Dirichlet character

Axiomatic definition

A Dirichlet character is any function χ from the integers Z to the complex numbers C such that χ 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.

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

1. 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 χ is a character to the modulus k. This is equivalent to saying that

1. 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

1. 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. A character is called real if it assumes real values only. A character which is not real is called complex.

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

Construction via residue classes

Dirichlet characters may be viewed in terms of the character group of the unit group of the ring Z/kZ, as extended residue class characters.

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: n ^ = { m ∣ m ≡ n mod k } . That is, the residue class 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 m n ^ = m ^ n ^ and φ again denotes Euler's phi function. The identity in this group is the residue class 1 ^ and the inverse of m ^ is the residue class n ^ where m ^ n ^ = 1 ^ , i.e., m n ≡ 1 mod k . For example, for k=6, the set of units is { 1 ^ , 5 ^ } 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^*.

Dirichlet characters

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

χ : ( Z / k Z ) ∗ → C ∗ ,

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 χ 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.

The principal character χ 0 modulo k has the properties

χ 0 ( n ) = 1 if gcd(n, k) = 1 and χ 0 ( n ) = 0 if gcd(n, k) > 1.

The associated character of the multiplicative group (Z/kZ)^* is the principal character which always takes the value 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.

There are φ(n) Dirichlet characters modulo n.

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 12. The characters χ₀ are the principal characters.

Modulus 1

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

Note that χ is wholly determined by χ(0) since 0 generates the group of units modulo 1.

This is the trivial character.

The Dirichlet L-series for χ 0 ( n ) is the Riemann zeta function

ζ ( s ) = ∑ n = 1 ∞ 1 n s = 1 1 s + 1 2 s + 1 3 s + 1 4 s + ⋯ .

Modulus 2

There is φ ( 2 ) = 1 character modulo 2:

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

Modulus 3

There are φ ( 3 ) = 2 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 φ ( 4 ) = 2 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 χ 0 ( n ) is the Dirichlet lambda function (closely related to the Dirichlet eta function)

L ( χ 0 , s ) = ( 1 − 2 − s ) ζ ( s )

where ζ ( s ) is the Riemann zeta-function. The L-series for χ 1 ( n ) is the Dirichlet beta-function

L ( χ 1 , s ) = β ( s ) .

Modulus 5

There are φ ( 5 ) = 4 characters modulo 5. In the table below, i is the imaginary unit.

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

Modulus 6

There are φ ( 6 ) = 2 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 φ ( 7 ) = 6 characters modulo 7. In the table below, ω = exp ⁡ ( π i 3 ) = 1 2 + 3 2 i .

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

Modulus 8

There are φ ( 8 ) = 4 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 φ ( 9 ) = 6 characters modulo 9. In the table below, ω = exp ⁡ ( π i 3 ) = 1 2 + 3 2 i .

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

Modulus 10

There are φ ( 10 ) = 4 characters modulo 10. In the table below, i is the imaginary unit.

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

Modulus 11

There are φ ( 11 ) = 10 characters modulo 11. In the table below, ω = exp ⁡ ( π i 5 ) = 5 + 1 4 + 10 − 2 5 4 i .

Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 11.

Modulus 12

There are φ ( 12 ) = 4 characters modulo 12.

Note that χ is wholly determined by χ(5) and χ(7) since 5 and 7 generate the group of units modulo 12.

Examples

If p is an odd prime number, then the function

χ ( n ) = ( n p ) ,   where ( n p ) is the Legendre symbol, is a primitive Dirichlet character modulo p.

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

χ ( n ) = ( n m ) ,   where ( n m ) is the Jacobi symbol, is a Dirichlet character modulo m.

These are examples of real characters. In general, all real characters arise from the Kronecker symbol.

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.

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: equivalently, χ(a) = χ(b) whenever a, b are congruent mod d and each coprime to n. A character is primitive if there is no smaller induced modulus.

We can formalize this differently by defining characters χ₁ mod N₁ and χ₂ mod N₂ to be co-trained if for some modulus N such that N₁ and N₂ both divide N we have χ₁(n) = χ₂(n) for all n coprime to N: that is, there is some character χ* induced by each of χ₁ and χ₂. 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. If we fix a character χ modulo n then the sum

∑ a mod n χ ( a ) = 0  

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

∑ χ χ ( a ) = 0  

unless a ≡ 1 ( mod n ) 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.

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.