Character (number theory)

A character (or a numerical character , or a Dirichlet character ) is a certain arithmetic function that arises from characters on reversible elements $\mathbb {Z} /k\mathbb {Z}$ ${\ displaystyle \ mathbb {Z} / k \ mathbb {Z}}$ ${\ displaystyle \ mathbb {Z} / k \ mathbb {Z}}$ . Dirichlet characters are used to determine Dirichlet L- functions , which are meromorphic functions with many interesting analytic properties. If $\chi$ ${\ displaystyle \ chi}$ $\ chi$ is a Dirichlet character; its L -Dirichlet series is defined by the equality

L(s,\chi )=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}

{\ displaystyle L (s, \ chi) = \ sum _ {n = 1} ^ {\ infty} {\ frac {\ chi (n)} {n ^ {s}}}}

{\ displaystyle L (s, \ chi) = \ sum _ {n = 1} ^ {\ infty} {\ frac {\ chi (n)} {n ^ {s}}}}

where s is a complex number with the real part> 1. By analytic continuation, this function can be extended to a meromorphic function on the entire complex plane . Dirichlet L -functions are a generalization of the Riemann zeta function and are noticeably manifested in the generalized Riemann hypotheses .

The characters of Dirichlet are named after Peter Gustav Lejeune Dirichlet .

Content

1 Axiomatic definition
2 Construction through residue classes
- 2.1 Deduction classes
- 2.2 Dirichlet characters
3 Examples
4 Some character tables
- 4.1 Modulo 1
- 4.2 Modulo 2
- 4.3 Modulo 3
- 4.4 Modulo 4
- 4.5 Modulo 5
- 4.6 Modulo 6
- 4.7 Modulo 7
- 4.8 Modulo 8
- 4.9 Modulo 9
- 4.10 Modulo 10
5 Examples
6 Primitive characters and conductor
7 Orthogonality of characters
8 History
9 See also
10 notes
11 Literature
12 Literature

Axiomatic definition

Dirichlet character is any function $\chi$ ${\ displaystyle \ chi}$ $\chi$ on a set of integers $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ into complex numbers $\mathbb {C}$ ${\ displaystyle \ mathbb {C}}$ such that $\chi$ ${\ displaystyle \ chi}$ $\chi$ has the following properties ^[1] :

There is a positive integer k such that $\chi (n)=\chi (n+k)$ ${\ displaystyle \ chi (n) = \ chi (n + k)}$ for any n .
If n and k are not coprime , then $\chi (n)=0$ ${\ displaystyle \ chi (n) = 0}$ ; if they are mutually simple, $\chi (n)\neq 0$ ${\ displaystyle \ chi (n) \ neq 0}$ .
$\chi (mn)=\chi (m)\chi (n)$ ${\ displaystyle \ chi (mn) = \ chi (m) \ chi (n)}$ for any integers m and n .

Some other properties can be deduced from this definition. According to property 3) $\chi (1)=\chi (1\times 1)=\chi (1)\chi (1)$ ${\ displaystyle \ chi (1) = \ chi (1 \ times 1) = \ chi (1) \ chi (1)}$ . Since GCD (1, k ) = 1, property 2) states that $\chi (1)\neq 0$ ${\ displaystyle \ chi (1) \ neq 0}$ , so that

$\chi (1)=1$ ${\ displaystyle \ chi (1) = 1}$ .

Properties 3) and 4) show that any Dirichlet character $\chi$ ${\ displaystyle \ chi}$ $\chi$ is character .

