Dirichlet character

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In number theory, Dirichlet characters are certain arithmetic functions that capture some important properties of the cyclic group. 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 about the infinitude of primes in 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

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