Dirichlet character

In analytic number theory and related branches of mathematics, a complex-valued arithmetic function ${\displaystyle \chi :\mathbb {Z} \rightarrow \mathbb {C} }$ is a Dirichlet character of modulus ${\displaystyle m}$ (where ${\displaystyle m}$ is a positive integer) if for all integers ${\displaystyle a}$ and ${\displaystyle b}$:^[1]

1. ${\displaystyle \chi (ab)=\chi (a)\chi (b);}$ that is, ${\displaystyle \chi }$ is completely multiplicative.
2. ${\displaystyle \chi (a){\begin{cases}=0&{\text{if }}\gcd(a,m)>1\\\neq 0&{\text{if }}\gcd(a,m)=1.\end{cases}}}$ (gcd is the greatest common divisor)
3. ${\displaystyle \chi (a+m)=\chi (a)}$; that is, ${\displaystyle \chi }$ is periodic with period ${\displaystyle m}$.

The simplest possible character, called the principal character, usually denoted ${\displaystyle \chi _{0}}$, (see Notation below) exists for all moduli:^[2]

${\displaystyle \chi _{0}(a)={\begin{cases}0&{\text{if }}\gcd(a,m)>1\\1&{\text{if }}\gcd(a,m)=1.\end{cases}}}$

The German mathematician Peter Gustav Lejeune Dirichlet—for whom the character is named—introduced these functions in his 1837 paper on primes in arithmetic progressions.^[3][4]

${\displaystyle \phi (n)}$ is Euler's totient function.^[5]

${\displaystyle \zeta _{n}}$ is a complex primitive n-th root of unity:

${\displaystyle \zeta _{n}^{n}=1,}$ but ${\displaystyle \zeta _{n}\neq 1,\zeta _{n}^{2}\neq 1,...\zeta _{n}^{n-1}\neq 1.}$

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

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

${\displaystyle p,p_{k},}$ etc. are prime numbers.

${\displaystyle (m,n)}$ is a standard^[6] abbreviation^[7] for ${\displaystyle \gcd(m,n)}$

${\displaystyle \chi (a),\chi '(a),\chi _{r}(a),}$ 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 Conrey labeling (introduced by Brian Conrey and used by the LMFDB).

In this labeling characters for modulus ${\displaystyle m}$ are denoted ${\displaystyle \chi _{m,t}(a)}$ where the index ${\displaystyle t}$ is described in the section the group of characters below. In this labeling, ${\displaystyle \chi _{m,\_}(a)}$ denotes an unspecified character and ${\displaystyle \chi _{m,1}(a)}$ denotes the principal character mod ${\displaystyle m}$.

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

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

The set of characters is denoted ${\displaystyle {\widehat {G}}.}$ If the product of two characters is defined by pointwise multiplication ${\displaystyle \eta \theta (a)=\eta (a)\theta (a),}$ the identity by the trivial character ${\displaystyle \eta _{0}(a)=1}$ and the inverse by complex inversion ${\displaystyle \eta ^{-1}(a)=\eta (a)^{-1}}$ then ${\displaystyle {\widehat {G}}}$ becomes an abelian group.^[8]

If ${\displaystyle A}$ is a finite abelian group then^[9] there is an isomorphism ${\displaystyle A\cong {\widehat {A}}}$, and the orthogonality relations:^[10]

${\displaystyle \sum _{a\in A}\eta (a)={\begin{cases}|A|&{\text{ if }}\eta =\eta _{0}\\0&{\text{ if }}\eta \neq \eta _{0}\end{cases}}}$     and     ${\displaystyle \sum _{\eta \in {\widehat {A}}}\eta (a)={\begin{cases}|A|&{\text{ if }}a=1\\0&{\text{ if }}a\neq 1.\end{cases}}}$

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

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

${\displaystyle \chi (a)={\begin{cases}0&{\text{if }}[a]\not \in (\mathbb {Z} /m\mathbb {Z} )^{\times }&{\text{i.e. }}(a,m)>1\\\rho ([a])&{\text{if }}[a]\in (\mathbb {Z} /m\mathbb {Z} )^{\times }&{\text{i.e. }}(a,m)=1,\end{cases}}}$

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

Paraphrasing Davenport,^[11] 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 ${\displaystyle \gcd(1,m)=1,}$ property 2) says ${\displaystyle \chi (1)\neq 0}$ so it can be canceled from both sides of ${\displaystyle \chi (1)\chi (1)=\chi (1\times 1)=\chi (1)}$:

${\displaystyle \chi (1)=1.}$^[12]

5) Property 3) is equivalent to

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

6) Property 1) implies that, for any positive integer ${\displaystyle n}$

${\displaystyle \chi (a^{n})=\chi (a)^{n}.}$

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

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

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

${\displaystyle \chi (a)={\begin{cases}0&{\text{if }}\gcd(a,m)>1\\\zeta _{\phi (m)}^{r}&{\text{if }}\gcd(a,m)=1\end{cases}}}$

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

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

${\displaystyle \chi \chi '(a)=\chi (a)\chi '(a)}$   (${\displaystyle \chi \chi '}$ obviously satisfies 1-3).^[13]

The principal character is an identity:

${\displaystyle \chi \chi _{0}(a)=\chi (a)\chi _{0}(a)={\begin{cases}0\times 0&=\chi (a)&{\text{if }}\gcd(a,m)>1\\\chi (a)\times 1&=\chi (a)&{\text{if }}\gcd(a,m)=1.\end{cases}}}$

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

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

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

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

Thus for all integers ${\displaystyle a}$

${\displaystyle \chi (a){\overline {\chi }}(a)={\begin{cases}0&{\text{if }}\gcd(a,m)>1\\1&{\text{if }}\gcd(a,m)=1\end{cases}};}$   in other words ${\displaystyle \chi {\overline {\chi }}=\chi _{0}}$. 

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 ${\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }}$ have different structures depending on whether ${\displaystyle m}$ is a power of 2, a power of an odd prime, or the product of prime powers.^[14]

If ${\displaystyle q=p^{k}}$ is an odd number ${\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{\times }}$ is cyclic of order ${\displaystyle \phi (q)}$; a generator is called a primitive root mod ${\displaystyle q}$.^[15] Let ${\displaystyle g_{q}}$ be a primitive root and for ${\displaystyle (a,q)=1}$ define the function ${\displaystyle \nu _{q}(a)}$ (the index of ${\displaystyle a}$) by

${\displaystyle a\equiv g_{q}^{\nu _{q}(a)}{\pmod {q}},}$

${\displaystyle 0\leq \nu _{q}<\phi (q).}$

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

${\displaystyle \chi (a)=\chi (g_{q}^{\nu _{q}(a)})=\chi (g_{q})^{\nu _{q}(a)},}$   ${\displaystyle \chi }$ is determined by its value at ${\displaystyle g_{q}.}$

Let ${\displaystyle \omega _{q}=\zeta _{\phi (q)}}$ be a primitive ${\displaystyle \phi (q)}$-th root of unity. From property 7) above the possible values of ${\displaystyle \chi (g_{q})}$ are ${\displaystyle \omega _{q},\omega _{q}^{2},...\omega _{q}^{\phi (q)}=1.}$ These distinct values give rise to ${\displaystyle \phi (q)}$ Dirichlet characters mod ${\displaystyle q.}$ For ${\displaystyle (r,q)=1}$ define ${\displaystyle \chi _{q,r}(a)}$ as

${\displaystyle \chi _{q,r}(a)={\begin{cases}0&{\text{if }}\gcd(a,q)>1\\\omega _{q}^{\nu _{q}(r)\nu _{q}(a)}&{\text{if }}\gcd(a,q)=1.\end{cases}}}$

Then for ${\displaystyle (rs,q)=1}$ and all ${\displaystyle a}$ and ${\displaystyle b}$

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

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

2 is a primitive root mod 3.   (${\displaystyle \phi (3)=2}$)

${\displaystyle 2^{1}\equiv 2,\;2^{2}\equiv 2^{0}\equiv 1{\pmod {3}},}$

so the values of ${\displaystyle \nu _{3}}$ are

${\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}a&1&2\\\hline \nu _{3}(a)&0&1\\\end{array}}}$.

The nonzero values of the characters mod 3 are

${\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}&1&2\\\hline \chi _{3,1}&1&1\\\chi _{3,2}&1&-1\\\end{array}}}$

2 is a primitive root mod 5.   (${\displaystyle \phi (5)=4}$)

${\displaystyle 2^{1}\equiv 2,\;2^{2}\equiv 4,\;2^{3}\equiv 3,\;2^{4}\equiv 2^{0}\equiv 1{\pmod {5}},}$

so the values of ${\displaystyle \nu _{5}}$ are

${\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}a&1&2&3&4\\\hline \nu _{5}(a)&0&1&3&2\\\end{array}}}$.

