In
analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diri ...
and related branches of mathematics, a complex-valued
arithmetic function is a Dirichlet character of modulus
(where
is a positive integer) if for all integers
and
:
:1)
i.e.
is
completely multiplicative.
:2)
(gcd is the
greatest common divisor)
:3)
; i.e.
is periodic with period
.
The simplest possible character, called the principal character, usually denoted
, (see
Notation below) exists for all moduli:
:
The German mathematician
Peter Gustav Lejeune Dirichlet—for whom the character is named—introduced these functions in his 1837 paper on
primes in arithmetic progressions.
Notation
is
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ...
.
is a complex primitive
n-th root of unity:
:
but
is the
group of units mod . It has order
is the group of Dirichlet characters mod
.
etc. are
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s.
is a standard abbreviation for
etc. are Dirichlet characters. (the lowercase Greek letter chi for character)
There is no standard notation for Dirichlet characters that includes the modulus. In many contexts (such as in the proof of Dirichlet's theorem) the modulus is fixed. In other contexts, such as this article, characters of different moduli appear. Where appropriate this article employs a variation o
Conrey labeling(introduced by
Brian Conrey and used by th
LMFDB.
In this labeling characters for modulus
are denoted
where the index
is described in the section
the group of characters below. In this labeling,
denotes an unspecified character and
denotes the principal character mod
.
Relation to group characters
The word "
character
Character or Characters may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
" is used several ways in mathematics. In this section it refers to a
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
from a group
(written multiplicatively) to the multiplicative group of the field of complex numbers:
:
The set of characters is denoted
If the product of two characters is defined by pointwise multiplication
the identity by the trivial character
and the inverse by complex inversion
then
becomes an abelian group.
If
is a
finite abelian group then
[Ireland and Rosen p. 253-254] there are 1) an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
and 2) the orthogonality relations:
:
and
The elements of the finite abelian group
are the residue classes
where
A group character
can be extended to a Dirichlet character
by defining
:
and conversely, a Dirichlet character mod
defines a group character on
Paraphrasing Davenport Dirichlet characters can be regarded as a particular case of Abelian group characters. But this article follows Dirichlet in giving a direct and constructive account of them. This is partly for historical reasons, in that Dirichlet's work preceded by several decades the development of group theory, and partly for a mathematical reason, namely that the group in question has a simple and interesting structure which is obscured if one treats it as one treats the general Abelian group.
Elementary facts
4) Since
property 2) says
so it can be canceled from both sides of
:
:
5) Property 3) is equivalent to
:if
then
6) Property 1) implies that, for any positive integer
:
7)
Euler's theorem
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if and are coprime positive integers, and \varphi(n) is Euler's totient function, then raised to the power \varphi(n) is congr ...
states that if
then
Therefore,
:
That is, the nonzero values of
are
-th
roots of unity:
:
for some integer
which depends on
and
. This implies there are only a finite number of characters for a given modulus.
8) If
and
are two characters for the same modulus so is their product
defined by pointwise multiplication:
:
(
obviously satisfies 1-3).
The principal character is an identity:
:
9) Let
denote the inverse of
in
.
Then
:
so
which extends 6) to all integers.
The
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of a root of unity is also its inverse (see
here
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here
Television
* Here TV (formerly "here!"), a ...
for details), so for
:
(
also obviously satisfies 1-3).
Thus for all integers
:
in other words
.
10) The multiplication and identity defined in 8) and the inversion defined in 9) turn the set of Dirichlet characters for a given modulus into a
finite abelian group.
The group of characters
There are three different cases because the groups
have different structures depending on whether
is a power of 2, a power of an odd prime, or the product of prime powers.
Powers of odd primes
If
is an odd number
is cyclic of order
; a generator is called a
primitive root mod
.
Let
be a primitive root and for
define the function
(the index of
) by
:
:
For
if and only if
Since
:
is determined by its value at
Let
be a primitive
-th root of unity. From property 7) above the possible values of
are
These distinct values give rise to
Dirichlet characters mod
For
define
as
:
Then for
and all
and
:
showing that
is a character and
:
which gives an explicit isomorphism
Examples ''m'' = 3, 5, 7, 9
2 is a primitive root mod 3. (
)
:
so the values of
are
:
.
The nonzero values of the characters mod 3 are
:
2 is a primitive root mod 5. (
)
:
so the values of
are
:
.
The nonzero values of the characters mod 5 are
:
3 is a primitive root mod 7. (
)
:
so the values of
are
:
.
The nonzero values of the characters mod 7 are (
)
:
.
2 is a primitive root mod 9. (
)
:
so the values of
are
:
.
The nonzero values of the characters mod 9 are (
)
:
.
Powers of 2
is the trivial group with one element.
is cyclic of order 2. For 8, 16, and higher powers of 2, there is no primitive root; the powers of 5 are the units
and their negatives are the units
For example
:
:
:
Let
; then
is the direct product of a cyclic group of order 2 (generated by −1) and a cyclic group of order
(generated by 5).
For odd numbers
define the functions
and
by
:
:
For odd
and
if and only if
and
For odd
the value of
is determined by the values of
and
Let
be a primitive
-th root of unity. The possible values of
are
These distinct values give rise to
Dirichlet characters mod
For odd
define
by
:
Then for odd
and
and all
and
:
showing that
is a character and
:
showing that
Examples ''m'' = 2, 4, 8, 16
The only character mod 2 is the principal character
.
−1 is primitive root mod 4 (
)
:
The nonzero values of the characters mod 4 are
:
−1 is and 5 generate the units mod 8 (
)
:
.
The nonzero values of the characters mod 8 are
:
−1 and 5 generate the units mod 16 (
)
:
.
The nonzero values of the characters mod 16 are
:
.
Products of prime powers
Let