Dirichlet character

$\backslash phi(n)$ is Euler's totient function.{{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}}

$\backslash zeta\_n$ is a complex primitive n-th root of unity:

:$$

\zeta_n^n=1, but $\backslash zeta\_n\backslash ne\; 1,\; \backslash zeta\_n^2\backslash ne\; 1,\; ...\; \backslash zeta\_n^\{n-1\}\backslash ne\; 1.$

$(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times$ is the group of units mod $m$. It has order $\backslash phi(m).$

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

$p,\; p\_k,$ etc. are prime numbers.

$(m,n)$ is a standardUsed in Davenport, Landau, Ireland and Rosen abbreviation$(rs,m)=1$ is equivalent to $\backslash gcd(r,m)=\backslash gcd(s,m)=1$ for $\backslash gcd(m,n)$

$\backslash chi(a),\; \backslash chi\text{'}(a),\; \backslash chi\_r(a),$ etc. are Dirichlet characters. (the lowercase 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 $m$ are denoted $\backslash chi\_\{m,\; t\}(a)$ where the index $t$ is described in the section the group of characters below. In this labeling, $\backslash chi\_\{m,\backslash \_\}(a)$ denotes an unspecified character and

$\backslash chi\_\{m,1\}(a)$ denotes the principal character mod $m$.

The word "character" is used several ways in mathematics. In this section it refers to a homomorphism from a group $G$ (written multiplicatively) to the multiplicative group of the field of complex numbers:

:$\backslash eta:\; G\backslash rightarrow\; \backslash mathbb\{C\}^\backslash times,\backslash ;\backslash ;\backslash eta(gh)=\backslash eta(g)\backslash eta(h),\backslash ;\backslash ;\backslash eta(g^\{-1\})=\backslash eta(g)^\{-1\}.$

The set of characters is denoted $\backslash widehat\{G\}.$ If the product of two characters is defined by pointwise multiplication $\backslash eta\backslash theta(a)=\backslash eta(a)\backslash theta(a),$ the identity by the trivial character $\backslash eta\_0(a)=1$ and the inverse by complex inversion $\backslash eta^\{-1\}(a)=\backslash eta(a)^\{-1\}$ then $\backslash widehat\{G\}$ becomes an abelian group.See Multiplicative character

If $A$ is a finite abelian group thenIreland and Rosen p. 253-254 there is an isomorphism $A\backslash cong\backslash widehat\{A\}$, and the orthogonality relations:See Character group#Orthogonality of characters

:$\backslash sum\_\{a\backslash in\; A\}\; \backslash eta(a)=$

\begin{cases}

|A|&\text{ if } \eta=\eta_0\\

0&\text{ if } \eta\ne\eta_0

\end{cases}

    and     $\backslash sum\_\{\backslash eta\backslash in\backslash widehat\{A\}\}\backslash eta(a)=$

\begin{cases}

|A|&\text{ if } a=1\\

0&\text{ if } a\ne 1.

\end{cases}

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

A group character $\backslash rho:(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times\backslash rightarrow\; \backslash mathbb\{C\}^\backslash times$ can be extended to a Dirichlet character $\backslash chi:\backslash mathbb\{Z\}\backslash rightarrow\; \backslash mathbb\{C\}$ by defining

:$$

\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}

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

Paraphrasing Davenport,Davenport p. 27 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.

4) Since $\backslash gcd(1,m)=1,$ property 2) says $\backslash chi(1)\backslash ne\; 0$ so it can be canceled from both sides of $\backslash chi(1)\backslash chi(1)=\backslash chi(1\backslash times\; 1)\; =\backslash chi(1)$:

:$\backslash chi(1)=1.$These properties are derived in all introductions to the subject, e.g. Davenport p. 27, Landau p. 109.

5) Property 3) is equivalent to

:if $a\; \backslash equiv\; b\; \backslash pmod\{m\}$   then $\backslash chi(a)\; =\backslash chi(b).$

6) Property 1) implies that, for any positive integer $n$

:$\backslash chi(a^n)=\backslash chi(a)^n.$

7) Euler's theorem states that if $(a,m)=1$ then $a^\{\backslash phi(m)\}\backslash equiv\; 1\; \backslash pmod\{m\}.$ Therefore,

:$\backslash chi(a)^\{\backslash phi(m)\}=\backslash chi(a^\{\backslash phi(m)\})=\backslash chi(1)=1.$

That is, the nonzero values of $\backslash chi(a)$ are $\backslash phi(m)$-th roots of unity:

:$$

\chi(a)=

\begin{cases}

0 &\text{if } \gcd(a,m)>1\\

\zeta_{\phi(m)}^r&\text{if } \gcd(a,m)=1

\end{cases}

for some integer $r$ which depends on $\backslash chi,\; \backslash zeta,$ and $a$. This implies there are only a finite number of characters for a given modulus.

8) If $\backslash chi$ and $\backslash chi\text{'}$ are two characters for the same modulus so is their product $\backslash chi\backslash chi\text{'},$ defined by pointwise multiplication:

:$\backslash chi\backslash chi\text{'}(a)\; =\; \backslash chi(a)\backslash chi\text{'}(a)$   ($\backslash chi\backslash chi\text{'}$ obviously satisfies 1-3).In general, the product of a character mod $m$ and a character mod $n$ is a character mod $\backslash operatorname\{lcm\}(m,n)$

The principal character is an identity:

:$$

\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}

9) Let $a^\{-1\}$ denote the inverse of $a$ in $(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times$.