Property 1) says that the character is a periodic function with period k . We say that $\chi$ ${\ displaystyle \ chi}$ $\chi$ is a character modulo k . This is equivalent to saying that

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

If GCD ( a , k ) = 1, Euler's theorem states that $a^{\varphi (k)}\equiv 1{\pmod {k}}$ ${\ displaystyle a ^ {\ varphi (k)} \ equiv 1 {\ pmod {k}}}$ (Where $\varphi (k)$ ${\ displaystyle \ varphi (k)}$ is a Euler function ). Thus, according to properties 5) and 4) $\chi (a^{\varphi (k)})=\chi (1)=1$ ${\ displaystyle \ chi (a ^ {\ varphi (k)}) = \ chi (1) = 1}$ , and by property 3) $\chi (a^{\varphi (k)})=\chi (a)^{\varphi (k)}$ ${\ displaystyle \ chi (a ^ {\ varphi (k)}) = \ chi (a) ^ {\ varphi (k)}}$ . So we have

For all a , coprime to k , $\chi (a)$ ${\ displaystyle \ chi (a)}$ is an $\varphi (k)$ ${\ displaystyle \ varphi (k)}$ complex root of unity ,

i.e $e^{2ri\pi /\varphi (k)}$ ${\ displaystyle e ^ {2ri \ pi / \ varphi (k)}}$ for some whole $0\leqslant r<\varphi (k)$ ${\ displaystyle 0 \ leqslant r <\ varphi (k)}$ .

The only character with period 1 is called the trivial character . Note that any character becomes 0 at point 0, except for the trivial one, which is 1 for all integers.

A character assuming the value 1 on all numbers coprime to $k$ {\ displaystyle k} is called the main one :
$\chi _{0}(n)=\left\{{\begin{array}{ll}1,&{\text{НОД}}(n,\;k)=1;\\0,&{\text{НОД}}(n,\;k)\neq 1.\end{array}}\right.$ ${\ displaystyle \ chi _ {0} (n) = \ left \ {{\ begin {array} {ll} 1, & {\ text {GCD}} (n, \; k) = 1; \\ 0, & {\ text {GCD}} (n, \; k) \ neq 1. \ end {array}} \ right.}$ ^[2] .
- In the group of characters modulo $k$ ${\ displaystyle k}$ he plays the role of unit.

A character is called real if it takes only real values. A character that is not real is called complex ^[3]

Character sign $\chi$ ${\ displaystyle \ chi}$ depends on its value at the point −1. They say that $\chi$ ${\ displaystyle \ chi}$ odd if $\chi (-1)=-1$ ${\ displaystyle \ chi (-1) = - 1}$ , and even if $\chi (-1)=1$ ${\ displaystyle \ chi (-1) = 1}$ .

Building through Deduction Classes

Dirichlet characters can be considered in terms of a $\mathbb {Z} /k\mathbb {Z}$ ${\ displaystyle \ mathbb {Z} / k \ mathbb {Z}}$ as extended characters of residue classes ^[4] .

Deduction classes

Given an integer k , we can define the residue class of the integer n as the set of all integers comparable to n modulo k : ${\hat {n}}=\{m\mid m\equiv n{\pmod {k}}\}.$ ${\ displaystyle {\ hat {n}} = \ {m \ mid m \ equiv n {\ pmod {k}} \}.}$ That is a class of deductions ${\hat {n}}$ ${\ displaystyle {\ hat {n}}}$ is the adjacency class n in the quotient ring $\mathbb {Z} /k\mathbb {Z}$ ${\ displaystyle \ mathbb {Z} / k \ mathbb {Z}}$ .

The set of invertible elements modulo k forms an abelian group of order $\varphi (k)$ ${\ displaystyle \ varphi (k)}$ where the multiplication in the group is given by the equality ${\widehat {mn}}={\hat {m}}{\hat {n}}$ ${\ displaystyle {\ widehat {mn}} = {\ hat {m}} {\ hat {n}}}$ , but $\varphi$ ${\ displaystyle \ varphi}$ again means Euler function . The unit in this group is the deduction class. ${\hat {1}}$ ${\ displaystyle {\ hat {1}}}$ , and the inverse element for ${\hat {m}}$ ${\ displaystyle {\ hat {m}}}$ is a deduction class ${\hat {n}}$ ${\ displaystyle {\ hat {n}}}$ where ${\hat {m}}{\hat {n}}={\hat {1}}$ ${\ displaystyle {\ hat {m}} {\ hat {n}} = {\ hat {1}}}$ , i.e $mn\equiv 1{\pmod {k}}$ ${\ displaystyle mn \ equiv 1 {\ pmod {k}}}$ . For example, for k = 6, the set of reversible elements is $\{{\hat {1}},{\hat {5}}\}$ ${\ displaystyle \ {{\ hat {1}}, {\ hat {5}} \}}$ , since 0, 2, 3, and 4 are not coprime with 6.

