principal character

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(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.

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 ab (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(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.

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.

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(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     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 ω2 ω −ω −ω2 −1
chi_3(n) 0 1 −ω ω2 ω2 −ω 1
chi_4(n) 0 1 1 −1 1 −1 −1
chi_5(n) 0 1 ω2 −ω −ω ω2 1
chi_6(n) 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(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 ω2 −ω2 0 −ω −1
chi_3(n) 0 1 ω2 0 −ω −ω 0 ω2 1
chi_4(n) 0 1 −1 0 1 −1 0 1 −1
chi_5(n) 0 1 −ω 0 ω2 ω2 0 −ω 1
chi_6(n) 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(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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