Then

:$\backslash chi(a)\backslash chi(a^\{-1\})=\backslash chi(aa^\{-1\})=\backslash chi(1)=1,$

so $\backslash chi(a^\{-1\})=\backslash chi(a)^\{-1\},$ which extends 6) to all integers.

The complex conjugate of a root of unity is also its inverse (see here for details), so for $(a,m)=1$

:$\backslash overline\{\backslash chi\}(a)=\backslash chi(a)^\{-1\}=\backslash chi(a^\{-1\}).$   ($\backslash overline\backslash chi$ also obviously satisfies 1-3).

Thus for all integers $a$

:$$

\chi(a)\overline{\chi}(a)=

\begin{cases}

0 &\text{if } \gcd(a,m)>1\\

1 &\text{if } \gcd(a,m)=1

\end{cases};   in other words $\backslash chi\backslash overline\{\backslash chi\}=\backslash chi\_0$. 

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 finite abelian group.

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

If $q=p^k$ is an odd number $(\backslash mathbb\{Z\}/q\backslash mathbb\{Z\})^\backslash times$ is cyclic of order $\backslash phi(q)$; a generator is called a primitive root mod $q$.There is a primitive root mod $p$ which is a primitive root mod $p^2$ and all higher powers of $p$. See, e.g., Landau p. 106

Let $g\_q$ be a primitive root and for $(a,q)=1$ define the function $\backslash nu\_q(a)$ (the index of $a$) by

:$a\backslash equiv\; g\_q^\{\backslash nu\_q(a)\}\backslash pmod\; \{q\},$

:

For $(ab,q)=1,\backslash ;\backslash ;a\; \backslash equiv\; b\backslash pmod\{q\}$ if and only if $\backslash nu\_q(a)=\backslash nu\_q(b).$ Since

:$\backslash chi(a)=\backslash chi(g\_q^\{\backslash nu\_q(a)\})=\backslash chi(g\_q)^\{\backslash nu\_q(a)\},$   $\backslash chi$ is determined by its value at $g\_q.$

Let $\backslash omega\_q=\; \backslash zeta\_\{\backslash phi(q)\}$ be a primitive $\backslash phi(q)$-th root of unity. From property 7) above the possible values of $\backslash chi(g\_q)$ are

$\backslash omega\_q,\; \backslash omega\_q^2,\; ...\; \backslash omega\_q^\{\backslash phi(q)\}=1.$ These distinct values give rise to $\backslash phi(q)$ Dirichlet characters mod $q.$ For $(r,q)=1$ define $\backslash chi\_\{q,r\}(a)$ as

:$$

\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}

Then for $(rs,q)=1$ and all $a$ and $b$

:$\backslash chi\_\{q,r\}(a)\backslash chi\_\{q,r\}(b)=\backslash chi\_\{q,r\}(ab),$ showing that $\backslash chi\_\{q,r\}$ is a character and

:$\backslash chi\_\{q,r\}(a)\backslash chi\_\{q,s\}(a)=\backslash chi\_\{q,rs\}(a),$ which gives an explicit isomorphism $\backslash widehat\{(\backslash mathbb\{Z\}/p^k\backslash mathbb\{Z\})^\backslash times\}\backslash cong(\backslash mathbb\{Z\}/p^k\backslash mathbb\{Z\})^\backslash times.$

2 is a primitive root mod 3.   ($\backslash phi(3)=2$)

:$2^1\backslash equiv\; 2,\backslash ;2^2\backslash equiv2^0\backslash equiv\; 1\backslash pmod\{3\},$

so the values of $\backslash nu\_3$ are

:$$

\begin{array}

a & 1 & 2 \\

\hline

\nu_3(a) & 0 & 1\\

\end{array}

.

The nonzero values of the characters mod 3 are

:$$

\begin{array}

& 1 & 2 \\

\hline

\chi_{3,1} & 1 & 1 \\

\chi_{3,2} & 1 & -1 \\

\end{array}

2 is a primitive root mod 5.   ($\backslash phi(5)=4$)

:$2^1\backslash equiv\; 2,\backslash ;2^2\backslash equiv\; 4,\backslash ;2^3\backslash equiv\; 3,\backslash ;2^4\backslash equiv2^0\backslash equiv\; 1\backslash pmod\{5\},$

so the values of $\backslash nu\_5$ are

:$$

\begin{array}

a & 1 & 2 & 3 & 4 \\

\hline

\nu_5(a) & 0 & 1 & 3 & 2 \\

\end{array}

.

The nonzero values of the characters mod 5 are

:$$

\begin{array}

& 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}

3 is a primitive root mod 7.   ($\backslash phi(7)=6$)

:$3^1\backslash equiv\; 3,\backslash ;3^2\backslash equiv\; 2,\backslash ;3^3\backslash equiv\; 6,\backslash ;3^4\backslash equiv\; 4,\backslash ;3^5\backslash equiv\; 5,\backslash ;3^6\backslash equiv3^0\backslash equiv\; 1\backslash pmod\{7\},$

so the values of $\backslash nu\_7$ are

:$$

\begin{array}

a & 1 & 2 & 3 & 4 & 5 & 6 \\

\hline

\nu_7(a) & 0 & 2 & 1 & 4 & 5 & 3 \\

\end{array}

.

The nonzero values of the characters mod 7 are ($\backslash omega=\backslash zeta\_6,\; \backslash ;\backslash ;\backslash omega^3=-1$)

:$$

\begin{array}

& 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}

.