Character group $(\mathbb {Z} /k)^{*}$ ${\ displaystyle (\ mathbb {Z} / k) ^ {*}}$ consists of characters of residue classes . The nature of the deduction class $\theta$ ${\ displaystyle \ theta}$ on $(\mathbb {Z} /k)^{*}$ ${\ displaystyle (\ mathbb {Z} / k) ^ {*}}$ primitive if there is no proper divisor d for k such that $\theta$ ${\ displaystyle \ theta}$ factorized as $(\mathbb {Z} /k)^{*}\to (\mathbb {Z} /d)^{*}\to \mathbb {C} ^{*}$ ${\ displaystyle (\ mathbb {Z} / k) ^ {*} \ to (\ mathbb {Z} / d) ^ {*} \ to \ mathbb {C} ^ {*}}$ ^[5] .

Dirichlet characters

Determining the Dirichlet character modulo k ensures that it is bounded by the group of invertible elements modulo k ^[6] : the group of homomorphisms $\chi$ ${\ displaystyle \ chi}$ of $(\mathbb {Z} /k)^{*}$ ${\ displaystyle (\ mathbb {Z} / k) ^ {*}}$ into nonzero complex numbers

\chi :(\mathbb {Z} /k\mathbb {Z} )^{*}\to \mathbb {C} ^{*}

{\ displaystyle \ chi: (\ mathbb {Z} / k \ mathbb {Z}) ^ {*} \ to \ mathbb {C} ^ {*}}

,

with values that will necessarily be roots of unity, since reversible elements modulo k form a finite group. In the opposite direction, if a group of homomorphisms is given $\chi$ ${\ displaystyle \ chi}$ on a group of invertible elements modulo k , we can to a function on integers coprime to k , and then extend this function to all integers by assigning the value 0 on all integers having nontrivial divisors common with k . The resulting function will then be the Dirichlet character ^[7] .

Main character $\chi _{1}$ ${\ displaystyle \ chi _ {1}}$ modulo k has the properties ^[7]

\chi _{1}(n)=1

{\ displaystyle \ chi _ {1} (n) = 1}

with GCD ( n , k ) = 1 and

\chi _{1}(n)=0

{\ displaystyle \ chi _ {1} (n) = 0}

for GCD ( n , k )> 1.

Associative nature of the multiplicative group $(\mathbb {Z} /k)^{*}$ ${\ displaystyle (\ mathbb {Z} / k) ^ {*}}$ is the main character that always takes the value 1 ^[8] .

When k is 1, the main character modulo k is 1 on all integers. For k greater than 1, the main characters modulo k vanish in integers that have non-zero common factors with k , and is equal to 1 on other integers.

Is available $\varphi (n)$ ${\ displaystyle \ varphi (n)}$ Dirichlet characters modulo n ^[7] .

Examples

For any odd module $k$ ${\ displaystyle k}$ Jacobi symbol $\left({\frac {n}{k}}\right)$ ${\ displaystyle \ left ({\ frac {n} {k}} \ right)}$ is a modulo character $k$ ${\ displaystyle k}$ .
A power-law deduction of a degree above 2 is a non-material character.

Some character tables

The tables below help illustrate the nature of Dirichlet characters. They represent characters modulo 1 to 10. Characters $\chi _{1}$ ${\ displaystyle \ chi _ {1}}$ are the main characters.

Modulo 1

Judges $\varphi (1)=1$ ${\ displaystyle \ varphi (1) = 1}$ character modulo 1:

$\chi \setminus n$ ${\ displaystyle \ chi \ setminus n}$	0
$\chi _{1}(n)$ ${\ displaystyle \ chi _ {1} (n)}$	one

This is a trivial character.

