In
number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
,
Dirichlet characters are certain
arithmetic functionIn number theory, an arithmetic function is a real or complex valued function ƒ defined on the set of natural numbers In number theory, an arithmetic (or arithmetical) function is a real or complex valued function ƒ(n) defined on the set of natural numbers In number theory, an arithmetic (or...
s which arise from completely multiplicative
charactersIn mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
on the units of
. Dirichlet characters are used to define
Dirichlet L-functions, which are
meromorphic functionIn complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
s with a variety of interesting analytic properties.
If
is a Dirichlet character, one defines its Dirichlet
L-series by
where
s is a
complex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
with real part > 1. By
analytic continuationIn complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
, this function can be extended to a
meromorphic functionIn complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
on the whole
complex planeIn mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
. Dirichlet
L-functions are generalizations of the
Riemann zeta-function and appear prominently in the
generalized Riemann hypothesisThe Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function...
.
Dirichlet characters are named in honour of
Johann Peter Gustav Lejeune DirichletJohann Peter Gustav Lejeune Dirichlet was a German mathematician with deep contributions to number theory , as well as to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a...
.
Axiomatic definition
A Dirichlet character is any
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
χ from the
integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s to the
complex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s which has the following properties:
- There exists a positive integer k such that χ(n) = χ(n + k) for all n.
- If gcd
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
(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
periodicIn mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
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
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, χ(a) is a φ(k)-th complex root of unity
In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
.
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 groupIn mathematics, a character group is the group of representations of a group by complex-valued functions. These functions can be thought of as one-dimensional matrix representations and so are special cases of the group characters which arises in the related context of character theory...
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
moduloIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
k
:
That is, the residue class 
is the cosetIn mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
of n
in the quotient ringIn ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...
Z/kZ.
The set of units modulo
k forms an
abelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
of order
, where group multiplication is given by
and
again denotes Euler's phi function.
The identity in this group is the residue class
and the inverse of
is the residue class
where
, i.e.,
. For example, for
k=6, the set of units is
because 0, 2, 3, and 4 are not coprime to 6.
Dirichlet characters
A Dirichlet character modulo
k is a group homomorphism
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
liftIn the branch of mathematics called category theory, given a morphism f from an object X to an object Y, and a morphism g from an object Z to Y, a lift of f to Z is a morphism h from X to Z such that gh = f.A basic example in topology is lifting a path in one space to a path in a covering space...
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
modulo
k has the properties
if gcd(
n,
k) = 1 and
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 χ
1 are the principal characters.
Modulus 1
There is
character modulo 1:
| χ \ n |
0 |
 |
1 |
This is the trivial character.
Modulus 2
There is
character modulo 2:
| χ \ n |
0 |
1 |
 |
0 |
1 |
Note that χ is wholly determined by χ(1) since 1 generates the group of units modulo 2.
Modulus 3
There are
characters modulo 3:
| χ \ n |
0 |
1 |
2 |
 |
0 |
1 |
1 |
 |
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
characters modulo 4:
| χ \ n |
0 |
1 |
2 |
3 |
 |
0 |
1 |
0 |
1 |
 |
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
is
the Dirichlet lambda function (closely related to the
Dirichlet eta function)
where
is the Riemann zeta-function. The
L-series for
is the
Dirichlet beta-functionIn mathematics, the Dirichlet beta function is a special function, closely related to the Riemann zeta function...
Modulus 5
There are
characters modulo 5. In the tables,
i is a square root of
.
| χ \ n |
0 |
1 |
2 |
3 |
4 |
 |
0 |
1 |
1 |
1 |
1 |
 |
0 |
1 |
i |
−i |
−1 |
 |
0 |
1 |
−1 |
−1 |
1 |
 |
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
characters modulo 6:
| χ \ n |
0 |
1 |
2 |
3 |
4 |
5 |
 |
0 |
1 |
0 |
0 |
0 |
1 |
 |
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
characters modulo 7. In the table below,
| χ \ n |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
 |
0 |
1 |
ω2 |
ω |
−ω |
−ω2 |
−1 |
 |
0 |
1 |
−ω |
ω2 |
ω2 |
−ω |
1 |
 |
0 |
1 |
1 |
−1 |
1 |
−1 |
−1 |
 |
0 |
1 |
ω2 |
−ω |
−ω |
ω2 |
1 |
 |
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
characters modulo 8.
| χ \ n |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
 |
0 |
1 |
0 |
1 |
0 |
−1 |
0 |
−1 |
 |
0 |
1 |
0 |
−1 |
0 |
1 |
0 |
−1 |
 |
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
characters modulo 9. In the table below,
| χ \ n |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
 |
0 |
1 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
 |
0 |
1 |
ω |
0 |
ω2 |
−ω2 |
0 |
−ω |
−1 |
 |
0 |
1 |
ω2 |
0 |
−ω |
−ω |
0 |
ω2 |
1 |
 |
0 |
1 |
−1 |
0 |
1 |
−1 |
0 |
1 |
−1 |
 |
0 |
1 |
−ω |
0 |
ω2 |
ω2 |
0 |
−ω |
1 |
 |
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
characters modulo 10.
| χ \ n |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
1 |
 |
0 |
1 |
0 |
i |
0 |
0 |
0 |
−i |
0 |
−1 |
 |
0 |
1 |
0 |
−1 |
0 |
0 |
0 |
−1 |
0 |
1 |
 |
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 numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
, then the function
where
is the
Legendre symbolIn number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo a prime number p: its value on a quadratic residue mod p is 1 and on a quadratic non-residue is −1....
, is a Dirichlet character modulo
p.
More generally, if
m is an odd number the function
where
is the
Jacobi symbolThe Jacobi symbol is a generalization of the Legendre symbol. Introduced by Jacobi in 1837, it is of theoretical interest in modular arithmetic and other branches of number theory, but its main use is in computational number theory, especially primality testing and integer factorization; these in...
, 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-functionThe theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
s.
History
Dirichlet characters and their
L-series were introduced by
Johann Peter Gustav Lejeune DirichletJohann Peter Gustav Lejeune Dirichlet was a German mathematician with deep contributions to number theory , as well as to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a...
, in 1831, in order to prove
Dirichlet's theorem on arithmetic progressionsIn number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n ≥ 0. In other words, there are infinitely many primes which are...
. 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 RiemannGeorg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
in 1859.
See also
- Character sum
In mathematics, a character sum is a sum\Sigma \chi\,of values of a Dirichlet character χ modulo N, taken over a given range of values of n. Such sums are basic in a number of questions, for example in the distribution of quadratic residues, and in particular in the classical question of finding an...
- Dirichlet L-function
- Gaussian sum
- Primitive root modulo n
In modular arithmetic, a branch of number theory, a primitive root modulo n is any number g with the property that any number coprime to n is congruent to a power of g modulo n. In other words, g is a generator of the multiplicative group of integers modulo n...
- Selberg class
In mathematics, the Selberg class S is an axiomatic definition of a class of L-functions. The members of the class are Dirichlet series which obey four axioms that seem to capture the essential properties satisfied by most functions that are commonly called L-functions or zeta functions...