2 is a primitive root mod 9.   ($\backslash phi(9)=6$)

:$2^1\backslash equiv\; 2,\backslash ;2^2\backslash equiv\; 4,\backslash ;2^3\backslash equiv\; 8,\backslash ;2^4\backslash equiv\; 7,\backslash ;2^5\backslash equiv\; 5,\backslash ;2^6\backslash equiv2^0\backslash equiv\; 1\backslash pmod\{9\},$

so the values of $\backslash nu\_9$ are

:$$

\begin{array}

a & 1 & 2 &4 & 5&7&8 \\

\hline

\nu_9(a) & 0 & 1 & 2 & 5&4&3 \\

\end{array}

.

The nonzero values of the characters mod 9 are ($\backslash omega=\backslash zeta\_6,\; \backslash ;\backslash ;\backslash omega^3=-1$)

:$$

\begin{array}

& 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}

.

$(\backslash mathbb\{Z\}/2\backslash mathbb\{Z\})^\backslash times$ is the trivial group with one element. $(\backslash mathbb\{Z\}/4\backslash mathbb\{Z\})^\backslash times$ 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 $\backslash equiv\; 1\backslash pmod\{4\}$ and their negatives are the units $\backslash equiv\; 3\backslash pmod\{4\}.$Landau pp. 107-108

For example

:$5^1\backslash equiv\; 5,\backslash ;5^2\backslash equiv5^0\backslash equiv\; 1\backslash pmod\{8\}$

:$5^1\backslash equiv\; 5,\backslash ;5^2\backslash equiv\; 9,\backslash ;5^3\backslash equiv\; 13,\backslash ;5^4\backslash equiv5^0\backslash equiv\; 1\backslash pmod\{16\}$

:$5^1\backslash equiv\; 5,\backslash ;5^2\backslash equiv\; 25,\backslash ;5^3\backslash equiv\; 29,\backslash ;5^4\backslash equiv\; 17,\backslash ;5^5\backslash equiv\; 21,\backslash ;5^6\backslash equiv\; 9,\backslash ;5^7\backslash equiv\; 13,\backslash ;5^8\backslash equiv5^0\backslash equiv\; 1\backslash pmod\{32\}.$

Let $q=2^k,\; \backslash ;\backslash ;k\backslash ge3$; then $(\backslash mathbb\{Z\}/q\backslash mathbb\{Z\})^\backslash times$ is the direct product of a cyclic group of order 2 (generated by −1) and a cyclic group of order $\backslash frac\{\backslash phi(q)\}\{2\}$ (generated by 5).

For odd numbers $a$ define the functions $\backslash nu\_0$ and $\backslash nu\_q$ by

:$a\backslash equiv(-1)^\{\backslash nu\_0(a)\}5^\{\backslash nu\_q(a)\}\backslash pmod\{q\},$

:

For odd $a$ and $b,\; \backslash ;\backslash ;a\backslash equiv\; b\backslash pmod\{q\}$ if and only if $\backslash nu\_0(a)=\backslash nu\_0(b)$ and $\backslash nu\_q(a)=\backslash nu\_q(b).$

For odd $a$ the value of $\backslash chi(a)$ is determined by the values of $\backslash chi(-1)$ and $\backslash chi(5).$

Let $\backslash omega\_q\; =\; \backslash zeta\_\{\backslash frac\{\backslash phi(q)\}\{2\}\}$ be a primitive $\backslash frac\{\backslash phi(q)\}\{2\}$-th root of unity. The possible values of $\backslash chi((-1)^\{\backslash nu\_0(a)\}5^\{\backslash nu\_q(a)\})$ are

$\backslash pm\backslash omega\_q,\; \backslash pm\backslash omega\_q^2,\; ...\; \backslash pm\backslash omega\_q^\{\backslash frac\{\backslash phi(q)\}\{2\}\}=\backslash pm1.$ These distinct values give rise to $\backslash phi(q)$ Dirichlet characters mod $q.$ For odd $r$ define $\backslash chi\_\{q,r\}(a)$ by

:$$

\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}

Then for odd $r$ and $s$ and all $a$ and $b$

:$\backslash chi\_\{q,r\}(a)\backslash chi\_\{q,r\}(b)=\backslash chi\_\{q,r\}(ab)$ showing that $\backslash chi\_\{q,r\}$ is a character and

:$\backslash chi\_\{q,r\}(a)\backslash chi\_\{q,s\}(a)=\backslash chi\_\{q,rs\}(a)$ showing that $\backslash widehat\{(\backslash mathbb\{Z\}/2^\{k\}\backslash mathbb\{Z\})^\backslash times\}\backslash cong\; (\backslash mathbb\{Z\}/2^\{k\}\backslash mathbb\{Z\})^\backslash times.$

The only character mod 2 is the principal character $\backslash chi\_\{2,1\}$.

−1 is a primitive root mod 4 ($\backslash phi(4)=2$)

:$$

\begin{array}

a & 1 & 3 \\

\hline

\nu_0(a) & 0 & 1 \\

\end{array}

The nonzero values of the characters mod 4 are

:$$

\begin{array}

& 1 & 3 \\

\hline

\chi_{4,1} & 1 & 1 \\

\chi_{4,3} & 1 & -1 \\

\end{array}

−1 is and 5 generate the units mod 8 ($\backslash phi(8)=4$)

:$$

\begin{array}

a & 1 & 3 & 5 & 7 \\

\hline

\nu_0(a) & 0 & 1 & 0 & 1 \\

\nu_8(a) & 0 & 1 & 1 & 0 \\

\end{array}

.

The nonzero values of the characters mod 8 are

:$$

\begin{array}

& 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}

−1 and 5 generate the units mod 16 ($\backslash phi(16)=8$)

:$$

\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}

.