Modulo 2

Exist $\varphi (2)=1$ ${\ displaystyle \ varphi (2) = 1}$ character modulo 2:

$\chi \setminus n$ ${\ displaystyle \ chi \ setminus n}$	0	one
$\chi _{1}(n)$ ${\ displaystyle \ chi _ {1} (n)}$	0	one

notice, that $\chi$ ${\ displaystyle \ chi}$ fully determined by $\chi (1)$ ${\ displaystyle \ chi (1)}$ , since 1 generates a group of invertible elements modulo 2.

Modulo 3

there is $\varphi (3)=2$ ${\ displaystyle \ varphi (3) = 2}$ character modulo 3:

$\chi \setminus n$ ${\ displaystyle \ chi \ setminus n}$	one	2
$\chi _{1}(n)$ ${\ displaystyle \ chi _ {1} (n)}$	one	one
$\chi _{2}(n)$ ${\ displaystyle \ chi _ {2} (n)}$	one	−1

notice, that $\chi$ ${\ displaystyle \ chi}$ fully determined by $\chi (2)$ ${\ displaystyle \ chi (2)}$ , since 2 generates a group of invertible elements modulo 3.

Modulo 4

Exist $\varphi (4)=2$ ${\ displaystyle \ varphi (4) = 2}$ character modulo 4:

$\chi \setminus n$ ${\ displaystyle \ chi \ setminus n}$	one	2	3
$\chi _{1}(n)$ ${\ displaystyle \ chi _ {1} (n)}$	one	0	one
$\chi _{2}(n)$ ${\ displaystyle \ chi _ {2} (n)}$	one	0	−1

notice, that $\chi$ ${\ displaystyle \ chi}$ fully determined by $\chi (3)$ ${\ displaystyle \ chi (3)}$ , since 3 generates a group of invertible elements modulo 4.

L -Dirichlet series for $\chi _{1}(n)$ ${\ displaystyle \ chi _ {1} (n)}$ equal to the Dirichlet lambda function (closely related to the Dirichlet eta-function )

L(\chi _{1},s)=(1-2^{-s})\zeta (s)

{\ displaystyle L (\ chi _ {1}, s) = (1-2 ^ {- s}) \ zeta (s)}

,

Where $\zeta (s)$ ${\ displaystyle \ zeta (s)}$ is the Riemann zeta function. L- row for $\chi _{2}(n)$ ${\ displaystyle \ chi _ {2} (n)}$ is a Dirichlet beta function

L(\chi _{2},s)=\beta (s).\,

{\ displaystyle L (\ chi _ {2}, s) = \ beta (s). \,}

Modulo 5

Exist $\varphi (5)=4$ ${\ displaystyle \ varphi (5) = 4}$ characters modulo 5. In tables i is the square root of $-1$ ${\ displaystyle -1}$ .

$\chi \setminus n$ ${\ displaystyle \ chi \ setminus n}$	one	2	3	four
$\chi _{1}(n)$ ${\ displaystyle \ chi _ {1} (n)}$	one	one	one	one
$\chi _{2}(n)$ ${\ displaystyle \ chi _ {2} (n)}$	one	i	−i	−1
$\chi _{3}(n)$ ${\ displaystyle \ chi _ {3} (n)}$	one	−1	−1	one
$\chi _{4}(n)$ ${\ displaystyle \ chi _ {4} (n)}$	one	- i	i	−1

notice, that $\chi$ ${\ displaystyle \ chi}$ fully defined value $\chi (2)$ ${\ displaystyle \ chi (2)}$ , since 2 generates a group of invertible elements modulo 5.

Modulo 6

Exist $\varphi (6)=2$ ${\ displaystyle \ varphi (6) = 2}$ characters modulo 6:

$\chi \setminus n$ ${\ displaystyle \ chi \ setminus n}$	one	2	3	four	5
$\chi _{1}(n)$ ${\ displaystyle \ chi _ {1} (n)}$	one	0	0	0	one
$\chi _{2}(n)$ ${\ displaystyle \ chi _ {2} (n)}$	one	0	0	0	−1

notice, that $\chi$ ${\ displaystyle \ chi}$ fully determined by $\chi (5)$ ${\ displaystyle \ chi (5)}$ , since 5 generates a group of invertible elements modulo 6.

