Dirichlet 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 χ 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.

Contents

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.

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 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

  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 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 φ(k), where group multiplication is given by \widehat{mn}=\hat{m}\hat{n} and ϕ 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 χ 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 χ 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 χ1 modulo k has the properties

χ1(n) = 1 if gcd(n, k) = 1 and
χ1(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 ϕ(1) = 1 character modulo 1:

χ \ n     0  
χ1(n) 1

This is the trivial character.

Modulus 2

There is ϕ(2) = 1 character modulo 2:

χ \ n     0     1  
χ1(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 ϕ(3) = 2 characters modulo 3:

χ \ n     0     1     2  
χ1(n) 0 1 1
χ2(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 ϕ(4) = 2 characters modulo 4:

χ \ n     0     1     2     3  
χ1(n) 0 1 0 1
χ2(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 χ1(n) is the Dirichlet lambda function (closely related to the Dirichlet eta function)

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

where ζ(s) is the Riemann zeta-function. The L-series for χ2(n) is the Dirichlet beta-function

L(\chi_2, s)=\beta(s).\,

Modulus 5

There are ϕ(5) = 4 characters modulo 5. In the tables, i is a square root of − 1.

χ \ n     0     1     2     3     4  
χ1(n) 0 1 1 1 1
χ2(n) 0 1 i −i −1
χ3(n) 0 1 −1 −1 1
χ4(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 ϕ(6) = 2 characters modulo 6:

χ \ n     0     1     2     3     4     5  
χ1(n) 0 1 0 0 0 1
χ2(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 ϕ(7) = 6 characters modulo 7. In the table below, ω = exp(πi / 3).

χ \ n     0     1     2     3     4     5     6  
χ1(n) 0 1 1 1 1 1 1
χ2(n) 0 1 ω2 ω −ω −ω2 −1
χ3(n) 0 1 −ω ω2 ω2 −ω 1
χ4(n) 0 1 1 −1 1 −1 −1
χ5(n) 0 1 ω2 −ω −ω ω2 1
χ6(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 ϕ(8) = 4 characters modulo 8.

χ \ n     0     1     2     3     4     5     6     7  
χ1(n) 0 1 0 1 0 1 0 1
χ2(n) 0 1 0 1 0 −1 0 −1
χ3(n) 0 1 0 −1 0 1 0 −1
χ4(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 ϕ(9) = 6 characters modulo 9. In the table below, ω = exp(πi / 3).

χ \ n     0     1     2     3     4     5     6     7     8  
χ1(n) 0 1 1 0 1 1 0 1 1
χ2(n) 0 1 ω 0 ω2 −ω2 0 −ω −1
χ3(n) 0 1 ω2 0 −ω −ω 0 ω2 1
χ4(n) 0 1 −1 0 1 −1 0 1 −1
χ5(n) 0 1 −ω 0 ω2 ω2 0 −ω 1
χ6(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 ϕ(10) = 4 characters modulo 10.

χ \ n     0     1     2     3     4     5     6     7     8     9  
χ1(n) 0 1 0 1 0 0 0 1 0 1
χ2(n) 0 1 0 i 0 0 0 i 0 −1
χ3(n) 0 1 0 −1 0 0 0 −1 0 1
χ4(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 Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR0434929 
  • Spira, Robert (1969). "Calculation of Dirichlet L-Functions". Mathematics of Computation 23 (107): 489–497. doi:10.1090/S0025-5718-1969-0247742-X. MR0247742. 
  • 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. MR0279053. 
  • Hasse, Helmut (1964). Vorlesungen über Zahlentheorie. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. Springer. MR0188128.  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]. 

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