The nonzero values of the characters mod 16 are

:$$

\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}

.

Let $m=p\_1^\{m\_1\}p\_2^\{m\_2\}\; \backslash cdots\; p\_k^\{m\_k\}\; =\; q\_1q\_2\; \backslash cdots\; q\_k$ where

:$(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times\; \backslash cong(\backslash mathbb\{Z\}/q\_1\backslash mathbb\{Z\})^\backslash times\; \backslash times(\backslash mathbb\{Z\}/q\_2\backslash mathbb\{Z\})^\backslash times\; \backslash times\; \backslash dots\; \backslash times(\backslash mathbb\{Z\}/q\_k\backslash mathbb\{Z\})^\backslash times\; .$

This means that 1) there is a one-to-one correspondence between $a\backslash in\; (\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times$ and $k$-tuples $(a\_1,\; a\_2,\backslash dots,\; a\_k)$ where $a\_i\backslash in(\backslash mathbb\{Z\}/q\_i\backslash mathbb\{Z\})^\backslash times$

and 2) multiplication mod $m$ corresponds to coordinate-wise multiplication of $k$-tuples:

:$ab\backslash equiv\; c\backslash pmod\{m\}$ corresponds to

:$(a\_1,a\_2,\backslash dots,a\_k)\backslash times(b\_1,b\_2,\backslash dots,b\_k)=(c\_1,c\_2,\backslash dots,c\_k)$ where $c\_i\backslash equiv\; a\_ib\_i\backslash pmod\{q\_i\}.$

The Chinese remainder theorem (CRT) implies that the $a\_i$ are simply $a\_i\backslash equiv\; a\backslash pmod\{q\_i\}.$

There are subgroups such that To construct the $G\_i,$ for each $a\backslash in\; (\backslash mathbb\{Z\}/q\_i\backslash mathbb\{Z\})^\backslash times$ use the CRT to find $a\_i\backslash in\; (\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times$ where

:$a\_i\backslash equiv$

\begin{cases}

a &\mod q_i\\

1&\mod q_j, j\ne i.

\end{cases}

:$G\_i\backslash cong(\backslash mathbb\{Z\}/q\_i\backslash mathbb\{Z\})^\backslash times$ and

:$G\_i\backslash equiv$

\begin{cases}

(\mathbb{Z}/q_i\mathbb{Z})^\times &\mod q_i\\

\{1\}&\mod q_j, j\ne i.

\end{cases}

Then $(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times\; \backslash cong\; G\_1\backslash times\; G\_2\backslash times...\backslash times\; G\_k$

and every $a\backslash in\; (\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times$ corresponds to a $k$-tuple $(a\_1,\; a\_2,...a\_k)$ where $a\_i\backslash in\; G\_i$ and $a\_i\backslash equiv\; a\backslash pmod\{q\_i\}.$

Every $a\backslash in\; (\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times$ can be uniquely factored as $a\; =a\_1a\_2...a\_k.$

Assume $a$ corresponds to $(a\_1,a\_2,\; ...)$. By construction $a\_1$ corresponds to $(a\_1,1,1,...)$, $a\_2$ to $(1,a\_2,1,...)$ etc. whose coordinate-wise product is $(a\_1,a\_2,\; ...).$

For example let $m=40,\; q\_1=8,\; q\_2=5.$ Then $G\_1=\backslash \{1,11,21,31\backslash \}$ and $G\_2=\backslash \{1,9,17,33\backslash \}.$ The factorization of the elements of $(\backslash mathbb\{Z\}/40\backslash mathbb\{Z\})^\backslash times$ is

:$$

\begin{array}

& 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}

If $\backslash chi\_\{m,\backslash \_\}$ is a character mod $m,$ on the subgroup $G\_i$ it must be identical to some $\backslash chi\_\{q\_i,\backslash \_\}$ mod $q\_i$ Then

:$\backslash chi\_\{m,\backslash \_\}(a)=\backslash chi\_\{m,\backslash \_\}(a\_1a\_2...)=\backslash chi\_\{m,\backslash \_\}(a\_1)\backslash chi\_\{m,\backslash \_\}(a\_2)...=\backslash chi\_\{q\_1,\backslash \_\}(a\_1)\backslash chi\_\{q\_2,\backslash \_\}(a\_2)...,$

showing that every character mod $m$ is the product of characters mod the $q\_i$.

For $(t,m)=1$ defineSee [https://lmfdb.org/knowledge/show/character.dirichlet.conrey Conrey labeling].

:$\backslash chi\_\{m,t\}=\backslash chi\_\{q\_1,t\}\backslash chi\_\{q\_2,t\}...$

Then for $(rs,m)=1$ and all $a$ and $b$Because these formulas are true for each factor.

:$\backslash chi\_\{m,r\}(a)\backslash chi\_\{m,r\}(b)=\backslash chi\_\{m,r\}(ab),$ showing that $\backslash chi\_\{m,r\}$ is a character and

:$\backslash chi\_\{m,r\}(a)\backslash chi\_\{m,s\}(a)=\backslash chi\_\{m,rs\}(a),$ showing an isomorphism $\backslash widehat\{(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times\}\backslash cong(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times.$

$(\backslash mathbb\{Z\}/15\backslash mathbb\{Z\})^\backslash times\backslash cong(\backslash mathbb\{Z\}/3\backslash mathbb\{Z\})^\backslash times\backslash times(\backslash mathbb\{Z\}/5\backslash mathbb\{Z\})^\backslash times.$

The factorization of the characters mod 15 is

:$$

\begin{array}

& \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}

The nonzero values of the characters mod 15 are

:$$

\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}

.

