and functions ===
{{Main|Modular form}}
Dirichlet characters appear several places in the theory of modular forms and functions.  A typical example is<ref>Koblittz, prop. 17b p. 127</ref>

Let <math>\chi\in\widehat{(\mathbb{Z}/M\mathbb{Z})^\times}</math> and let <math>\chi_1\in\widehat{(\mathbb{Z}/N\mathbb{Z})^\times}</math> be primitive.

If
:<math>f(z)=\sum a_n z^n\in M_k(M,\chi)</math><ref><math>f(z)\in M_k(M,\chi)</math> means

1) <math>f(\frac{az+b}{cz+d})(cz+d)^{-k}=f(z) </math> where <math>ad-bc=1</math> and

<math> a\equiv d\equiv 1,\;\;c\equiv 0\pmod{M}.</math>
and 2) <math>f(\frac{az+b}{cz+d})(cz+d)^{-k}=\chi(d)f(z) </math> where <math>ad-bc=1</math> and <math>c\equiv 0\pmod{M}.
</math> See Koblitz Ch. III.</ref>
define
:<math>f_{\chi_1}(z)=\sum\chi_1(n)a_nz^n</math>,<ref>the '''twist''' of <math>f</math> by <math>\chi_1</math></ref>  &nbsp;
Then
:<math>f_{\chi_1}(z)\in M_k(MN^2,\chi\chi_1^2)</math>. If <math>f</math> is a [[cusp form]] so is <math>f_{\chi_1}.</math>
 
See [[Theta function#Theta series of a Dirichlet character|theta series of a Dirichlet character]] for another example.

=== Gauss sum ===
{{Main|Gauss sum}}

The Gauss sum of a Dirichlet character modulo {{mvar|N}} is
:<math>G(\chi)=\sum_{a=1}^N\chi(a)e^\frac{2\pi ia}{N}.</math>
It appears in the [[Dirichlet L-function#Functional equation|functional equation]] of the [[Dirichlet L-function]].

=== Jacobi sum ===
{{Main|Jacobi sum}}

If  <math> \chi</math> and <math>\psi</math> are Dirichlet characters mod a prime <math>p</math> their Jacobi sum is

: <math> J(\chi,\psi) = \sum_{a=2}^{p-1} \chi(a) \psi(1 - a). </math>
Jacobi sums can be factored into products of Gauss sums.

=== Kloosterman sum ===
{{Main|Kloosterman sum}}

If <math>\chi</math> is a Dirichlet character mod <math>q</math> and <math>\zeta = e^\frac{2\pi i}{q}</math> the Kloosterman sum <math> K(a,b,\chi)</math> is defined as<ref>[https://www.lmfdb.org/knowledge/show/character.dirichlet.kloosterman_sum LMFDB definition of Kloosterman sum]</ref>

:<math>K(a,b,\chi)=\sum_{r\in (\mathbb{Z}/q\mathbb{Z})^\times}\chi(r)\zeta^{ar+\frac{b}{r}}.</math>

If <math>b=0</math> it is a Gauss sum.

== Sufficient conditions ==
It is not necessary to establish the defining  properties 1) – 3) to show that a function is a Dirichlet character.

=== From Davenport's book ===

If <math>\Chi:\mathbb{Z}\rightarrow\mathbb{C}</math> such that
:1) &nbsp; <math>\Chi(ab) = \Chi(a)\Chi(b),</math>
:2) &nbsp; <math>\Chi(a + m) = \Chi(a)</math>,
:3) &nbsp; If <math>\gcd(a,m)>1</math> then <math>\Chi(a)=0</math>, but
:4) &nbsp; <math>\Chi(a)</math> is not always 0,
then <math>\Chi(a)</math> is one of the <math>\phi(m)</math> characters mod <math>m</math><ref>Davenport p. 30</ref>

