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

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

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.

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.
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 χ 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. 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 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.

The set of units modulo k forms an abelian group of order $\phi(k)$ , where group multiplication is given by $\widehat{mn}=\hat{m}\hat{n}$ 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.

Dirichlet characters

A Dirichlet character modulo k is a group homomorphism $\chi$ from the unit group modulo k to the non-zero complex numbers

$\chi�: (\mathbb{Z}/k\mathbb{Z})^* \to \mathbb{C}$ ,

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

$\chi_1(n)=1$ 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.

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 $\phi(1)=1$ character modulo 1:

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	1	2
$\chi_1(n)$	1	1
$\chi_2(n)$	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	1	2	3
$\chi_1(n)$	1	0	1
$\chi_2(n)$	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 the Dirichlet lambda function (closely related to the Dirichlet eta function)

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

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	1	2	3	4
$\chi_1(n)$	1	1	1	1
$\chi_2(n)$	1	i	−i	−1
$\chi_3(n)$	1	−1	−1	1
$\chi_4(n)$	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	1	2	3	4	5
$\chi_1(n)$	1	0	0	0	1
$\chi_2(n)$	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	1	2	3	4	5	6
$\chi_1(n)$	1	1	1	1	1	1
$\chi_2(n)$	1	ω²	ω	−ω	−ω²	−1
$\chi_3(n)$	1	−ω	ω²	ω²	−ω	1
$\chi_4(n)$	1	1	−1	1	−1	−1
$\chi_5(n)$	1	ω²	−ω	−ω	ω²	1
$\chi_6(n)$	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 $\phi(8)=4$ characters modulo 8.

χ \ n	1	2	3	4	5	6	7
$\chi_1(n)$	1	0	1	0	1	0	1
$\chi_2(n)$	1	0	1	0	−1	0	−1
$\chi_3(n)$	1	0	−1	0	1	0	−1
$\chi_4(n)$	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	1	2	3	4	5	6	7	8
$\chi_1(n)$	1	1	0	1	1	0	1	1
$\chi_2(n)$	1	ω	0	ω²	−ω²	0	−ω	−1
$\chi_3(n)$	1	ω²	0	−ω	−ω	0	ω²	1
$\chi_4(n)$	1	−1	0	1	−1	0	1	−1
$\chi_5(n)$	1	−ω	0	ω²	ω²	0	−ω	1
$\chi_6(n)$	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 $\phi(10)=4$ characters modulo 10.

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

References

See chapter 6 of Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929
Spira, Robert (1969). "Calculation of Dirichlet L-Functions". Mathematics of Computation 23 (107): 489–497. doi:10.1090/S0025-5718-1969-0247742-X. MR 0247742.
Apostol, T. M. (1971). "Some properties of completely multiplicative arithmetical functions". The American Mathematical Monthly 78 (3): 266–271. doi:10.2307/2317522. JSTOR 2317522. MR 0279053.
Hasse, Helmut (1964). Vorlesungen über Zahlentheorie. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. Springer. MR 0188128. see chapter 13.

Mathar, R. J. (2010). "Table of Dirichlet L-series and prime zeta modulo functions for small moduli". arXiv:1008.2547 [math.NT].