# Dirichlet character

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

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 Peter Gustav Lejeune Dirichlet.

## Axiomatic definition

A Dirichlet character is any function from the integers to the complex numbers such that has the following properties:^{[1]}

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

From this definition, several other properties can be deduced.
By property 3), χ(1)=χ(1×1)=χ(1)χ(1). Since gcd(1, *k*) = 1, property 2) says χ(1) ≠ 0, so

- χ(1) = 1.

Properties 3) and 4) show that every Dirichlet character χ is completely multiplicative.

Property 1) says that a character is periodic with period *k*; we say that is a character to the **modulus** *k*. This is equivalent to saying that

- If
*a*≡*b*(mod*k*) then χ(*a*) = χ(*b*).

If gcd(*a*,*k*) = 1, Euler's theorem says that *a*^{φ(k)} ≡ 1 (mod *k*) (where φ(*k*) is the totient function). Therefore by 5) and 4), χ(*a*^{φ(k)}) = χ(1) = 1, and by 3), χ(*a*^{φ(k)}) =χ(*a*)^{φ(k)}. So

- For all
*a*relatively prime to*k*, χ(*a*) is a φ(*k*)-th complex root of unity.

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.^{[2]} A character is called **real** if it assumes real values only. A character which is not real is called **complex**.^{[3]}

The **sign** of the character depends on its value at −1. Specifically, is said to be **odd** if and **even** if .

## Construction via residue classes

Dirichlet characters may be viewed in terms of the character group of the
unit group of the ring **Z**/*k***Z**, as *extended residue class characters*.^{[4]}

### 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*:
That is, the residue class is the coset of *n* in the quotient ring **Z**/*k***Z**.

The set of units modulo *k* forms an abelian group of order , where group multiplication is given by
and
again denotes Euler's phi function.
The identity in this group is the residue class and the inverse of is the residue class where
, i.e., . For example, for *k*=6, the set of units is because 0, 2, 3, and 4 are not coprime to 6.

The character group of (**Z**/*k*)^{*} consists of the *residue class characters*. A residue class character θ on (**Z**/*k*)^{*} is **primitive** if there is no proper divisor *d* of *k* such that θ factors as a map (**Z**/*k*)^{*} → (**Z**/*d*)^{*} → **C**^{*}.^{[5]}

### Dirichlet characters

The definition of a Dirichlet character modulo *k* ensures that it restricts to a character of the unit group modulo *k*:^{[6]} a group homomorphism from (**Z**/*k***Z**)^{*} to the non-zero complex numbers

with values that are necessarily roots of unity since the units modulo *k* form a finite group. In the opposite direction, given a group homomorphism on the unit group modulo *k*, we can lift to a completely multiplicative function on integers relatively prime to *k* and then extend this function to all integers by defining it to be 0 on integers having a non-trivial factor in common with *k*. The resulting function will then be a Dirichlet character.^{[7]}

The **principal character** modulo *k* has the properties^{[7]}

The associated character of the multiplicative group (**Z**/*k***Z**)^{*} is the *principal* character which always takes the value 1.^{[8]}

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.

There are φ(*n*) Dirichlet characters modulo *n*.^{[7]}

## 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 χ_{1} are the principal characters.

### Modulus 1

This is the trivial character.

### Modulus 2

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

### Modulus 3

There are characters modulo 3:

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

### Modulus 4

There are characters modulo 4:

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

The Dirichlet *L*-series for is
the Dirichlet lambda function (closely related to the Dirichlet eta function)

where is the Riemann zeta-function. The *L*-series for is the Dirichlet beta-function

### Modulus 5

There are characters modulo 5. In the tables, *i* is the imaginary constant.

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

### Modulus 6

There are characters modulo 6:

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

### Modulus 7

There are characters modulo 7. In the table below,

χ \ *n***0****1****2****3****4****5****6**0 1 1 1 1 1 1 0 1 ω ^{2}ω −ω −ω ^{2}−1 0 1 −ω ω ^{2}ω ^{2}−ω 1 0 1 1 −1 1 −1 −1 0 1 ω ^{2}−ω −ω ω ^{2}1 0 1 −ω −ω ^{2}ω ^{2}ω −1

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

### Modulus 8

There are characters modulo 8.

