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

$\chi (a)$ or decorated versions such as $\chi '(a)$ or $\chi _{r}(a)$ are Dirichlet characters.

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.

8) and 9) show that the set of Dirichlet characters for a given modulus are a finite abelian group.

The group of characters

Abstract overview

Let $A$ be a finite abelian group written multiplicatively with identity $e.$ A character $\theta$ of the group $A$ is a homomorphism from $A$ to the nonzero complex numbers:

$\theta :A\to \mathbb {C} ^{*},$

$\theta (ab)=\theta (a)\theta (b),\;\;\theta (a^{-1})=\theta (a)^{-1}{\text{ for all }}a,b\in A.$

Reasoning as in the preceding section (using Lagrange's theorem rather than Euler's), it is easy to prove that the values of the characters of $A$ are roots of unity and that the characters themselves form group under pointwise multiplication, with the principal character $\chi _{0}$ (identically 1) as identity. This "dual group" is denoted ${\hat {A}}.$

Any character $\theta$ of the finite abelian group $(\mathbb {Z} /m\mathbb {Z} )^{\times }$ (the multiplicative group of invertible residue classes modulo $m$) defines a Dirichlet character $\chi$ of modulus $m$:

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

(where ${\bar {a}}$ is the residue class modulo $m$ containing $a$), and conversely, a Dirichlet character defines a character of $(\mathbb {Z} /m\mathbb {Z} )^{\times }$.

The basis theorem states that $A$ is the direct sum of cyclic subgroups of prime power order. That is, there is a set of generators $g_{1},g_{2},...g_{t}\in A$ and a matching set of powers of prime numbers $n_{1},n_{2},...n_{t}$ with the properties that

$n_{1}n_{2}...n_{t}=|A|$

$g_{i}^{n_{i}}=e,\;g_{i}\neq e,\;g_{i}^{2}\neq e,\;...\;g_{i}^{n_{i}-1}\neq e.$

and every for every $a\in A$ there is a unique set of exponents $\nu _{i}=\nu _{i}(a),\;\;0\leq \nu _{i}<n_{i},$

$a=g_{1}^{\nu _{1}}g_{2}^{\nu _{2}}\;...\;g_{t}^{\nu _{t}}.$

For each $i,\;1\leq i\leq t,$ let $\omega _{i}=\zeta _{n_{i}}$ be a primitive $n_{i}$-th root of unity and define the function $\chi _{i}$

$\chi _{i}(g_{j}^{k})={\begin{cases}\omega _{i}^{k}&{\text{if }}\;i=j\\1&{\text{if }}\;i\neq j.\end{cases}}$

Clearly $\chi _{i}$ is multiplicative, and for all $a\in A$

$\chi _{i}(a)=\chi _{i}(g_{1}^{\nu _{1}}g_{2}^{\nu _{2}}\;...\;g_{t}^{\nu _{t}})=\chi _{i}(g_{1}^{\nu _{1}})\chi _{i}(g_{2}^{\nu _{2}})\;...\chi _{i}(g_{i}^{\nu _{i}})\;...\;\chi _{i}(g_{t}^{\nu _{t}})=\omega _{i}^{\nu _{i}(a)}.$

The powers of $\chi _{i},\;\;\chi _{i},\chi _{i}^{2},\chi _{i}^{3},\;...$ are distinct functions (they have different values at $g_{i}$) and $\chi _{i}^{n_{i}}=\chi _{i}^{0}$ is identically 1. In other words the powers of $\chi _{i}$ are a cyclic group of order $n_{i}$. The direct sum of all these cyclic groups is a group isomorphic to $A.$ Any character of $A$ is a product of $\chi _{i}$s (work one $i$ at a time), demonstrating the fundamental theorem

$A\cong {\hat {A}}.$

An explicit isomorphism is given by defining for $r\in A$

$\chi _{r}=\chi _{1}^{\nu _{1}(r)}\chi _{2}^{\nu _{2}(r)}...\chi _{t}^{\nu _{t}(r)},$

$\chi _{r}\chi _{s}=\chi _{rs}.$

Applying this function to $a\in A$ gives a formula which is symmetric in the index $r$ and the argument $a$:

$\chi _{r}(a)=\chi _{a}(r)=\omega _{1}^{\nu _{1}(r)\nu _{1}(a)}\omega _{2}^{\nu _{2}(r)\nu _{2}(a)}...\omega _{t}^{\nu _{t}(r)\nu _{t}(a)}.$

Note that under this notation the principal character is denoted $\chi _{e}$.

Orthogonality

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

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

Explicit construction

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

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

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 $r\in (\mathbb {Z} /q\mathbb {Z} )^{\times }$ define $\chi _{q,\;r}(a)$ by

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

Then for $a,b,r,s\in (\mathbb {Z} /q\mathbb {Z} )^{\times }$

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

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 $r\in (\mathbb {Z} /q\mathbb {Z} )^{\times }$ 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, just as for odd prime powers, for $a,b,r,s\in (\mathbb {Z} /q\mathbb {Z} )^{\times }$

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

and

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

again showing an explicit 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)

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

online

d's 0riginal in eng.

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