Dirichlet character

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In number theory, Dirichlet characters are certain arithmetic functions which arise from multiplicative characters on the units of $\mathbb Z / n \mathbb Z$ . Dirichlet characters are used to define Dirichlet L-functions, which are meromorphic functions with a variety of interesting analytic properties. Dirichlet characters are named in honour of Johann Peter Gustav Lejeune Dirichlet.

1 Axiomatic definition
2 Construction via residue classes
- 2.1 Residue classes
- 2.2 Dirichlet characters
3 A few character tables
4 Examples
5 History
6 See also
7 References

[edit] Axiomatic definition

A Dirichlet character is any function χ from the integers to the complex numbers which has the following properties:

There exists a positive integer k such that χ(n) = χ(n + k) for all n.
χ(n) = 0 for every n with gcd(n,k) > 1.
χ(mn) = χ(m)χ(n) for all integers m and n.
χ(1) = 1.

Condition 1 says that a character is periodic with period k; we say that χ is a character to the modulus k. Condition 3 says that a character is completely multiplicative. The unique character of period 1 is called the trivial character. Note that any character vanishes at 0 except the trivial one, which is 1 on all integers. A character is called principal if it assumes the value 1 for arguments coprime to its modulus and otherwise is 0. A character is called real if it assumes real values only. A character which is not real is called complex.

[edit] Construction via residue classes

The last two properties show that every Dirichlet character χ is completely multiplicative. One can show that χ(n) is a φ(k)th root of unity whenever n and k are coprime, and where φ(k) is the totient function. Dirichlet characters may be viewed in terms of the character group of the unit group of the ring Z/kZ, as given below.

[edit] Residue classes

Given an integer k, one defines the residue class of an integer n as the set of all integers congruent to n modulo k: $\hat{n}=\{m | m \equiv n \mod k \}.$ That is, the residue class $\hat{n}$ is the coset of n in the quotient ring Z/kZ.

The set of units modulo k forms an abelian group of order $φ(k)$ , where group multiplication is given by $\hat{mn}=\hat{m}\hat{n}$ and $φ$ denoted Euler's phi function. The identity in this group is the residue class $\hat{1}$ and the inverse of $\hat{m}$ is the residue class $\hat{n}$ where $mn=1 \mod k$ . For example, for k=6, the set of units is $\{\hat{1}, \hat{5}\}$ because 0, 2, 3, and 4 are not coprime to 6.

[edit] Dirichlet characters

A Dirichlet character modulo k is a group homomorphism $χ$ from the unit group modulo k to the non-zero complex numbers (necessarily with values that are roots of unity since the units modulo k form a finite group). We can lift $χ$ to a completely multiplicative function on integers relatively prime to k and then to all integers by extending the function to be 0 on integers having a non-trivial factor in common with k. The principal character $χ 1$ modulo k has the properties

χ 1 (n) = 1

if gcd(n, k) = 1 and

χ 1 (n) = 0

if gcd(n, k) > 1.

When k is 1, the principal character modulo k is equal to 1 at all integers. For k greater than 1, the principal character modulo k vanishes at integers having a non-trivial common factor with k and is 1 at other integers.

[edit] A few character tables

The tables below help illustrate the nature of a Dirichlet character. They present all of the characters from modulus 1 to modulus 7. The characters χ₁ are the principal characters.

[edit] Modulus 1

There is one (1 = φ(1)) character modulo 1:

χ \ n	1
$χ 1 (n)$	1

This is the trivial character.

[edit] Modulus 2

There is one (1=φ(2)) character to the modulus 2:

χ \ n	1	2
$χ 1 (n)$	1	0

[edit] Modulus 3

There are $φ(3) = 2$ characters modulo 3:

χ \ n	1	2	3
$χ 1 (n)$	1	1	0
$χ 2 (n)$	1	−1	0

[edit] Modulus 4

There are $φ(4) = 2$ characters modulo 4:

χ \ n	1	2	3	4
$χ 1 (n)$	1	0	1	0
$χ 2 (n)$	1	0	−1	0

The Dirichlet L-series (defined below) for $χ 1 (n)$ is

$L(\chi_1, s)= (1-2^{-s})\zeta(s)\,$

where $ζ(s)$ is the Riemann zeta-function. The L-series for $χ 2 (n)$ is the Dirichlet beta-function

$L(\chi_2, s)=\beta(s).\,$

[edit] Modulus 5

There are $φ(5) = 4$ characters modulo 5. In the tables, i is a square root of $- 1$ .

χ \ n	1	2	3	4	5
$χ 1 (n)$	1	1	1	1	0
$χ 2 (n)$	1	−1	−1	1	0
$χ 3 (n)$	1	i	−i	−1	0
$χ 4 (n)$	1	−i	i	−1	0

[edit] Modulus 6

There are $φ(6) = 2$ characters modulo 6:

χ \ n	1	2	3	4	5	6
$χ 1 (n)$	1	0	0	0	1	0
$χ 2 (n)$	1	0	0	0	−1	0

[edit] Modulus 7

There are $φ(7) = 6$ characters modulo 7. In the table below, $ω = exp(π i / 3).$

χ \ n	1	2	3	4	5	6	7
$χ 1 (n)$	1	1	1	1	1	1	0
$χ 2 (n)$	1	1	−1	1	−1	−1	0
$χ 3 (n)$	1	ω²	−ω	−ω	−ω²	1	0
$χ 4 (n)$	1	ω²	ω	−ω	−ω²	−1	0
$χ 5 (n)$	1	−ω	ω²	ω²	−ω	1	0
$χ 6 (n)$	1	−ω	−ω²	ω²	ω	−1	0

[edit] Examples

If p is a prime number, then the function

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

where $\left(\frac{n}{p}\right)$ is the Legendre symbol, is a Dirichlet character modulo p.

[edit] History

Dirichlet characters and their L-series were introduced by Johann Peter Gustav Lejeune Dirichlet, in 1831, in order to prove Dirichlet's theorem on arithmetic progressions. He only studied them for real s and especially as s tends to 1. The extension of these functions to complex s in the whole complex plane was obtained by Bernhard Riemann in 1859.

[edit] See also

[edit] References

Tom M. Apostol Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9 See chapter 6.

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Category: Zeta and L-functions