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 characters modulo 9. In the table below,

χ \ *n***0****1****2****3****4****5****6****7****8**0 1 1 0 1 1 0 1 1 0 1 ω 0 ω ^{2}−ω ^{2}0 −ω −1 0 1 ω ^{2}0 −ω −ω 0 ω ^{2}1 0 1 −1 0 1 −1 0 1 −1 0 1 −ω 0 ω ^{2}ω ^{2}0 −ω 1 0 1 −ω ^{2}0 −ω ω 0 ω ^{2}−1

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

### Modulus 10

There are characters modulo 10.

χ \ *n***0****1****2****3****4****5****6****7****8****9**0 1 0 1 0 0 0 1 0 1 0 1 0 *i*0 0 0 − *i*0 −1 0 1 0 −1 0 0 0 −1 0 1 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 an odd prime number, then the function

- where is the Legendre symbol, is a primitive Dirichlet character modulo
*p*.^{[9]}

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

- where is the Jacobi symbol, is a Dirichlet character modulo
*m*.^{[9]}

These are *quadratic characters*: in general, the primitive quadratic characters arise precisely from the Kronecker symbol.^{[10]}

## Primitive characters and conductor

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 *induces* a character χ* mod *N* for any multiple *N* of *M*. A character is **primitive** if it is not induced by any character of smaller modulus.^{[3]}

If χ is a character mod *n* and *d* divides *n*, then we say that the modulus *d* is an *induced modulus* for χ if *a* coprime to *n* and 1 mod *d* implies χ(*a*)=1:^{[11]} equivalently, χ(*a*) = χ(*b*) whenever *a*, *b* are congruent mod *d* and each coprime to *n*.^{[12]} A character is primitive if there is no smaller induced modulus.^{[12]}

We can formalize this differently by defining characters χ_{1} mod *N*_{1} and χ_{2} mod *N*_{2} to be **co-trained** if for some modulus *N* such that *N*_{1} and *N*_{2} both divide *N* we have χ_{1}(*n*) = χ_{2}(*n*) for all *n* coprime to *N*: that is, there is some character χ* induced by each of χ_{1} and χ_{2}. This is an equivalence relation on characters. A character with the smallest modulus in an equivalence class is primitive and this smallest modulus is the **conductor** of the characters in the class.

Imprimitivity of characters can lead to missing Euler factors in their L-functions.

## Character orthogonality

The orthogonality relations for characters of a finite group transfer to Dirichlet characters.^{[13]} If we fix a character χ modulo *n* then the sum

unless χ is principal, in which case the sum is φ(*n*). Similarly, if we fix a residue class *a* modulo *n* and sum over all characters we have

unless in which case the sum is φ(*n*). We deduce that any periodic function with period *n* supported on the residue classes prime to *n* is a linear combination of Dirichlet characters.^{[14]}

## History

Dirichlet characters and their *L*-series were introduced by 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.

## See also

- Hecke character (also known as grössencharacter)
- Character sum
- Gaussian sum
- Primitive root modulo
*n* - Selberg class

## References

- ↑ Montgomery & Vaughan (2007) pp.117–8
- ↑ Montgomery & Vaughan (2007) p.115
- ↑
^{3.0}^{3.1}Montgomery & Vaughan (2007) p.123 - ↑ Fröhlich & Taylor (1991) p.218
- ↑ Frohlich & Taylor (1991) p.215
- ↑ Apostol (1976) p.139
- ↑
^{7.0}^{7.1}^{7.2}Apostol (1976) p.138 - ↑ Apostol (1976) p.134
- ↑
^{9.0}^{9.1}Montgomery & Vaughan (2007) p.295 - ↑ Montgomery & Vaughan (2007) p.296
- ↑ Apostol (1976) p.166
- ↑
^{12.0}^{12.1}Apostol (1976) p.168 - ↑ Apostol (1976) p.140
- ↑ Davenport (1967) pp.31–32

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