=== Sárközy's Condition ===
A Dirichlet character is a completely multiplicative function <math>f: \mathbb{N} \rightarrow \mathbb{C}</math> that satisfies a [[linear recurrence relation]]: that is, if <math>a_1 f(n+b_1) + \cdots + a_kf(n+b_k) = 0

</math>

for all positive integers <math>n</math>, where <math>a_1,\ldots,a_k</math> are not all zero and <math>b_1,\ldots,b_k</math> are distinct then <math>f</math> is a Dirichlet character.<ref>Sarkozy</ref>

=== Chudakov's Condition ===
A Dirichlet character is a completely multiplicative function <math>f: \mathbb{N} \rightarrow \mathbb{C}</math> satisfying the following three properties: a) <math>f</math> takes only finitely many values; b) <math>f</math> vanishes at only finitely many primes; c) there is an <math>\alpha \in \mathbb{C}</math> for which the remainder

<math>\left|\sum_{n \leq x} f(n)- \alpha  x\right| </math>

is uniformly bounded, as <math>x \rightarrow \infty</math>. This equivalent definition of Dirichlet characters was conjectured by Chudakov<ref>Chudakov</ref> in 1956, and proved in 2017 by Klurman and Mangerel.<ref>Klurman</ref>

== See also ==
{{col div|colwidth=30em}}
* [[Character sum]]
* [[Multiplicative group of integers modulo n|Multiplicative group of integers modulo ''n'']]
* [[Primitive root modulo n|Primitive root modulo ''n'']]
* [[Multiplicative character]]
{{colend}}

== Notes ==
{{reflist}}

== References ==
*{{Cite journal
|last=Chudakov|first=N.G.
|title=Theory of the characters of number semigroups
|journal=J. Indian Math. Soc.|volume=20|pages=11–15}}
* {{cite book
| last=Davenport | first=Harold | author-link=Harold Davenport
| title=Multiplicative number theory | publisher=Markham | series=Lectures in advanced mathematics | volume=1 | location=Chicago | year=1967 | zbl=0159.06303 }}
*{{citation
  | last1 = Ireland  | first1 = Kenneth
  | last2 = Rosen  | first2 = Michael
  | title = A Classical Introduction to Modern Number Theory (Second edition)
  | publisher = [[Springer Science+Business Media|Springer]]
  | location = New York
  | date = 1990
  | isbn = 0-387-97329-X}}
*{{Cite journal|last1=Klurman|first1=Oleksiy|last2=Mangerel|first2=Alexander P.|title=Rigidity Theorems for Multiplicative Functions|journal=Math. Ann.|volume=372|issue=1|pages=651–697|doi=10.1007/s00208-018-1724-6|bibcode=2017arXiv170707817K|year=2017|arxiv=1707.07817|s2cid=119597384}}
* {{Cite book
|first=Neal
|last=Koblitz
|author-link=Neal Koblitz
|title=Introduction to Elliptic Curves and Modular Forms
|edition=2nd revised
|series=Graduate Texts in Mathematics
|volume=97
|publisher=[[Springer-Verlag]]
|year=1993
|isbn=0-387-97966-2
}}
*{{citation
  | last1 = Landau | first1 = Edmund
  | title = Elementary Number Theory
  | publisher = Chelsea
  | location = New York
  | date = 1966}}
*{{Cite journal
|last=Sarkozy|first=Andras
|title=On multiplicative arithmetic functions satisfying a linear recursion
|journal=Studia Sci. Math. Hung.|volume=13|issue=1–2|pages=79–104}}

== External links ==

* [https://arxiv.org/abs/0808.1408#:~:text=Dirichlet's%20proof%20of%20infinitely%20many,and%20the%20distribution%20of%20primes. English translation of Dirichlet's 1837 paper on primes in arithmetic progressions]
* [https://www.lmfdb.org/ LMFDB] Lists 30,397,486 Dirichlet characters of modulus up to 10,000 and their L-functions

{{Peter Gustav Lejeune Dirichlet}}

[[Category:Analytic number theory]]
[[Category:Zeta and L-functions]]