Dirichlet character


In analytic number theory and related branches of mathematics, a complex-valued arithmetic function is a Dirichlet character of modulus if for all integers and :
The simplest possible character, called the principal character and usually denoted, exists for all moduli:
Dirichlet characters were named after German mathematician Peter Gustav Lejeune Dirichlet, who introduced these functions in his 1837 paper on primes in arithmetic progressions.

Notation

is Euler's totient function.
is a complex primitive n-th root of unity:
is the group of units mod . It has order
is the group of Dirichlet characters mod.
etc. are prime numbers.
is a standard abbreviation for
etc. are Dirichlet characters.
There is no standard notation for Dirichlet characters that includes the modulus. In many contexts the modulus is fixed. In other contexts, such as this article, characters of different moduli appear. Where appropriate this article employs a variation of .
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" is used several ways in mathematics. In this section it refers to a homomorphism from a group 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 there is an isomorphism, and the orthogonality relations:
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
6) Property 1) implies that, for any positive integer
7) Euler's theorem 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:
The principal character is an identity:
9) Let denote the inverse of in.
Then
The complex conjugate of a root of unity is also its inverse, so for
Thus for all integers
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 by
For if and only if Since
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

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 and a cyclic group of order .
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

Examples ''m'' = 2, 4, 8, 16

The only character mod 2 is the principal character.
−1 is a 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 where be the factorization of into prime powers. The group of units mod is isomorphic to the direct product of the groups mod the :
This means that 1) there is a one-to-one correspondence between and -tuples where
and 2) multiplication mod corresponds to coordinate-wise multiplication of -tuples:
The Chinese remainder theorem implies that the are simply
There are subgroups such that
Then
and every corresponds to a -tuple where and
Every can be uniquely factored as
If is a character mod on the subgroup it must be identical to some mod Then
showing that every character mod is the product of characters mod the.
For define
Then for and all and

Examples ''m'' = 15, 24, 40

The factorization of the characters mod 15 is
The nonzero values of the characters mod 15 are
The factorization of the characters mod 24 is
The nonzero values of the characters mod 24 are
The factorization of the characters mod 40 is
The nonzero values of the characters mod 40 are

Summary

Let, be the factorization of and assume
There are Dirichlet characters mod They are denoted by where is equivalent to
The identity is an isomorphism
Each character mod has a unique factorization as the product of characters mod the prime powers dividing :
If the product is a character where is given by and
Also,

Orthogonality

The two orthogonality relations are
The relations can be written in the symmetric form
The first relation is easy to prove: If there are non-zero summands each equal to 1. If there is some Then
The second relation can be proven directly in the same way, but requires a lemma
The second relation has an important corollary: if define the function
That is the indicator function of the residue class. It is basic in the proof of Dirichlet's theorem.

Classification of characters

Conductor; Primitive and induced characters

Any character mod a prime power is also a character mod every larger power. For example, mod 16
has period 16, but has period 8 and has period 4: and
We say that a character of modulus has a quasiperiod of if for all, coprime to satisfying mod. For example,, the only Dirichlet character of modulus, has a quasiperiod of, but not a period of . The smallest positive integer for which is quasiperiodic is the conductor of. So, for instance, has a conductor of.
The conductor of is 16, the conductor of is 8 and that of and is 4. If the modulus and conductor are equal the character is primitive, otherwise imprimitive. An imprimitive character is induced by the character for the smallest modulus: is induced from and and are induced from.
A related phenomenon can happen with a character mod the product of primes; its nonzero values may be periodic with a smaller period.
For example, mod 15,
The nonzero values of have period 15, but those of have period 3 and those of have period 5. This is easier to see by juxtaposing them with characters mod 3 and 5:
If a character mod is defined as
its nonzero values are determined by the character mod and have period.
The smallest period of the nonzero values is the conductor of the character. For example, the conductor of is 15, the conductor of is 3, and that of is 5.
As in the prime-power case, if the conductor equals the modulus the character is primitive, otherwise imprimitive. If imprimitive it is induced from the character with the smaller modulus. For example, is induced from and is induced from
The principal character is not primitive.
The character is primitive if and only if each of the factors is primitive.
Primitive characters often simplify formulas in the theories of L-functions and modular forms.

