User:Virginia-American/Sandbox/Dirichlet character

In analytic number theory and related branches of mathematics, Dirichlet characters are certain complex-valued arithmetic functions. Specifically, given a positive integer $m$, a function $\chi :\mathbb {Z} \rightarrow \mathbb {C}$ is a Dirichlet character of modulus $m$ if for all integers $a$ and $b$:

1) $\chi (ab)=\chi (a)\chi (b);$   i.e. $\chi$ is completely multiplicative.

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

3) $\chi (a+m)=\chi (a)$; i.e. $\chi$ is periodic with period $m$.

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

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

Dirichlet introduced these functions in his 1837 paper on primes in arithmetic progressions.

Notation

$\phi (n)$ is the Euler totient function.

$\operatorname {E} (x):=e^{2\pi ix}.$   Note that $\operatorname {E} (x)\operatorname {E} (y)=\operatorname {E} (x+y).$

$\zeta _{n}=\operatorname {E} \left({\frac {1}{n}}\right)$ is a primitive n-th root of unity:

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

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

$\chi (a)$ (or decorated versions such as $\chi '(a)$ or $\chi _{r}(a)$) is a Dirichlet 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 $m$ are denoted $\chi _{m,\;t}(a)$ where the index $t$ is based on the group structure of the characters mod $q$ and is described in the section Explicit construction below. Note that the principal character for modulus $m$ is labeled $\chi _{m,\;1}(a)$.

Elementary facts

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

$\chi (1)=1.$

5) Property 3) is equivalent to

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

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

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

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

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

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

$\chi (a)={\begin{cases}0&{\text{if }}\;\gcd(a,m)>1\\e^{2\pi i{\frac {r}{\phi (m)}}}=\operatorname {E} \left({\frac {r}{\phi (m)}}\right)&{\text{if }}\;\gcd(a,m)=1\end{cases}}$

for some integer $r$ which depends on $\chi$ and $a$.

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

$\chi \chi '(a)=\chi (a)\chi '(a),$   ($\chi \chi '$ obviously satisfies 1-3).

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) The complex conjugate of a root of unity is its inverse (see here for details):

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

In other words

$\chi {\overline {\chi }}=\chi _{0}$.

Note that this implies for $\gcd(a,m)=1,\;\chi (a^{-1})=\chi (a)^{-1},$ extending 6) to all integers.

The multiplication and identity defined in 8) and the inversion defined in 9) turns the set of Dirichlet characters for a given modulus into a finite abelian group.

The group of characters

Construction

There are three cases to consider: powers of odd primes, powers of 2, and products of prime powers.

Powers of odd primes

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

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

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

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

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

Then for $r,s$ relatively prime to $q$ (i.e. $\gcd(r,q)=\gcd(s,q)=1$)

$\chi _{q,\;r}(a)\chi _{q,\;r}(b)=\chi _{q,\;r}(ab),$

and

$\chi _{q,\;r}(a)\chi _{q,\;s}(a)=\chi _{q,\;rs}(a)$

where the latter formula shows an explicit isomorphism between the group of characters mod $q$ and $(\mathbb {Z} /q\mathbb {Z} )^{\times }.$

For example, 2 is a primitive root mod 9   ($\phi (9)=6$)

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

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

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

The characters mod 9 are ($\omega =\zeta _{6}$)

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

Powers of 2

$(\mathbb {Z} /2\mathbb {Z} )^{\times }$ is the trivial group with one element. $(\mathbb {Z} /4\mathbb {Z} )^{\times }$ is cyclic of order 2 (−1 is a primitive root). For 8, 16, and higher powers of 2, there is no primitive root; the powers of 5 are the units $\equiv 1{\pmod {4}}$ and their negatives are the ones $\equiv 3{\pmod {4}}.$

For example

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

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

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

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

$a\equiv (-1)^{\nu _{0}(a)}5^{\nu _{2}(a)}{\pmod {q}},$

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

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

$\chi _{q,\;r}(a)=(-1)^{\nu _{0}(r)\nu _{0}(a)}\omega _{2}^{\nu _{2}(r)\nu _{2}(a)}.$

Then for odd $r,s$

$\chi _{q,\;r}(a)\chi _{q,\;r}(b)=\chi _{q,\;r}(ab)$

and

$\chi _{q,\;r}(a)\chi _{q,\;s}(a)=\chi _{q,\;rs}(a)$

and the later formula is an isomorphism between the group of characters mod $2^{k}$ and $(\mathbb {Z} /2^{k}\mathbb {Z} )^{\times }.$

For example, mod 16 ($\phi (16)=8$)

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

The characters mod 16 are ($i=\zeta _{4}$ is the imaginary unit)

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

Products of prime powers

Under multiplication the residue classes mod $m$ which are relatively prime to $m$ form a finite abelian group $(\mathbb {Z} /m\mathbb {Z} )^{\times }$of order $\phi (m)$ called the group of units mod $m$. Let $m=p_{1}^{k_{1}}p_{2}^{k_{2}}...p_{r}^{k_{r}}$ be the factorization of $m$ into powers of distinct primes. Then as explained here

$(\mathbb {Z} /m\mathbb {Z} )^{\times }\cong (\mathbb {Z} /p_{1}^{k_{1}}\mathbb {Z} )^{\times }\times (\mathbb {Z} /p_{2}^{k_{2}}\mathbb {Z} )^{\times }...\times (\mathbb {Z} /p_{r}^{k_{r}}\mathbb {Z} )^{\times }.$

Summary and consequences

Isomorphism

The group of Dirichlet characters mod $q$ is isomorphic to the group of units mod q.

Unique factorization

If $m=p_{1}^{k_{1}}p_{2}^{k_{2}}...p_{r}^{k_{r}}$ is the factorization of m into powers of distinct primes, (to make the formula more readable) let $q_{1}=p_{1}^{k_{1}},...q_{r}=p_{r}^{k_{r}}.$ Then for $\gcd(m,t)=1$

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

Orthogonality

$\sum _{a\in A}\chi (a)={\begin{cases}|A|&{\text{ if }}\;\chi =\chi _{0}\\0&{\text{ if }}\;\chi \neq \chi _{0}\end{cases}}$

$\sum _{\chi \in {\hat {A}}}\chi (a)={\begin{cases}|A|&{\text{ if }}\;a=e\\0&{\text{ if }}\;a\neq e\end{cases}}$

online

d's 0riginal in eng.

https://arxiv.org/abs/0808.1408#:~:text=Dirichlet's%20proof%20of%20infinitely%20many,and%20the%20distribution%20of%20primes.