Modulo 7

Exist $\varphi (7)=6$ ${\ displaystyle \ varphi (7) = 6}$ characters modulo 7. In the table below $\omega =\exp(\pi i/3).$ ${\ displaystyle \ omega = \ exp (\ pi i / 3).}$

$\chi \setminus n$ ${\ displaystyle \ chi \ setminus n}$	one	2	3	four	5	6
$\chi _{1}(n)$ ${\ displaystyle \ chi _ {1} (n)}$	one	one	one	one	one	one
$\chi _{2}(n)$ ${\ displaystyle \ chi _ {2} (n)}$	one	$\omega ^{2}$ ${\ displaystyle \ omega ^ {2}}$	$\omega$ ${\ displaystyle \ omega}$	$-\omega$ ${\ displaystyle - \ omega}$	$-\omega ^{2}$ ${\ displaystyle - \ omega ^ {2}}$	−1
$\chi _{3}(n)$ ${\ displaystyle \ chi _ {3} (n)}$	one	- $\omega$ ${\ displaystyle \ omega}$	$\omega ^{2}$ ${\ displaystyle \ omega ^ {2}}$	$\omega ^{2}$ ${\ displaystyle \ omega ^ {2}}$	$-\omega$ ${\ displaystyle - \ omega}$	one
$\chi _{4}(n)$ ${\ displaystyle \ chi _ {4} (n)}$	one	one	−1	one	−1	−1
$\chi _{5}(n)$ ${\ displaystyle \ chi _ {5} (n)}$	one	$\omega ^{2}$ ${\ displaystyle \ omega ^ {2}}$	$-\omega$ ${\ displaystyle - \ omega}$	$-\omega$ ${\ displaystyle - \ omega}$	$\omega ^{2}$ ${\ displaystyle \ omega ^ {2}}$	one
$\chi _{6}(n)$ ${\ displaystyle \ chi _ {6} (n)}$	one	$-\omega$ ${\ displaystyle - \ omega}$	$-\omega ^{2}$ ${\ displaystyle - \ omega ^ {2}}$	$\omega ^{2}$ ${\ displaystyle \ omega ^ {2}}$	$\omega$ ${\ displaystyle \ omega}$	−1

notice, that $\chi$ ${\ displaystyle \ chi}$ fully determined by $\chi (3)$ ${\ displaystyle \ chi (3)}$ , since 3 generates a group of invertible elements modulo 7.

Modulo 8

Exist $\varphi (8)=4$ ${\ displaystyle \ varphi (8) = 4}$ characters modulo 8.

$\chi \setminus n$ ${\ displaystyle \ chi \ setminus n}$	one	2	3	four	5	6	7
$\chi _{1}(n)$ ${\ displaystyle \ chi _ {1} (n)}$	one	0	one	0	one	0	one
$\chi _{2}(n)$ ${\ displaystyle \ chi _ {2} (n)}$	one	0	one	0	−1	0	−1
$\chi _{3}(n)$ ${\ displaystyle \ chi _ {3} (n)}$	one	0	−1	0	one	0	−1
$\chi _{4}(n)$ ${\ displaystyle \ chi _ {4} (n)}$	one	0	−1	0	−1	0	one

notice, that $\chi$ ${\ displaystyle \ chi}$ fully determined by the values $\chi (3)$ ${\ displaystyle \ chi (3)}$ and $\chi (5)$ ${\ displaystyle \ chi (5)}$ , since 3 and 5 generate a group of reversible elements modulo 8.

