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 :[1]
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.[7]
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[10] 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 :
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.[13]
Powers of odd primes
If is an odd number is cyclic of order ; a generator is called a
primitive root mod .[14]
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 [15]
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 :[16]
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:
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 lemma[28]
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.[29][30]
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[31]
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 .[32] 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 .[33] 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.
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 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 (they must be ); 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.[37]
Dirichlet's original proof that (which was only valid for prime moduli) 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.[38][39]
If the modulus is the absolute value of a
fundamental discriminant there is a real primitive character (there are two if the modulus is a multiple of 8); otherwise if there are any primitive characters[35] they are imaginary.[43]
A Dirichlet character is a completely multiplicative function that satisfies a
linear recurrence relation: that is, if
for all positive integer , where are not all zero and are distinct then is a Dirichlet character.[49]
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[50] in 1956, and proved in 2017 by Klurman and Mangerel.[51]
^because multiplying every element in a group by a constant element merely permutes the elements. See
Group (mathematics)
^Davenport 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]
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 :[1]
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.[7]
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[10] 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 :
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.[13]
Powers of odd primes
If is an odd number is cyclic of order ; a generator is called a
primitive root mod .[14]
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 [15]
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 :[16]
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:
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 lemma[28]
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.[29][30]
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[31]
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 .[32] 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 .[33] 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.
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 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 (they must be ); 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.[37]
Dirichlet's original proof that (which was only valid for prime moduli) 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.[38][39]
If the modulus is the absolute value of a
fundamental discriminant there is a real primitive character (there are two if the modulus is a multiple of 8); otherwise if there are any primitive characters[35] they are imaginary.[43]
A Dirichlet character is a completely multiplicative function that satisfies a
linear recurrence relation: that is, if
for all positive integer , where are not all zero and are distinct then is a Dirichlet character.[49]
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[50] in 1956, and proved in 2017 by Klurman and Mangerel.[51]
^because multiplying every element in a group by a constant element merely permutes the elements. See
Group (mathematics)
^Davenport 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]