The nonzero values of the characters mod 5 are

${\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}&1&2&3&4\\\hline \chi _{5,1}&1&1&1&1\\\chi _{5,2}&1&i&-i&-1\\\chi _{5,3}&1&-i&i&-1\\\chi _{5,4}&1&-1&-1&1\\\end{array}}}$

3 is a primitive root mod 7.   (${\displaystyle \phi (7)=6}$)

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

so the values of ${\displaystyle \nu _{7}}$ are

${\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}a&1&2&3&4&5&6\\\hline \nu _{7}(a)&0&2&1&4&5&3\\\end{array}}}$.

The nonzero values of the characters mod 7 are (${\displaystyle \omega =\zeta _{6},\;\;\omega ^{3}=-1}$)

${\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}&1&2&3&4&5&6\\\hline \chi _{7,1}&1&1&1&1&1&1\\\chi _{7,2}&1&-\omega &\omega ^{2}&\omega ^{2}&-\omega &1\\\chi _{7,3}&1&\omega ^{2}&\omega &-\omega &-\omega ^{2}&-1\\\chi _{7,4}&1&\omega ^{2}&-\omega &-\omega &\omega ^{2}&1\\\chi _{7,5}&1&-\omega &-\omega ^{2}&\omega ^{2}&\omega &-1\\\chi _{7,6}&1&1&-1&1&-1&-1\\\end{array}}}$.

2 is a primitive root mod 9.   (${\displaystyle \phi (9)=6}$)

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

so the values of ${\displaystyle \nu _{9}}$ are

${\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}a&1&2&4&5&7&8\\\hline \nu _{9}(a)&0&1&2&5&4&3\\\end{array}}}$.

The nonzero values of the characters mod 9 are (${\displaystyle \omega =\zeta _{6},\;\;\omega ^{3}=-1}$)

${\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}&1&2&4&5&7&8\\\hline \chi _{9,1}&1&1&1&1&1&1\\\chi _{9,2}&1&\omega &\omega ^{2}&-\omega ^{2}&-\omega &-1\\\chi _{9,4}&1&\omega ^{2}&-\omega &-\omega &\omega ^{2}&1\\\chi _{9,5}&1&-\omega ^{2}&-\omega &\omega &\omega ^{2}&-1\\\chi _{9,7}&1&-\omega &\omega ^{2}&\omega ^{2}&-\omega &1\\\chi _{9,8}&1&-1&1&-1&1&-1\\\end{array}}}$.

${\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{\times }}$ is the trivial group with one element. ${\displaystyle (\mathbb {Z} /4\mathbb {Z} )^{\times }}$ 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 ${\displaystyle \equiv 1{\pmod {4}}}$ and their negatives are the units ${\displaystyle \equiv 3{\pmod {4}}.}$^[16] For example

${\displaystyle 5^{1}\equiv 5,\;5^{2}\equiv 5^{0}\equiv 1{\pmod {8}}}$

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

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

Let ${\displaystyle q=2^{k},\;\;k\geq 3}$; then ${\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{\times }}$ is the direct product of a cyclic group of order 2 (generated by −1) and a cyclic group of order ${\displaystyle {\frac {\phi (q)}{2}}}$ (generated by 5). For odd numbers ${\displaystyle a}$ define the functions ${\displaystyle \nu _{0}}$ and ${\displaystyle \nu _{q}}$ by

${\displaystyle a\equiv (-1)^{\nu _{0}(a)}5^{\nu _{q}(a)}{\pmod {q}},}$

${\displaystyle 0\leq \nu _{0}<2,\;\;0\leq \nu _{q}<{\frac {\phi (q)}{2}}.}$

For odd ${\displaystyle a}$ and ${\displaystyle b,\;\;a\equiv b{\pmod {q}}}$ if and only if ${\displaystyle \nu _{0}(a)=\nu _{0}(b)}$ and ${\displaystyle \nu _{q}(a)=\nu _{q}(b).}$ For odd ${\displaystyle a}$ the value of ${\displaystyle \chi (a)}$ is determined by the values of ${\displaystyle \chi (-1)}$ and ${\displaystyle \chi (5).}$