Modulo 9

Exist $\varphi (9)=6$ ${\ displaystyle \ varphi (9) = 6}$ characters modulo 9. In the table below $\omega =\exp(\pi i/3).$ ${\ displaystyle \ omega = \ exp (\ pi i / 3).}$

$\chi \setminus n$ ${\ displaystyle \ chi \ setminus n}$	one	2	3	four	5	6	7	8
$\chi _{1}(n)$ ${\ displaystyle \ chi _ {1} (n)}$	one	one	0	one	one	0	one	one
$\chi _{2}(n)$ ${\ displaystyle \ chi _ {2} (n)}$	one	$\omega$ ${\ displaystyle \ omega}$	0	$\omega ^{2}$ ${\ displaystyle \ omega ^ {2}}$	$-\omega ^{2}$ ${\ displaystyle - \ omega ^ {2}}$	0	$-\omega$ ${\ displaystyle - \ omega}$	−1
$\chi _{3}(n)$ ${\ displaystyle \ chi _ {3} (n)}$	one	$\omega ^{2}$ ${\ displaystyle \ omega ^ {2}}$	0	$-\omega$ ${\ displaystyle - \ omega}$	$\omega$ ${\ displaystyle \ omega}$	0	$\omega ^{2}$ ${\ displaystyle \ omega ^ {2}}$	one
$\chi _{4}(n)$ ${\ displaystyle \ chi _ {4} (n)}$	one	−1	0	one	−1	0	one	−1
$\chi _{5}(n)$ ${\ displaystyle \ chi _ {5} (n)}$	one	$-\omega$ ${\ displaystyle - \ omega}$	0	$\omega ^{2}$ ${\ displaystyle \ omega ^ {2}}$	$\omega ^{2}$ ${\ displaystyle \ omega ^ {2}}$	0	$-\omega$ ${\ displaystyle - \ omega}$	one
$\chi _{6}(n)$ ${\ displaystyle \ chi _ {6} (n)}$	one	$-\omega ^{2}$ ${\ displaystyle - \ omega ^ {2}}$	0	$-\omega$ ${\ displaystyle - \ omega}$	$\omega$ ${\ displaystyle \ omega}$	0	$\omega ^{2}$ ${\ displaystyle \ omega ^ {2}}$	−1

notice, that $\chi$ ${\ displaystyle \ chi}$ fully determined by $\chi (2)$ ${\ displaystyle \ chi (2)}$ , since 2 generates a group of invertible elements modulo 9.

Modulo 10

Exist $\varphi (10)=4$ ${\ displaystyle \ varphi (10) = 4}$ characters modulo 10.

$\chi \setminus n$ ${\ displaystyle \ chi \ setminus n}$	one	2	3	four	5	6	7	8	9
$\chi _{1}(n)$ ${\ displaystyle \ chi _ {1} (n)}$	one	0	one	0	0	0	one	0	one
$\chi _{2}(n)$ ${\ displaystyle \ chi _ {2} (n)}$	one	0	i	0	0	0	- i	0	−1
$\chi _{3}(n)$ ${\ displaystyle \ chi _ {3} (n)}$	one	0	−1	0	0	0	−1	0	one
$\chi _{4}(n)$ ${\ displaystyle \ chi _ {4} (n)}$	one	0	- i	0	0	0	i	0	−1

notice, that $\chi$ ${\ displaystyle \ chi}$ fully determined by $\chi (3)$ ${\ displaystyle \ chi (3)}$ , since 3 generates a group of reversible elements modulo 10.

Examples

If p is an odd prime , then the function

\chi (n)=\left({\frac {n}{p}}\right),\

{\ displaystyle \ chi (n) = \ left ({\ frac {n} {p}} \ right), \}

Where

\left({\frac {n}{p}}\right)

{\ displaystyle \ left ({\ frac {n} {p}} \ right)}

is a Legendre symbol , is a primitive Dirichlet character modulo p ^[9] .

More generally, if m is a positive odd number, the function

\chi (n)=\left({\frac {n}{m}}\right),\

{\ displaystyle \ chi (n) = \ left ({\ frac {n} {m}} \ right), \}

Where

\left({\frac {n}{m}}\right)

{\ displaystyle \ left ({\ frac {n} {m}} \ right)}

is a Jacobi symbol , is a Dirichlet character modulo m ^[9] .