$(\backslash mathbb\{Z\}/24\backslash mathbb\{Z\})^\backslash times\backslash cong(\backslash mathbb\{Z\}/8\backslash mathbb\{Z\})^\backslash times\backslash times(\backslash mathbb\{Z\}/3\backslash mathbb\{Z\})^\backslash times.$

The factorization of the characters mod 24 is

:$$

\begin{array}

& \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}

The nonzero values of the characters mod 24 are

:$$

\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}

.

$(\backslash mathbb\{Z\}/40\backslash mathbb\{Z\})^\backslash times\backslash cong(\backslash mathbb\{Z\}/8\backslash mathbb\{Z\})^\backslash times\backslash times(\backslash mathbb\{Z\}/5\backslash mathbb\{Z\})^\backslash times.$

The factorization of the characters mod 40 is

:$$

\begin{array}

& \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}

The nonzero values of the characters mod 40 are

:$$

\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}

.

Let $m=p\_1^\{k\_1\}p\_2^\{k\_2\}\backslash cdots\; =\; q\_1q\_2\; \backslash cdots$,

There are $\backslash phi(m)$ Dirichlet characters mod $m.$ They are denoted by $\backslash chi\_\{m,r\},$ where $\backslash chi\_\{m,r\}=\backslash chi\_\{m,s\}$ is equivalent to $r\backslash equiv\; s\backslash pmod\{m\}.$

The identity $\backslash chi\_\{m,r\}(a)\backslash chi\_\{m,s\}(a)=\backslash chi\_\{m,rs\}(a)\backslash ;$ is an isomorphism $\backslash widehat\{(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times\}\backslash cong(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times.$This is true for all finite abelian groups: $A\backslash cong\backslash hat\{A\}$; See Ireland & Rosen pp. 253-254

Each character mod $m$ has a unique factorization as the product of characters mod the prime powers dividing $m$:

:$\backslash chi\_\{m,r\}=\backslash chi\_\{q\_1,r\}\backslash chi\_\{q\_2,r\}...$

If $m=m\_1m\_2,\; (m\_1,m\_2)=1$ the product $\backslash chi\_\{m\_1,r\}\backslash chi\_\{m\_2,s\}$ is a character $\backslash chi\_\{m,t\}$ where $t$ is given by $t\backslash equiv\; r\backslash pmod\{m\_1\}$ and $t\backslash equiv\; s\backslash pmod\{m\_2\}.$

Also,because the formulas for $\backslash chi$ mod prime powers are symmetric in $r$ and $s$ and the formula for products preserves this symmetry. See Davenport, p. 29.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.

$\backslash chi\_\{m,r\}(s)=\backslash chi\_\{m,s\}(r)$

The two orthogonality relations areSee #Relation to group characters above.

:$\backslash sum\_\{a\backslash in(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times\}\; \backslash chi(a)=$

\begin{cases}

\phi(m)&\text{ if }\;\chi=\chi_0\\

0&\text{ if }\;\chi\ne\chi_0

\end{cases}

    and     $\backslash sum\_\{\backslash chi\backslash in\backslash widehat\{(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times\}\}\backslash chi(a)=$

\begin{cases}

\phi(m)&\text{ if }\;a\equiv 1\pmod{m}\\

0&\text{ if }\;a\not\equiv 1\pmod{m}.

\end{cases}

The relations can be written in the symmetric form

:$\backslash sum\_\{a\backslash in(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times\}\; \backslash chi\_\{m,r\}(a)=$

\begin{cases}

\phi(m)&\text{ if }\;r\equiv 1\\

0&\text{ if }\;r\not\equiv 1

\end{cases}

    and     $\backslash sum\_\{r\backslash in(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times\}\; \backslash chi\_\{m,r\}(a)=$

\begin{cases}

\phi(m)&\text{ if }\;a\equiv 1\\

0&\text{ if }\;a\not\equiv 1.

\end{cases}

The first relation is easy to prove: If $\backslash chi=\backslash chi\_0$ there are $\backslash phi(m)$ non-zero summands each equal to 1. If $\backslash chi\backslash ne\backslash chi\_0$there isby the definition of $\backslash chi\_0$ some $a^*,\backslash ;\; (a^*,m)=1,\backslash ;\backslash chi(a^*)\backslash ne1.$  Then

:$\backslash chi(a^*)\backslash sum\_\{a\backslash in(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times\}\; \backslash chi(a)=\backslash sum\_\{a\}\backslash chi(a^*)\; \backslash chi(a)=\backslash sum\_\{a\}\; \backslash chi(a^*a)=\backslash sum\_\{a\}\; \backslash chi(a),$

because multiplying every element in a group by a constant element merely permutes the elements. See Group (mathematics)   implying

:$(\backslash chi(a^*)-1)\backslash sum\_\{a\}\; \backslash chi(a)=0.$   Dividing by the first factor gives $\backslash sum\_\{a\}\; \backslash chi(a)=0,$ QED. The identity $\backslash chi\_\{m,r\}(s)=\backslash chi\_\{m,s\}(r)$ for $(rs,m)=1$ shows that the relations are equivalent to each other.

