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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## Axiomatic definition

A Dirichlet character is any function χ from the integers to the complex numbers which has the following properties:## 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, $chi(n)^\{phi(k)\}\; =\; chi(n^\{phi(k)\})\; =\; chi(1)\; =\; 1$. 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:
$hat\{n\}=\{m\; |\; m\; equiv\; n\; mod\; k\; \}.$
That is, the residue class $hat\{n\}$ is the coset of n in the quotient ring Z/kZ.### 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 ## 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

There is $phi(1)=1$ character modulo 1:

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

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.

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 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,

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.

The set of units modulo k forms an abelian group of order $phi(k)$, where group multiplication is given by $hat\{mn\}=hat\{m\}hat\{n\}$ and $phi$ again denotes 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 $hat\{m\}\; hat\{n\}\; =\; hat\{1\}$, i.e., $m\; n\; equiv\; 1\; mod\; k$. 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.

- $chi\_1(n)=1$ if gcd(n, k) = 1 and

- $chi\_1(n)=0$ 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.

χ n 0 $chi\_1(n)$ 1 This is the trivial character.

### Modulus 2

There is $phi(2)=1$ character modulo 2:χ n 0 1 $chi\_1(n)$ 0 1 Note that χ is wholly determined by χ(1) since 1 generates the group of units modulo 2.

### Modulus 3

There are $phi(3)=2$ characters modulo 3:χ n 0 1 2 $chi\_1(n)$ 0 1 1 $chi\_2(n)$ 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 $phi(4)=2$ characters modulo 4:χ n 0 1 2 3 $chi\_1(n)$ 0 1 0 1 $chi\_2(n)$ 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 $chi\_1(n)$ is

- $L(chi\_1,\; s)=\; (1-2^\{-s\})zeta(s),$

where $zeta(s)$ is the Riemann zeta-function. The L-series for $chi\_2(n)$ is the Dirichlet beta-function

- $L(chi\_2,\; s)=beta(s).,$

### Modulus 5

There are $phi(5)=4$ characters modulo 5. In the tables, i is a square root of $-1$.χ n 0 1 2 3 4 $chi\_1(n)$ 0 1 1 1 1 $chi\_2(n)$ 0 1 i −i −1 $chi\_3(n)$ 0 1 −1 −1 1 $chi\_4(n)$ 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 $phi(6)=2$ characters modulo 6:χ n 0 1 2 3 4 5 $chi\_1(n)$ 0 1 0 0 0 1 $chi\_2(n)$ 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 $phi(7)=6$ characters modulo 7. In the table below, $omega\; =\; exp(pi\; i\; /3).$χ n 0 1 2 3 4 5 6 $chi\_1(n)$ 0 1 1 1 1 1 1 $chi\_2(n)$ 0 1 ω ^{2}ω −ω −ω ^{2}−1 $chi\_3(n)$ 0 1 −ω ω ^{2}ω ^{2}−ω 1 $chi\_4(n)$ 0 1 1 −1 1 −1 −1 $chi\_5(n)$ 0 1 ω ^{2}−ω −ω ω ^{2}1 $chi\_6(n)$ 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 $phi(8)=4$ characters modulo 8.χ n 0 1 2 3 4 5 6 7 $chi\_1(n)$ 0 1 0 1 0 1 0 1 $chi\_2(n)$ 0 1 0 1 0 −1 0 −1 $chi\_3(n)$ 0 1 0 −1 0 1 0 −1 $chi\_4(n)$ 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 $phi(9)=6$ characters modulo 9. In the table below, $omega\; =\; exp(pi\; i\; /3).$χ n 0 1 2 3 4 5 6 7 8 $chi\_1(n)$ 0 1 1 0 1 1 0 1 1 $chi\_2(n)$ 0 1 ω 0 ω ^{2}−ω ^{2}0 −ω −1 $chi\_3(n)$ 0 1 ω ^{2}0 −ω −ω 0 ω ^{2}1 $chi\_4(n)$ 0 1 −1 0 1 −1 0 1 −1 $chi\_5(n)$ 0 1 −ω 0 ω ^{2}ω ^{2}0 −ω 1 $chi\_6(n)$ 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 $phi(10)=4$ characters modulo 10.χ n 0 1 2 3 4 5 6 7 8 9 $chi\_1(n)$ 0 1 0 1 0 0 0 1 0 1 $chi\_2(n)$ 0 1 0 i 0 0 0 −i 0 −1 $chi\_3(n)$ 0 1 0 −1 0 0 0 −1 0 1 $chi\_4(n)$ 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

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

More generally, if m is an odd number the function

- $chi(n)\; =\; left(frac\{n\}\{m\}right),$ where $left(frac\{n\}\{m\}right)$ 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.## See also

## References

- See chapter 6 of

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