Definitions

# Dirichlet character

In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of $mathbb Z / k mathbb Z$. Dirichlet characters are used to define Dirichlet L-functions, which are meromorphic functions with a variety of interesting analytic properties. If $chi$ is a Dirichlet character, one defines its Dirichlet L-series by

$L\left(chi,s\right) = sum_\left\{n=1\right\}^infty frac\left\{chi\left(n\right)\right\}\left\{n^s\right\}$

where s is a complex number with real part > 1. By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane. Dirichlet L-functions are generalizations of the Riemann zeta-function and appear prominently in the generalized Riemann hypothesis.

Dirichlet characters are named in honour of Johann Peter Gustav Lejeune Dirichlet.

## Axiomatic definition

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

`1) There exists a positive integer k such that χ(n) = χ(n + k) for all n.`
`2) If gcd(n,k) > 1 then χ(n) = 0; if  gcd(n,k) =  1 then χ(n) ≠ 0.`
`3) χ(mn) = χ(m)χ(n) for all integers m and n.`

These consequences are important:

By property 3), χ(1)=χ(1×1)=χ(1)χ(1); since gcd(1, k) = 1, property 2) says χ(1) ≠ 0, so

`4) χ(1) = 1.`
1) says that a character is periodic with period k; we say that χ is a character to the modulus k. This is equivalent to saying that

`5) if a ≡ b (mod k) then χ(a) = χ(b).`

If gcd(a,k) = 1, Euler's theorem says that aφ(k) ≡ 1 (mod k) (where φ(k) is the totient function). Therefore by 5) and 4), χ(aφ(k)) = χ(1) = 1, and by 3), χ(aφ(k)) =χ(a)φ(k) . So for all a relatively prime to k,

`6) χ(a) is a φ(k)-th complex root of unity.`

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.

## 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: By simple group arithmetic, $chi\left(n\right)^\left\{phi\left(k\right)\right\} = chi\left(n^\left\{phi\left(k\right)\right\}\right) = chi\left(1\right) = 1$. Dirichlet characters may be viewed in terms of the character group of the unit group of the ring Z/kZ, as given below.

### 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: $hat\left\{n\right\}=\left\{m | m equiv n mod k \right\}.$ That is, the residue class $hat\left\{n\right\}$ is the coset of n in the quotient ring Z/kZ.

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

### Dirichlet characters

A Dirichlet character modulo k is a group homomorphism $chi$ 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 $chi$ 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 $chi_1$ modulo k has the properties

$chi_1\left(n\right)=1$ if gcd(n, k) = 1 and
$chi_1\left(n\right)=0$ 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.

## 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 10. The characters χ1 are the principal characters.

### Modulus 1

There is $phi\left(1\right)=1$ character modulo 1:

χ  n     0
$chi_1\left(n\right)$ 1

This is the trivial character.

### Modulus 2

There is $phi\left(2\right)=1$ character modulo 2:

χ  n     0     1
$chi_1\left(n\right)$ 0 1

Note that χ is wholly determined by χ(1) since 1 generates the group of units modulo 2.

### Modulus 3

There are $phi\left(3\right)=2$ characters modulo 3:

χ  n     0     1     2
$chi_1\left(n\right)$ 0 1 1
$chi_2\left(n\right)$ 0 1 −1

Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 3.

### Modulus 4

There are $phi\left(4\right)=2$ characters modulo 4:

χ  n     0     1     2     3
$chi_1\left(n\right)$ 0 1 0 1
$chi_2\left(n\right)$ 0 1 0 −1

Note that χ is wholly determined by χ(3) since 3 generates the group of units modulo 4.

The Dirichlet L-series for $chi_1\left(n\right)$ is

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

where $zeta\left(s\right)$ is the Riemann zeta-function. The L-series for $chi_2\left(n\right)$ is the Dirichlet beta-function

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

### Modulus 5

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

χ  n     0     1     2     3     4
$chi_1\left(n\right)$ 0 1 1 1 1
$chi_2\left(n\right)$ 0 1 i −i −1
$chi_3\left(n\right)$ 0 1 −1 −1 1
$chi_4\left(n\right)$ 0 1 i i −1

Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 5.