Let ${\displaystyle \omega _{q}=\zeta _{\frac {\phi (q)}{2}}}$ be a primitive ${\displaystyle {\frac {\phi (q)}{2}}}$-th root of unity. The possible values of ${\displaystyle \chi ((-1)^{\nu _{0}(a)}5^{\nu _{q}(a)})}$ are ${\displaystyle \pm \omega _{q},\pm \omega _{q}^{2},...\pm \omega _{q}^{\frac {\phi (q)}{2}}=\pm 1.}$ These distinct values give rise to ${\displaystyle \phi (q)}$ Dirichlet characters mod ${\displaystyle q.}$ For odd ${\displaystyle r}$ define ${\displaystyle \chi _{q,r}(a)}$ by

${\displaystyle \chi _{q,r}(a)={\begin{cases}0&{\text{if }}a{\text{ is even}}\\(-1)^{\nu _{0}(r)\nu _{0}(a)}\omega _{q}^{\nu _{q}(r)\nu _{q}(a)}&{\text{if }}a{\text{ is odd}}.\end{cases}}}$

Then for odd ${\displaystyle r}$ and ${\displaystyle s}$ and all ${\displaystyle a}$ and ${\displaystyle b}$

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

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

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

−1 is a primitive root mod 4 (${\displaystyle \phi (4)=2}$)

${\displaystyle {\begin{array}{|||}a&1&3\\\hline \nu _{0}(a)&0&1\\\end{array}}}$

The nonzero values of the characters mod 4 are

${\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}&1&3\\\hline \chi _{4,1}&1&1\\\chi _{4,3}&1&-1\\\end{array}}}$

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

${\displaystyle {\begin{array}{|||}a&1&3&5&7\\\hline \nu _{0}(a)&0&1&0&1\\\nu _{8}(a)&0&1&1&0\\\end{array}}}$.

The nonzero values of the characters mod 8 are

${\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}&1&3&5&7\\\hline \chi _{8,1}&1&1&1&1\\\chi _{8,3}&1&1&-1&-1\\\chi _{8,5}&1&-1&-1&1\\\chi _{8,7}&1&-1&1&-1\\\end{array}}}$

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

${\displaystyle {\begin{array}{|||}a&1&3&5&7&9&11&13&15\\\hline \nu _{0}(a)&0&1&0&1&0&1&0&1\\\nu _{16}(a)&0&3&1&2&2&1&3&0\\\end{array}}}$.

The nonzero values of the characters mod 16 are

${\displaystyle {\begin{array}{|||}&1&3&5&7&9&11&13&15\\\hline \chi _{16,1}&1&1&1&1&1&1&1&1\\\chi _{16,3}&1&-i&-i&1&-1&i&i&-1\\\chi _{16,5}&1&-i&i&-1&-1&i&-i&1\\\chi _{16,7}&1&1&-1&-1&1&1&-1&-1\\\chi _{16,9}&1&-1&-1&1&1&-1&-1&1\\\chi _{16,11}&1&i&i&1&-1&-i&-i&-1\\\chi _{16,13}&1&i&-i&-1&-1&-i&i&1\\\chi _{16,15}&1&-1&1&-1&1&-1&1&-1\\\end{array}}}$.

Let ${\displaystyle m=p_{1}^{m_{1}}p_{2}^{m_{2}}\cdots p_{k}^{m_{k}}=q_{1}q_{2}\cdots q_{k}}$ where ${\displaystyle p_{1}<p_{2}<\dots <p_{k}}$ be the factorization of ${\displaystyle m}$ into prime powers. The group of units mod ${\displaystyle m}$ is isomorphic to the direct product of the groups mod the ${\displaystyle q_{i}}$:^[17]

${\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }\cong (\mathbb {Z} /q_{1}\mathbb {Z} )^{\times }\times (\mathbb {Z} /q_{2}\mathbb {Z} )^{\times }\times \dots \times (\mathbb {Z} /q_{k}\mathbb {Z} )^{\times }.}$

This means that 1) there is a one-to-one correspondence between ${\displaystyle a\in (\mathbb {Z} /m\mathbb {Z} )^{\times }}$ and ${\displaystyle k}$-tuples ${\displaystyle (a_{1},a_{2},\dots ,a_{k})}$ where ${\displaystyle a_{i}\in (\mathbb {Z} /q_{i}\mathbb {Z} )^{\times }}$ and 2) multiplication mod ${\displaystyle m}$ corresponds to coordinate-wise multiplication of ${\displaystyle k}$-tuples:

${\displaystyle ab\equiv c{\pmod {m}}}$ corresponds to

${\displaystyle (a_{1},a_{2},\dots ,a_{k})\times (b_{1},b_{2},\dots ,b_{k})=(c_{1},c_{2},\dots ,c_{k})}$ where ${\displaystyle c_{i}\equiv a_{i}b_{i}{\pmod {q_{i}}}.}$

The Chinese remainder theorem (CRT) implies that the ${\displaystyle a_{i}}$ are simply ${\displaystyle a_{i}\equiv a{\pmod {q_{i}}}.}$

There are subgroups ${\displaystyle G_{i}<(\mathbb {Z} /m\mathbb {Z} )^{\times }}$ such that ^[18]

${\displaystyle G_{i}\cong (\mathbb {Z} /q_{i}\mathbb {Z} )^{\times }}$ and

${\displaystyle G_{i}\equiv {\begin{cases}(\mathbb {Z} /q_{i}\mathbb {Z} )^{\times }&\mod q_{i}\\\{1\}&\mod q_{j},j\neq i.\end{cases}}}$

Then ${\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }\cong G_{1}\times G_{2}\times ...\times G_{k}}$ and every ${\displaystyle a\in (\mathbb {Z} /m\mathbb {Z} )^{\times }}$ corresponds to a ${\displaystyle k}$-tuple ${\displaystyle (a_{1},a_{2},...a_{k})}$ where ${\displaystyle a_{i}\in G_{i}}$ and ${\displaystyle a_{i}\equiv a{\pmod {q_{i}}}.}$ Every ${\displaystyle a\in (\mathbb {Z} /m\mathbb {Z} )^{\times }}$ can be uniquely factored as ${\displaystyle a=a_{1}a_{2}...a_{k}.}$ ^{[19] [20]}

If ${\displaystyle \chi _{m,\_}}$ is a character mod ${\displaystyle m,}$ on the subgroup ${\displaystyle G_{i}}$ it must be identical to some ${\displaystyle \chi _{q_{i},\_}}$ mod ${\displaystyle q_{i}}$ Then

${\displaystyle \chi _{m,\_}(a)=\chi _{m,\_}(a_{1}a_{2}...)=\chi _{m,\_}(a_{1})\chi _{m,\_}(a_{2})...=\chi _{q_{1},\_}(a_{1})\chi _{q_{2},\_}(a_{2})...,}$

showing that every character mod ${\displaystyle m}$ is the product of characters mod the ${\displaystyle q_{i}}$.

For ${\displaystyle (t,m)=1}$ define^[21]

${\displaystyle \chi _{m,t}=\chi _{q_{1},t}\chi _{q_{2},t}...}$

Then for ${\displaystyle (rs,m)=1}$ and all ${\displaystyle a}$ and ${\displaystyle b}$^[22]

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

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

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

The factorization of the characters mod 15 is

${\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}&\chi _{5,1}&\chi _{5,2}&\chi _{5,3}&\chi _{5,4}\\\hline \chi _{3,1}&\chi _{15,1}&\chi _{15,7}&\chi _{15,13}&\chi _{15,4}\\\chi _{3,2}&\chi _{15,11}&\chi _{15,2}&\chi _{15,8}&\chi _{15,14}\\\end{array}}}$

The nonzero values of the characters mod 15 are

${\displaystyle {\begin{array}{|||}&1&2&4&7&8&11&13&14\\\hline \chi _{15,1}&1&1&1&1&1&1&1&1\\\chi _{15,2}&1&-i&-1&i&i&-1&-i&1\\\chi _{15,4}&1&-1&1&-1&-1&1&-1&1\\\chi _{15,7}&1&i&-1&i&-i&1&-i&-1\\\chi _{15,8}&1&i&-1&-i&-i&-1&i&1\\\chi _{15,11}&1&-1&1&1&-1&-1&1&-1\\\chi _{15,13}&1&-i&-1&-i&i&1&i&-1\\\chi _{15,14}&1&1&1&-1&1&-1&-1&-1\\\end{array}}}$.