The second relation can be proven directly in the same way, but requires a lemmaDavenport 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]

:Given $a\; \backslash not\backslash equiv\; 1\backslash pmod\{m\},\backslash ;(a,m)=1,$ there is a $\backslash chi^*,\backslash ;\; \backslash chi^*(a)\backslash ne1.$

The second relation has an important corollary: if $(a,m)=1,$ define the function

:$f\_a(n)=\backslash frac\{1\}\{\backslash phi(m)\}\; \backslash sum\_\{\backslash chi\}\; \backslash bar\{\backslash chi\}(a)\; \backslash chi(n).$   Then

:$f\_a(n)$

= \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}

That is $f\_a=\backslash mathbb\{1\}\_\{[a]\}$ the indicator function of the residue class $[a]=\backslash \{\; x:\backslash ;x\backslash equiv\; a\; \backslash pmod\{m\}\backslash \}$. It is basic in the proof of Dirichlet's theorem.Davenport chs. 1, 4; Landau p. 114Note that if $g:(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times\backslash rightarrow\backslash mathbb\{C\}$ is any function

$g(n)=\backslash sum\_\{a\backslash in(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times\}\; g(a)f\_a(n)$; see Fourier transform on finite groups#Fourier transform for finite abelian groups

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

:$$

\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}

$\backslash chi\_\{16,3\}$ has period 16, but $\backslash chi\_\{16,9\}$ has period 8 and $\backslash chi\_\{16,15\}$ has period 4:   $\backslash chi\_\{16,9\}=\backslash chi\_\{8,5\}$ and  $\backslash chi\_\{16,15\}=\backslash chi\_\{8,7\}=\backslash chi\_\{4,3\}.$

We say that a character $\backslash chi$ of modulus $q$ has a quasiperiod of $d$ if $\backslash chi(m)=\backslash chi(n)$ for all $m$, $n$ coprime to $q$ satisfying $m\backslash equiv\; n$ mod $d$.{{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}} For example, $\backslash chi\_\{2,1\}$, the only Dirichlet character of modulus $2$, has a quasiperiod of $1$, but not a period of $1$ (it has a period of $2$, though). The smallest positive integer for which $\backslash chi$ is quasiperiodic is the conductor of $\backslash chi$.{{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}} So, for instance, $\backslash chi\_\{2,1\}$ has a conductor of $1$.

The conductor of $\backslash chi\_\{16,3\}$ is 16, the conductor of $\backslash chi\_\{16,9\}$ is 8 and that of $\backslash chi\_\{16,15\}$ and $\backslash chi\_\{8,7\}$ 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: $\backslash chi\_\{16,9\}$ is induced from $\backslash chi\_\{8,5\}$ and $\backslash chi\_\{16,15\}$ and $\backslash chi\_\{8,7\}$ are induced from $\backslash chi\_\{4,3\}$.

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,

:$$

\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}

.

The nonzero values of $\backslash chi\_\{15,8\}$ have period 15, but those of $\backslash chi\_\{15,11\}$ have period 3 and those of $\backslash chi\_\{15,13\}$ have period 5. This is easier to see by juxtaposing them with characters mod 3 and 5:

:$$

\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}

.

If a character mod $m=qr,\backslash ;\backslash ;\; (q,r)=1,\; \backslash ;\backslash ;q>1,\backslash ;\backslash ;\; r>1$ is defined as

:$\backslash chi\_\{m,\backslash \_\}(a)=$

\begin{cases}

0&\text{ if }\gcd(a,m)>1\\

\chi_{q,\_}(a)&\text{ if }\gcd(a,m)=1

\end{cases}

,   or equivalently as $\backslash chi\_\{m,\backslash \_\}=\; \backslash chi\_\{q,\backslash \_\}\; \backslash chi\_\{r,1\},$

its nonzero values are determined by the character mod $q$ and have period $q$.

The smallest period of the nonzero values is the conductor of the character. For example, the conductor of $\backslash chi\_\{15,8\}$ is 15, the conductor of $\backslash chi\_\{15,11\}$ is 3, and that of $\backslash chi\_\{15,13\}$ 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, $\backslash chi\_\{15,11\}$ is induced from $\backslash chi\_\{3,2\}$ and $\backslash chi\_\{15,13\}$ is induced from $\backslash chi\_\{5,3\}$

The principal character is not primitive.Davenport classifies it as neither primitive nor imprimitive; the LMFDB induces it from $\backslash chi\_\{1,1\}.$

The character $\backslash chi\_\{m,r\}=\backslash chi\_\{q\_1,r\}\backslash chi\_\{q\_2,r\}...$ is primitive if and only if each of the factors is primitive.Note that if $m$ is two times an odd number, $m=2r$, all characters mod $m$ are imprimitive because $\backslash chi\_\{m,\backslash \_\}=\backslash chi\_\{r,\backslash \_\}\backslash chi\_\{2,1\}$

Primitive characters often simplify (or make possible) formulas in the theories of L-functionsFor example the functional equation of $L(s,\backslash chi)$ is only valid for primitive $\backslash chi$. See Davenport, p. 85 and modular forms.

$\backslash chi(a)$ is even if $\backslash chi(-1)=1$ and is odd if $\backslash chi(-1)=-1.$

This distinction appears in the functional equation of the Dirichlet L-function.

The order of a character is its order as an element of the group $\backslash widehat\{(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times\}$, i.e. the smallest positive integer $n$ such that $\backslash chi^n=\; \backslash chi\_0.$ Because of the isomorphism $\backslash widehat\{(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times\}\backslash cong(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times$ the order of $\backslash chi\_\{m,r\}$ is the same as the order of $r$ in $(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times.$ The principal character has order 1; other real characters have order 2, and imaginary characters have order 3 or greater. By Lagrange's theorem the order of a character divides the order of $\backslash widehat\{(\backslash mathbb\{Z\}/m\backslash mathbb\{Z\})^\backslash times\}$ which is $\backslash phi(m)$

$\backslash chi(a)$ is real or quadratic if all of its values are real (they must be $0,\backslash ;\backslash pm1$); otherwise it is complex or imaginary.