### Modulus 6

There are $phi\left(6\right)=2$ characters modulo 6:

χ  n     0     1     2     3     4     5
$chi_1\left(n\right)$ 0 1 0 0 0 1
$chi_2\left(n\right)$ 0 1 0 0 0 −1

Note that χ is wholly determined by χ(5) since 5 generates the group of units modulo 6.

### Modulus 7

There are $phi\left(7\right)=6$ characters modulo 7. In the table below, $omega = exp\left(pi i /3\right).$

χ  n     0     1     2     3     4     5     6
$chi_1\left(n\right)$ 0 1 1 1 1 1 1
$chi_2\left(n\right)$ 0 1 ω2 ω −ω −ω2 −1
$chi_3\left(n\right)$ 0 1 −ω ω2 ω2 −ω 1
$chi_4\left(n\right)$ 0 1 1 −1 1 −1 −1
$chi_5\left(n\right)$ 0 1 ω2 −ω −ω ω2 1
$chi_6\left(n\right)$ 0 1 −ω −ω2 ω2 ω −1

Note that χ is wholly determined by χ(3) since 3 generates the group of units modulo 7.

### Modulus 8

There are $phi\left(8\right)=4$ characters modulo 8.

χ  n     0     1     2     3     4     5     6     7
$chi_1\left(n\right)$ 0 1 0 1 0 1 0 1
$chi_2\left(n\right)$ 0 1 0 1 0 −1 0 −1
$chi_3\left(n\right)$ 0 1 0 −1 0 1 0 −1
$chi_4\left(n\right)$ 0 1 0 −1 0 −1 0 1

Note that χ is wholly determined by χ(3) and χ(5) since 3 and 5 generate the group of units modulo 8.

### Modulus 9

There are $phi\left(9\right)=6$ characters modulo 9. In the table below, $omega = exp\left(pi i /3\right).$

χ  n     0     1     2     3     4     5     6     7     8
$chi_1\left(n\right)$ 0 1 1 0 1 1 0 1 1
$chi_2\left(n\right)$ 0 1 ω 0 ω2 −ω2 0 −ω −1
$chi_3\left(n\right)$ 0 1 ω2 0 −ω −ω 0 ω2 1
$chi_4\left(n\right)$ 0 1 −1 0 1 −1 0 1 −1
$chi_5\left(n\right)$ 0 1 −ω 0 ω2 ω2 0 −ω 1
$chi_6\left(n\right)$ 0 1 −ω2 0 −ω ω 0 ω2 −1

Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 9.

### Modulus 10

There are $phi\left(10\right)=4$ characters modulo 10.

χ  n     0     1     2     3     4     5     6     7     8     9
$chi_1\left(n\right)$ 0 1 0 1 0 0 0 1 0 1
$chi_2\left(n\right)$ 0 1 0 i 0 0 0 i 0 −1
$chi_3\left(n\right)$ 0 1 0 −1 0 0 0 −1 0 1
$chi_4\left(n\right)$ 0 1 0 i 0 0 0 i 0 −1

Note that χ is wholly determined by χ(3) since 3 generates the group of units modulo 10.

## Examples

If p is a prime number, then the function

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

More generally, if m is an odd number the function

$chi\left(n\right) = left\left(frac\left\{n\right\}\left\{m\right\}right\right),$ where $left\left(frac\left\{n\right\}\left\{m\right\}right\right)$ is the Jacobi symbol, is a Dirichlet character modulo m. These are called the quadratic characters.

## Conductors

Residues mod N give rise to residues mod M, for any factor M of N, by discarding some information. The effect on Dirichlet characters goes in the opposite direction: if χ is a character mod M, it gives rise to a character χ* mod N for any multiple N of M. With some attention to the values at which characters take the value 0, one gets the concept of a primitive Dirichlet character, one that does not arise from a factor; and the associated idea of conductor, i.e. the natural (smallest) modulus for a character. Imprimitive characters can cause missing Euler factors in L-functions.

## 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.