${\displaystyle (\mathbb {Z} /24\mathbb {Z} )^{\times }\cong (\mathbb {Z} /8\mathbb {Z} )^{\times }\times (\mathbb {Z} /3\mathbb {Z} )^{\times }.}$ The factorization of the characters mod 24 is

${\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}&\chi _{8,1}&\chi _{8,3}&\chi _{8,5}&\chi _{8,7}\\\hline \chi _{3,1}&\chi _{24,1}&\chi _{24,19}&\chi _{24,13}&\chi _{24,7}\\\chi _{3,2}&\chi _{24,17}&\chi _{24,11}&\chi _{24,5}&\chi _{24,23}\\\end{array}}}$

The nonzero values of the characters mod 24 are

${\displaystyle {\begin{array}{|||}&1&5&7&11&13&17&19&23\\\hline \chi _{24,1}&1&1&1&1&1&1&1&1\\\chi _{24,5}&1&1&1&1&-1&-1&-1&-1\\\chi _{24,7}&1&1&-1&-1&1&1&-1&-1\\\chi _{24,11}&1&1&-1&-1&-1&-1&1&1\\\chi _{24,13}&1&-1&1&-1&-1&1&-1&1\\\chi _{24,17}&1&-1&1&-1&1&-1&1&-1\\\chi _{24,19}&1&-1&-1&1&-1&1&1&-1\\\chi _{24,23}&1&-1&-1&1&1&-1&-1&1\\\end{array}}}$.

${\displaystyle (\mathbb {Z} /40\mathbb {Z} )^{\times }\cong (\mathbb {Z} /8\mathbb {Z} )^{\times }\times (\mathbb {Z} /5\mathbb {Z} )^{\times }.}$ The factorization of the characters mod 40 is

${\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}&\chi _{8,1}&\chi _{8,3}&\chi _{8,5}&\chi _{8,7}\\\hline \chi _{5,1}&\chi _{40,1}&\chi _{40,11}&\chi _{40,21}&\chi _{40,31}\\\chi _{5,2}&\chi _{40,17}&\chi _{40,27}&\chi _{40,37}&\chi _{40,7}\\\chi _{5,3}&\chi _{40,33}&\chi _{40,3}&\chi _{40,13}&\chi _{40,23}\\\chi _{5,4}&\chi _{40,9}&\chi _{40,19}&\chi _{40,29}&\chi _{40,39}\\\end{array}}}$

The nonzero values of the characters mod 40 are

${\displaystyle {\begin{array}{|||}&1&3&7&9&11&13&17&19&21&23&27&29&31&33&37&39\\\hline \chi _{40,1}&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\\chi _{40,3}&1&i&i&-1&1&-i&-i&-1&-1&-i&-i&1&-1&i&i&1\\\chi _{40,7}&1&i&-i&-1&-1&-i&i&1&1&i&-i&-1&-1&-i&i&1\\\chi _{40,9}&1&-1&-1&1&1&-1&-1&1&1&-1&-1&1&1&-1&-1&1\\\chi _{40,11}&1&1&-1&1&1&-1&1&1&-1&-1&1&-1&-1&1&-1&-1\\\chi _{40,13}&1&-i&-i&-1&-1&-i&-i&1&-1&i&i&1&1&i&i&-1\\\chi _{40,17}&1&-i&i&-1&1&-i&i&-1&1&-i&i&-1&1&-i&i&-1\\\chi _{40,19}&1&-1&1&1&1&1&-1&1&-1&1&-1&-1&-1&-1&1&-1\\\chi _{40,21}&1&-1&1&1&-1&-1&1&-1&-1&1&-1&-1&1&1&-1&1\\\chi _{40,23}&1&-i&i&-1&-1&i&-i&1&1&-i&i&-1&-1&i&-i&1\\\chi _{40,27}&1&-i&-i&-1&1&i&i&-1&-1&i&i&1&-1&-i&-i&1\\\chi _{40,29}&1&1&-1&1&-1&1&-1&-1&-1&-1&1&-1&1&-1&1&1\\\chi _{40,31}&1&-1&-1&1&-1&1&1&-1&1&-1&-1&1&-1&1&1&-1\\\chi _{40,33}&1&i&-i&-1&1&i&-i&-1&1&i&-i&-1&1&i&-i&-1\\\chi _{40,37}&1&i&i&-1&-1&i&i&1&-1&-i&-i&1&1&-i&-i&-1\\\chi _{40,39}&1&1&1&1&-1&-1&-1&-1&1&1&1&1&-1&-1&-1&-1\\\end{array}}}$.