$\backslash chi$ is real if and only if $\backslash chi^2=\backslash chi\_0$; $\backslash chi\_\{m,k\}$ is real if and only if $k^2\backslash equiv1\backslash pmod\{m\}$; in particular, $\backslash chi\_\{m,-1\}$ is real and non-principal.In fact, for prime modulus $p\backslash ;\backslash ;\backslash chi\_\{p,-1\}$ is the Legendre symbol: $\backslash chi\_\{p,-1\}(a)=\backslash left(\backslash frac\{a\}\{p\}\backslash right).\backslash ;$ Sketch of proof: $\backslash nu\_p(-1)=\backslash frac\{p-1\}\{2\},\backslash ;\backslash ;\backslash omega^\{\backslash nu\_p(-1)\}=-1,\; \backslash ;\backslash ;\backslash nu\_p(a)$ is even (odd) if a is a quadratic residue (nonresidue)

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

Real characters are Kronecker symbols;Davenport p. 40 for example, the principal character can be writtenThe notation $\backslash chi\_\{m,1\}=\backslash left(\backslash frac\{m^2\}\{\backslash bullet\}\backslash right)$ is a shorter way of writing $\backslash chi\_\{m,1\}(a)=\backslash left(\backslash frac\{m^2\}\{a\}\backslash right)$

$\backslash chi\_\{m,1\}=\backslash left(\backslash frac\{m^2\}\{\backslash bullet\}\backslash right)$.

The real characters in the examples are:

If

$\backslash chi\_\{16,1\}=\backslash chi\_\{8,1\}=\backslash chi\_\{4,1\}=\backslash chi\_\{2,1\}=\backslash left(\backslash frac\{4\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{9,1\}=\backslash chi\_\{3,1\}=\backslash left(\backslash frac\{9\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{5,1\}=\backslash left(\backslash frac\{25\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{7,1\}=\backslash left(\backslash frac\{49\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{15,1\}=\backslash left(\backslash frac\{225\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{24,1\}=\backslash left(\backslash frac\{36\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{40,1\}=\backslash left(\backslash frac\{100\}\{\backslash bullet\}\backslash right)$  

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 they are imaginary.Davenport pp. 38-40

$\backslash chi\_\{3,2\}=\backslash left(\backslash frac\{-3\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{4,3\}=\backslash left(\backslash frac\{-4\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{5,4\}=\backslash left(\backslash frac\{5\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{7,6\}=\backslash left(\backslash frac\{-7\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{8,3\}=\backslash left(\backslash frac\{-8\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{8,5\}=\backslash left(\backslash frac\{8\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{15,14\}=\backslash left(\backslash frac\{-15\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{24,5\}=\backslash left(\backslash frac\{-24\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{24,11\}=\backslash left(\backslash frac\{24\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{40,19\}=\backslash left(\backslash frac\{-40\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{40,29\}=\backslash left(\backslash frac\{40\}\{\backslash bullet\}\backslash right)$

$\backslash chi\_\{8,7\}=\backslash chi\_\{4,3\}=\backslash left(\backslash frac\{-4\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{9,8\}=\backslash chi\_\{3,2\}=\backslash left(\backslash frac\{-3\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{15,4\}=\backslash chi\_\{5,4\}\backslash chi\_\{3,1\}=\backslash left(\backslash frac\{45\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{15,11\}=\backslash chi\_\{3,2\}\backslash chi\_\{5,1\}=\backslash left(\backslash frac\{-75\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{16,7\}=\backslash chi\_\{8,3\}=\backslash left(\backslash frac\{-8\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{16,9\}=\backslash chi\_\{8,5\}=\backslash left(\backslash frac\{8\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{16,15\}=\backslash chi\_\{4,3\}=\backslash left(\backslash frac\{-4\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{24,7\}=\backslash chi\_\{8,7\}\backslash chi\_\{3,1\}=\backslash chi\_\{4,3\}\backslash chi\_\{3,1\}=\backslash left(\backslash frac\{-36\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{24,13\}=\backslash chi\_\{8,5\}\backslash chi\_\{3,1\}=\backslash left(\backslash frac\{72\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{24,17\}=\backslash chi\_\{3,2\}\backslash chi\_\{8,1\}=\backslash left(\backslash frac\{-12\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{24,19\}=\backslash chi\_\{8,3\}\backslash chi\_\{3,1\}=\backslash left(\backslash frac\{-72\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{24,23\}=\backslash chi\_\{8,7\}\backslash chi\_\{3,2\}=\backslash chi\_\{4,3\}\backslash chi\_\{3,2\}=\backslash left(\backslash frac\{12\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{40,9\}=\backslash chi\_\{5,4\}\backslash chi\_\{8,1\}=\backslash left(\backslash frac\{20\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{40,11\}=\backslash chi\_\{8,3\}\backslash chi\_\{5,1\}=\backslash left(\backslash frac\{-200\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{40,21\}=\backslash chi\_\{8,5\}\backslash chi\_\{5,1\}=\backslash left(\backslash frac\{200\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{40,31\}=\backslash chi\_\{8,7\}\backslash chi\_\{5,1\}=\backslash chi\_\{4,3\}\backslash chi\_\{5,1\}=\backslash left(\backslash frac\{-100\}\{\backslash bullet\}\backslash right)$  

$\backslash chi\_\{40,39\}=\backslash chi\_\{8,7\}\backslash chi\_\{5,4\}=\backslash chi\_\{4,3\}\backslash chi\_\{5,4\}=\backslash left(\backslash frac\{-20\}\{\backslash bullet\}\backslash right)$  

{{Main|Dirichlet L-function}}

The Dirichlet L-series for a character $\backslash chi$ is

:$L(s,\backslash chi)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash chi(n)\}\{n^s\}.$

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

Dirichlet introduced the $L$-function along with the characters in his 1837 paper.