These are quadratic characters — in the general case, primitive quadratic characters arise exactly from the Kronecker – Jacobi symbol ^[10] .

Primitive characters and conductor

When passing from residues modulo N to residues modulo M, for any factor M of number N , information is lost. The effect of Dirichlet characters gives the opposite result - if $\chi *$ ${\ displaystyle \ chi *}$ is a character modulo M , it induces a character $\chi *$ ${\ displaystyle \ chi *}$ modulo N for any N multiple of M. A character is primitive if it is not induced by any character with a smaller modulus ^[3] .

If $\chi$ ${\ displaystyle \ chi}$ - the character modulo n and d divides n , we say that the module d is an induced module for $\chi$ ${\ displaystyle \ chi}$ , if $\chi (a)=1$ ${\ displaystyle \ chi (a) = 1}$ for all a mutually prime with n and 1 mod d ^[11] : the character is primitive if there is no smaller induced module ^[12] .

We can formalize this in various ways by defining characters. $\chi _{1}\mod N_{1}$ ${\ displaystyle \ chi _ {1} \ mod N_ {1}}$ and $\chi _{2}\mod N_{2}$ ${\ displaystyle \ chi _ {2} \ mod N_ {2}}$ as consistent , if for some module N such that N ₁ and N ₂ both share N , we have $\chi _{1}(n)=\chi _{2}(n)$ ${\ displaystyle \ chi _ {1} (n) = \ chi _ {2} (n)}$ for all n coprime with N , that is, there is some character $\chi *$ ${\ displaystyle \ chi *}$ originated as $\chi _{1}$ ${\ displaystyle \ chi _ {1}}$ so $\chi _{2}$ ${\ displaystyle \ chi _ {2}}$ . This is an equivalence relation on characters. The character with the smallest module in the equivalence class is primitive, and this smallest module is the conductor of characters in the class.

The non-primitiveness of characters can lead to the absence of in their L-functions .

Character Orthogonality

The orthogonality of the characters of the final group is transferred to the Dirichlet characters ^[13] .

If we fix the character $\chi$ ${\ displaystyle \ chi}$ modulo n , then

\sum _{a{\bmod {n}}}\chi (a)=0

{\ displaystyle \ sum _ {a {\ bmod {n}}} \ chi (a) = 0}

,

if $\chi$ ${\ displaystyle \ chi}$ not the main character, otherwise the amount is $\varphi (n)$ ${\ displaystyle \ varphi (n)}$ .

Similarly, if we fix the residue class a modulo n , then the sum over all characters gives

\sum _{\chi }\chi (a)=0

{\ displaystyle \ sum _ {\ chi} \ chi (a) = 0}

,

except for the case a = 1, when the sum is $\varphi (n)$ ${\ displaystyle \ varphi (n)}$ .

From this we conclude that any periodic function with period n over the class of residues coprime to n is a linear combination of Dirichlet characters ^[14] .

History

Dirichlet characters along with their $L$ ${\ displaystyle L}$ -dirits were introduced by Dirichlet in 1831, as part of the proof of the Dirichlet theorem on the infinity of the number of primes in arithmetic progressions. He studied them only for $s\in \mathbb {R}$ ${\ displaystyle s \ in \ mathbb {R}}$ and mostly when $s$ ${\ displaystyle s}$ tends to 1. The extension of these functions to the entire complex plane was obtained by Riemann in 1859.

Notes

↑ Montgomery, Vaughan, 2007 , p. 117-8.
↑ Montgomery, Vaughan, 2007 , p. 115.
↑ ¹ ² Montgomery, Vaughan, 2007 , p. 123.
↑ Fröhlich, Taylor, 1991 , p. 218.
↑ Fröhlich, Taylor, 1991 , p. 215.
↑ Apostol, 1976 , p. 139.
↑ ¹ ² ³ Apostol, 1976 , p. 138.
↑ Apostol, 1976 , p. 134.
↑ ¹ ² Montgomery, Vaughan, 2007 , p. 295.
↑ Montgomery, Vaughan, 2007 , p. 296.
↑ Apostol, 1976 , p. 166.
↑ Apostol, 1976 , p. 168.
↑ Apostol, 1976 , p. 140.
↑ Davenport, 1967 , p. 31–32.