Let ${\displaystyle m=p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots =q_{1}q_{2}\cdots }$, ${\displaystyle p_{1}<p_{2}<\dots }$ be the factorization of ${\displaystyle m}$ and assume ${\displaystyle (rs,m)=1.}$

There are ${\displaystyle \phi (m)}$ Dirichlet characters mod ${\displaystyle m.}$ They are denoted by ${\displaystyle \chi _{m,r},}$ where ${\displaystyle \chi _{m,r}=\chi _{m,s}}$ is equivalent to ${\displaystyle r\equiv s{\pmod {m}}.}$ The identity ${\displaystyle \chi _{m,r}(a)\chi _{m,s}(a)=\chi _{m,rs}(a)\;}$ is an isomorphism ${\displaystyle {\widehat {(\mathbb {Z} /m\mathbb {Z} )^{\times }}}\cong (\mathbb {Z} /m\mathbb {Z} )^{\times }.}$^[23]

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

${\displaystyle \chi _{m,r}=\chi _{q_{1},r}\chi _{q_{2},r}...}$

If ${\displaystyle m=m_{1}m_{2},(m_{1},m_{2})=1}$ the product ${\displaystyle \chi _{m_{1},r}\chi _{m_{2},s}}$ is a character ${\displaystyle \chi _{m,t}}$ where ${\displaystyle t}$ is given by ${\displaystyle t\equiv r{\pmod {m_{1}}}}$ and ${\displaystyle t\equiv s{\pmod {m_{2}}}.}$

Also,^[24][25] ${\displaystyle \chi _{m,r}(s)=\chi _{m,s}(r)}$

The two orthogonality relations are^[26]

${\displaystyle \sum _{a\in (\mathbb {Z} /m\mathbb {Z} )^{\times }}\chi (a)={\begin{cases}\phi (m)&{\text{ if }}\;\chi =\chi _{0}\\0&{\text{ if }}\;\chi \neq \chi _{0}\end{cases}}}$     and     ${\displaystyle \sum _{\chi \in {\widehat {(\mathbb {Z} /m\mathbb {Z} )^{\times }}}}\chi (a)={\begin{cases}\phi (m)&{\text{ if }}\;a\equiv 1{\pmod {m}}\\0&{\text{ if }}\;a\not \equiv 1{\pmod {m}}.\end{cases}}}$

The relations can be written in the symmetric form

${\displaystyle \sum _{a\in (\mathbb {Z} /m\mathbb {Z} )^{\times }}\chi _{m,r}(a)={\begin{cases}\phi (m)&{\text{ if }}\;r\equiv 1\\0&{\text{ if }}\;r\not \equiv 1\end{cases}}}$     and     ${\displaystyle \sum _{r\in (\mathbb {Z} /m\mathbb {Z} )^{\times }}\chi _{m,r}(a)={\begin{cases}\phi (m)&{\text{ if }}\;a\equiv 1\\0&{\text{ if }}\;a\not \equiv 1.\end{cases}}}$

The first relation is easy to prove: If ${\displaystyle \chi =\chi _{0}}$ there are ${\displaystyle \phi (m)}$ non-zero summands each equal to 1. If ${\displaystyle \chi \neq \chi _{0}}$there is^[27] some ${\displaystyle a^{*},\;(a^{*},m)=1,\;\chi (a^{*})\neq 1.}$  Then

${\displaystyle \chi (a^{*})\sum _{a\in (\mathbb {Z} /m\mathbb {Z} )^{\times }}\chi (a)=\sum _{a}\chi (a^{*})\chi (a)=\sum _{a}\chi (a^{*}a)=\sum _{a}\chi (a),}$^[28]   implying

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

The second relation can be proven directly in the same way, but requires a lemma^[29]

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

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

${\displaystyle f_{a}(n)={\frac {1}{\phi (m)}}\sum _{\chi }{\bar {\chi }}(a)\chi (n).}$   Then

${\displaystyle f_{a}(n)={\frac {1}{\phi (m)}}\sum _{\chi }\chi (a^{-1})\chi (n)={\frac {1}{\phi (m)}}\sum _{\chi }\chi (a^{-1}n)={\begin{cases}1,&n\equiv a{\pmod {m}}\\0,&n\not \equiv a{\pmod {m}},\end{cases}}}$

That is ${\displaystyle f_{a}=\mathbb {1} _{[a]}}$ the indicator function of the residue class ${\displaystyle [a]=\{x:\;x\equiv a{\pmod {m}}\}}$. It is basic in the proof of Dirichlet's theorem.^[30][31]