{{Main|Modular form}}

Dirichlet characters appear several places in the theory of modular forms and functions. A typical example isKoblittz, prop. 17b p. 127

Let $\backslash chi\backslash in\backslash widehat\{(\backslash mathbb\{Z\}/M\backslash mathbb\{Z\})^\backslash times\}$ and let $\backslash chi\_1\backslash in\backslash widehat\{(\backslash mathbb\{Z\}/N\backslash mathbb\{Z\})^\backslash times\}$ be primitive.

If

:$f(z)=\backslash sum\; a\_n\; z^n\backslash in\; M\_k(M,\backslash chi)$$f(z)\backslash in\; M\_k(M,\backslash chi)$ means

1) $f(\backslash frac\{az+b\}\{cz+d\})(cz+d)^\{-k\}=f(z)$ where $ad-bc=1$ and

$a\backslash equiv\; d\backslash equiv\; 1,\backslash ;\backslash ;c\backslash equiv\; 0\backslash pmod\{M\}.$

and 2) $f(\backslash frac\{az+b\}\{cz+d\})(cz+d)^\{-k\}=\backslash chi(d)f(z)$ where $ad-bc=1$ and $c\backslash equiv\; 0\backslash pmod\{M\}.$

See Koblitz Ch. III.

define

:$f\_\{\backslash chi\_1\}(z)=\backslash sum\backslash chi\_1(n)a\_nz^n$,the twist of $f$ by $\backslash chi\_1$  

Then

:$f\_\{\backslash chi\_1\}(z)\backslash in\; M\_k(MN^2,\backslash chi\backslash chi\_1^2)$. If $f$ is a cusp form so is $f\_\{\backslash chi\_1\}.$

See theta series of a Dirichlet character for another example.

{{Main|Gauss sum}}

The Gauss sum of a Dirichlet character modulo {{mvar|N}} is

:$G(\backslash chi)=\backslash sum\_\{a=1\}^N\backslash chi(a)e^\backslash frac\{2\backslash pi\; ia\}\{N\}.$

It appears in the functional equation of the Dirichlet L-function.

{{Main|Jacobi sum}}

If $\backslash chi$ and $\backslash psi$ are Dirichlet characters mod a prime $p$ their Jacobi sum is

: $J(\backslash chi,\backslash psi)\; =\; \backslash sum\_\{a=2\}^\{p-1\}\; \backslash chi(a)\; \backslash psi(1\; -\; a).$

Jacobi sums can be factored into products of Gauss sums.

{{Main|Kloosterman sum}}

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

:$K(a,b,\backslash chi)=\backslash sum\_\{r\backslash in\; (\backslash mathbb\{Z\}/q\backslash mathbb\{Z\})^\backslash times\}\backslash chi(r)\backslash zeta^\{ar+\backslash frac\{b\}\{r\}\}.$

If $b=0$ it is a Gauss sum.

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

If $\backslash Chi:\backslash mathbb\{Z\}\backslash rightarrow\backslash mathbb\{C\}$ such that

:1)   $\backslash Chi(ab)\; =\; \backslash Chi(a)\backslash Chi(b),$

:2)   $\backslash Chi(a\; +\; m)\; =\; \backslash Chi(a)$,

:3)   If $\backslash gcd(a,m)>1$ then $\backslash Chi(a)=0$, but

:4)   $\backslash Chi(a)$ is not always 0,

then $\backslash Chi(a)$ is one of the $\backslash phi(m)$ characters mod $m$Davenport p. 30

A Dirichlet character is a completely multiplicative function $f:\; \backslash mathbb\{N\}\; \backslash rightarrow\; \backslash mathbb\{C\}$ that satisfies a linear recurrence relation: that is, if $a\_1\; f(n+b\_1)\; +\; \backslash cdots\; +\; a\_kf(n+b\_k)\; =\; 0$

for all positive integers $n$, where $a\_1,\backslash ldots,a\_k$ are not all zero and $b\_1,\backslash ldots,b\_k$ are distinct then $f$ is a Dirichlet character.Sarkozy

A Dirichlet character is a completely multiplicative function $f:\; \backslash mathbb\{N\}\; \backslash rightarrow\; \backslash mathbb\{C\}$ satisfying the following three properties: a) $f$ takes only finitely many values; b) $f$ vanishes at only finitely many primes; c) there is an $\backslash alpha\; \backslash in\; \backslash mathbb\{C\}$ for which the remainder

$\backslash left|\backslash sum\_\{n\; \backslash leq\; x\}\; f(n)-\; \backslash alpha\; x\backslash right|$

is uniformly bounded, as $x\; \backslash rightarrow\; \backslash infty$. This equivalent definition of Dirichlet characters was conjectured by ChudakovChudakov in 1956, and proved in 2017 by Klurman and Mangerel.Klurman

{{col div|colwidth=30em}}

• Character sum
• Multiplicative group of integers modulo n
• Primitive root modulo n
• Multiplicative character

{{colend}}

{{reflist}}

• {{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

| 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}}

• [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