Literature

Apostol TM Introduction to analytic number theory. - New York-Heidelberg: Springer-Verlag, 1976. - (Undergraduate Texts in Mathematics). - ISBN 978-0-387-90163-3 .
ApostolTM Some properties of completely multiplicative arithmetical functions // The American Mathematical Monthly. - 1971. - T. 78 , no. 3 . - S. 266–271 . - DOI : 10.2307 / 2317522 .
Harold Davenport. Multiplicative number theory. - Chicago: Markham, 1967. - T. 1. - (Lectures in advanced mathematics).
- Davenport G. Multiplicative Number Theory. - M .: "Science", 1971.
Helmut Hasse. Vorlesungen über Zahlentheorie. - 2nd revised. - Springer-Verlag . - T. 59. - (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen). see chapter 13.
- Hasse G. Lectures on number theory. - M .: Foreign Literature, 1953.
Mathar, RJ (2010), "Table of Dirichlet L-series and prime zeta modulo functions for small moduli", arΧiv : 1008.2547 [math.NT]

Hugh L Montgomery, Robert C. Vaughan. Multiplicative number theory. I. Classical theory. - Cambridge University Press , 2007. - T. 97. - (Cambridge Studies in Advanced Mathematics). - ISBN 0-521-84903-9 .
- Montgomery G. Multiplicative Number Theory. - M .: "The World", 1974.
Robert Spira. Calculation of Dirichlet L-Functions // Mathematics of Computation. - 1969. - T. 23 , no. 107 . - S. 489–497 . - DOI : 10.1090 / S0025-5718-1969-0247742-X .
Fröhlich A., Taylor MJ Algebraic number theory. - Cambridge University Press , 1991. - T. 27. - (Cambridge studies in advanced mathematics). - ISBN 0-521-36664-X .

Literature

Galochkin A. I., Nesterenko Yu. V., Shidlovsky A. B. Introduction to number theory. - M .: Publishing house of Moscow University, 1984.
Karatsuba A. A. Fundamentals of analytic number theory. - 3rd ed. - M .: URSS, 2004.

[_e432d67eb054af6b-1] Montgomery, Vaughan, 2007 , p. 117-8.

[_f3b949ef7bb7e706-2] Montgomery, Vaughan, 2007 , p. 115.

[_f3b94aef7bb7e8d5-3] ¹ ² Montgomery, Vaughan, 2007 , p. 123.

[_33e29a09afb16ce0-4] Fröhlich, Taylor, 1991 , p. 218.

[_33e29a09afb16ced-5] Fröhlich, Taylor, 1991 , p. 215.

[_c30ac224927b0cfd-6] Apostol, 1976 , p. 139.

[_c30ac224927b0cfc-7] ¹ ² ³ Apostol, 1976 , p. 138.

[_c30ac224927b0cf0-8] Apostol, 1976 , p. 134.

[_f3c3d1ef7bc122e5-9] ¹ ² Montgomery, Vaughan, 2007 , p. 295.

[_f3c3d1ef7bc122e6-10] Montgomery, Vaughan, 2007 , p. 296.

[_c30abf24927b079d-11] Apostol, 1976 , p. 166.

[_c30abf24927b0793-12] Apostol, 1976 , p. 168.

[_c30ac124927b0b21-13] Apostol, 1976 , p. 140.

[_09a5d9ce8595eb27-14] Davenport, 1967 , p. 31–32.

Character (number theory)

Content

Axiomatic definition

Building through Deduction Classes

Deduction classes

Dirichlet characters

Examples

Some character tables

Modulo 1

Modulo 2

Modulo 3

Modulo 4

Modulo 5

Modulo 6

Modulo 7

Modulo 8

Modulo 9

Modulo 10

Examples

Primitive characters and conductor

Character Orthogonality

History

See also

Notes

Literature

Literature

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