Parity

is even if and is odd if
This distinction appears in the functional equation of the Dirichlet L-function.

Order

The order of a character is its order as an element of the group, i.e. the smallest positive integer such that Because of the isomorphism the order of is the same as the order of in The principal character has order 1; other real characters have order 2, and imaginary characters have order 3 or greater. By Lagrange's theorem the order of a character divides the order of which is

Real characters

is real or quadratic if all of its values are real ; otherwise it is complex or imaginary.
is real if and only if ; is real if and only if ; in particular, is real and non-principal.
Dirichlet's original proof that took two different forms depending on whether was real or not. His later proof, valid for all moduli, was based on his class number formula.
Real characters are Kronecker symbols; for example, the principal character can be written
The [|real characters] in the examples are:

Principal

If the principal character is







Primitive

If the modulus is the absolute value of a fundamental discriminant there is a real primitive character ; otherwise if there are any primitive characters they are imaginary.










Imprimitive


















Applications

L-functions

The Dirichlet L-series for a character is
This series converges absolutely for. If the character is non-principal then furthermore it converges for and it can be analytically continued to an entire function, defined and differentiable on the whole complex plane. If the character is principal then the series it converges only for ; in this case, it can be analytically continued to a meromorphic function with simple pole at.
Dirichlet introduced the -function along with the characters in his 1837 paper.

Modular forms and functions

Dirichlet characters appear several places in the theory of modular forms and functions. A typical example is
Let and let be primitive.
If
define
Then
See theta series of a Dirichlet character for another example.

Gauss sum

The Gauss sum of a Dirichlet character modulo is
It appears in the functional equation of the Dirichlet L-function.

Jacobi sum

If and are Dirichlet characters mod a prime their Jacobi sum is
Jacobi sums can be factored into products of Gauss sums.

Kloosterman sum

If is a Dirichlet character mod and the Kloosterman sum is defined as
If it is a Gauss sum.

Sufficient conditions

It is not necessary to establish the defining properties 1) – 3) to show that a function is a Dirichlet character.

From Davenport's book

If such that
then is one of the characters mod

Sárközy's Condition

A Dirichlet character is a completely multiplicative function that satisfies a linear recurrence relation: that is, if
for all positive integers, where are not all zero and are distinct then is a Dirichlet character.

Chudakov's Condition

A Dirichlet character is a completely multiplicative function satisfying the following three properties: a) takes only finitely many values; b) vanishes at only finitely many primes; c) there is an for which the remainder
is uniformly bounded, as. This equivalent definition of Dirichlet characters was conjectured by Chudakov in 1956, and proved in 2017 by Klurman and Mangerel.

Some notable special modules

  • 8, the smallest module whose Dirichlet characters need more than one generator
  • 13, the smallest module whose Dirichlet characters contain numbers such that there is no primes p in which are still primes in
  • 19, the smallest module whose Dirichlet characters contain numbers whose real and imaginary parts are not constructible numbers
  • 24, the largest module whose Dirichlet characters are all real
  • 47, the smallest module whose Dirichlet characters contain numbers such that the class number of the cyclotomic field is greater than 1
  • 120, the smallest module whose Dirichlet characters need more than three generators
  • 149, the smallest module whose Dirichlet characters contain numbers such that the full class number of the cyclotomic field is not coprime to the smallest number such that
  • 240, the largest module whose Dirichlet characters are all Gaussian integers
  • 383, the smallest module whose Dirichlet characters contain numbers such that the class number of the cyclotomic field is greater than 1
  • 504, the largest module whose Dirichlet characters are all Eisenstein integers
  • 840, the smallest module whose Dirichlet characters need more than four generators