In analytic number theory 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 :This is the standard definition; e.g. Davenport p.27; Landau p. 109; Ireland and Rosen p. 253
1. that is, is completely multiplicative.
2. .
3. ; that is, is periodic with period .
The simplest possible character, called the principal character and usually denoted , exists for all moduli:Note the special case of modulus 1: the unique character mod 1 is the constant 1; all other characters are 0 at 0
Dirichlet characters were named after German mathematician Peter Gustav Lejeune Dirichlet, who introduced these functions in his 1837 paper on primes in arithmetic progressions.Davenport p. 1An English translation is in External Links They are a prominent example of the general idea of a character in mathematics.
Notation
is Euler's totient function.
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 numbers.
is a standardUsed in Davenport, Landau, Ireland and Rosen abbreviation is equivalent to 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 of Conrey labeling (introduced by Brian Conrey and used by the 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" is used several ways in mathematics. In this section it refers to a homomorphism 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.See Multiplicative character
If is a finite abelian group thenIreland and Rosen p. 253-254 there is an isomorphism , and the orthogonality relations:See Character group#Orthogonality of characters
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,Davenport p. 27 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 :
These properties are derived in all introductions to the subject, e.g. Davenport p. 27, Landau p. 109.
5) Property 3) is equivalent to
if then
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: ( obviously satisfies 1-3).In general, the product of a character mod and a character mod is a character mod
The principal character is an identity:
9) Let denote the inverse of in . Then so which extends 6) to all integers.
The complex conjugate of a root of unity is also its inverse (see here 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.Except for the use of the modified Conrie labeling, this section follows Davenport pp. 1-3, 27-30
Powers of odd primes
If is an odd number is cyclic of order ; a generator is called a primitive root mod .There is a primitive root mod which is a primitive root mod and all higher powers of . See, e.g., Landau p. 106 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 Landau pp. 107-108 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 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 :See group of units for details
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: corresponds to where
The Chinese remainder theorem (CRT) implies that the are simply
There are subgroups such that To construct the for each use the CRT to find where and
Then and every corresponds to a -tuple where and Every can be uniquely factored as Assume corresponds to . By construction corresponds to , to etc. whose coordinate-wise product is For example let Then and The factorization of the elements of is
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 defineSee Conrey labeling.
Then for and all and Because these formulas are true for each factor. showing that is a character and showing an isomorphism
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 This is true for all finite abelian groups: ; See Ireland & Rosen pp. 253-254
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,because the formulas for mod prime powers are symmetric in and and the formula for products preserves this symmetry. See Davenport, p. 29.This is the same thing as saying that the n-th column and the n-th row in the tables of nonzero values are the same.
Orthogonality
The two orthogonality relations areSee #Relation to group characters above. and
The relations can be written in the symmetric form and
The first relation is easy to prove: If there are non-zero summands each equal to 1. If there isby the definition of some Then because multiplying every element in a group by a constant element merely permutes the elements. See Group (mathematics) implying Dividing by the first factor gives QED. The identity for shows that the relations are equivalent to each other.
The second relation can be proven directly in the same way, but requires a lemmaDavenport p. 30 (paraphrase) To prove [the second relation] one has to use ideas that we have used in the construction [as in this article or Landau pp. 109-114], or appeal to the basis theorem for abelian groups [as in Ireland & Rosen pp. 253-254] Given there is a
The second relation has an important corollary: if define the function Then
That is the indicator function of the residue class . It is basic in the proof of Dirichlet's theorem.Davenport chs. 1, 4; Landau p. 114Note that if is any function ; see Fourier transform on finite groups#Fourier transform for finite abelian groups
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 16This section follows Davenport pp. 35-36,
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 (it has a period of , though). 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 , or equivalently 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.