Complex-valued arithmetic function
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]
that is,
is completely multiplicative.
(gcd is the greatest common divisor)
; that is,
is periodic with period
.
The simplest possible character, called the principal character, usually denoted
, (see Notation below) exists for all moduli:[2]
![{\displaystyle \chi _{0}(a)={\begin{cases}0&{\text{if }}\gcd(a,m)>1\\1&{\text{if }}\gcd(a,m)=1.\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57b4eaf6b62362b1a48821e65c55f3b9e9c9ecc4)
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.[3][4]
Notation
is Euler's totient function.
is a complex primitive n-th root of unity:
but ![{\displaystyle \zeta _{n}\neq 1,\zeta _{n}^{2}\neq 1,...\zeta _{n}^{n-1}\neq 1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/333b2efa4d33a58106957361eddfed5df889c91e)
is the group of units mod
. It has order
is the group of Dirichlet characters mod
.
etc. are prime numbers.
is a standard[5] abbreviation[6] 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:
![{\displaystyle \eta :G\rightarrow \mathbb {C} ^{\times },\;\;\eta (gh)=\eta (g)\eta (h),\;\;\eta (g^{-1})=\eta (g)^{-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abae68b9134e63ebf8c6aa55375437e005965f9b)
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]
If
is a finite abelian group then[8] there are 1) an isomorphism
and 2) the orthogonality relations:[9]
and ![{\displaystyle \sum _{\eta \in {\widehat {A}}}\eta (a)={\begin{cases}|A|&{\text{ if }}a=1\\0&{\text{ if }}a\neq 1.\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23aadad4b7c0ba913e492bf6394cfce4f02d7040)
The elements of the finite abelian group
are the residue classes
where
A group character
can be extended to a Dirichlet character
by defining
![{\displaystyle \chi (a)={\begin{cases}0&{\text{if }}[a]\not \in (\mathbb {Z} /m\mathbb {Z} )^{\times }&{\text{i.e. }}(a,m)>1\\\rho ([a])&{\text{if }}[a]\in (\mathbb {Z} /m\mathbb {Z} )^{\times }&{\text{i.e. }}(a,m)=1,\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e63216fd11801a3183ccac899e4a141b3481071)
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
:
[11]
5) Property 3) is equivalent to
- if
then ![{\displaystyle \chi (a)=\chi (b).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91975cf08976dea1e71333c5df8baf25a2ac6a7e)
6) Property 1) implies that, for any positive integer
![{\displaystyle \chi (a^{n})=\chi (a)^{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38838f48aa03849466bb776a9d2b3c67d9b49d5b)
7) Euler's theorem states that if
then
Therefore,
![{\displaystyle \chi (a)^{\phi (m)}=\chi (a^{\phi (m)})=\chi (1)=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83a6cd290e2fc4d274b6d011f247217e4d7cc99e)
That is, the nonzero values of
are
-th roots of unity:
![{\displaystyle \chi (a)={\begin{cases}0&{\text{if }}\gcd(a,m)>1\\\zeta _{\phi (m)}^{r}&{\text{if }}\gcd(a,m)=1\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fdbf7fab19fa9e2c379cad406567073c33b4a64)
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).[12]
The principal character is an identity:
![{\displaystyle \chi \chi _{0}(a)=\chi (a)\chi _{0}(a)={\begin{cases}0\times 0&=\chi (a)&{\text{if }}\gcd(a,m)>1\\\chi (a)\times 1&=\chi (a)&{\text{if }}\gcd(a,m)=1.\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32c274cc38f63c7bef25b6f0a21d9d0982f213d1)
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.[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
![{\displaystyle a\equiv g_{q}^{\nu _{q}(a)}{\pmod {q}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/458aa2e6726f8940890df8f13c5117862c2d730d)
![{\displaystyle 0\leq \nu _{q}<\phi (q).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c528758a876525428496f6fae3529fd7f5d7381)
For
if and only if
Since
is determined by its value at ![{\displaystyle g_{q}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e867873dcdd0b155c6aa63ad05eef4f1162871f9)
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
![{\displaystyle \chi _{q,r}(a)={\begin{cases}0&{\text{if }}\gcd(a,q)>1\\\omega _{q}^{\nu _{q}(r)\nu _{q}(a)}&{\text{if }}\gcd(a,q)=1.\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/920bb6d412e646459addd94086c55c551a29dbb6)
Then for
and all
and
showing that
is a character and
which gives an explicit isomorphism ![{\displaystyle {\widehat {(\mathbb {Z} /p^{k}\mathbb {Z} )^{\times }}}\cong (\mathbb {Z} /p^{k}\mathbb {Z} )^{\times }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96f7b929ec60afcc3a26d12c0cf320e4ac83f4a3)
Examples m = 3, 5, 7, 9
2 is a primitive root mod 3. (
)
![{\displaystyle 2^{1}\equiv 2,\;2^{2}\equiv 2^{0}\equiv 1{\pmod {3}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b833fa6c51de526e25cd9dc38cc17c66d3b55109)
so the values of
are
.
The nonzero values of the characters mod 3 are
![{\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}&1&2\\\hline \chi _{3,1}&1&1\\\chi _{3,2}&1&-1\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aacae90e91abe7835f09072805d2c48da8657787)
2 is a primitive root mod 5. (
)
![{\displaystyle 2^{1}\equiv 2,\;2^{2}\equiv 4,\;2^{3}\equiv 3,\;2^{4}\equiv 2^{0}\equiv 1{\pmod {5}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0398d7365efdb2a288ff0c086d06333933e8b608)
so the values of
are
.
The nonzero values of the characters mod 5 are
![{\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}&1&2&3&4\\\hline \chi _{5,1}&1&1&1&1\\\chi _{5,2}&1&i&-i&-1\\\chi _{5,3}&1&-i&i&-1\\\chi _{5,4}&1&-1&-1&1\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a0527fa362e168bfbba6f5799e919faa1a0e991)
3 is a primitive root mod 7. (
)
![{\displaystyle 3^{1}\equiv 3,\;3^{2}\equiv 2,\;3^{3}\equiv 6,\;3^{4}\equiv 4,\;3^{5}\equiv 5,\;3^{6}\equiv 3^{0}\equiv 1{\pmod {7}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe7728e0b5d1d4fda71462133c87545bd06aab7)
so the values of
are
.
The nonzero values of the characters mod 7 are (
)
.
2 is a primitive root mod 9. (
)
![{\displaystyle 2^{1}\equiv 2,\;2^{2}\equiv 4,\;2^{3}\equiv 8,\;2^{4}\equiv 7,\;2^{5}\equiv 5,\;2^{6}\equiv 2^{0}\equiv 1{\pmod {9}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/369fe52dfcd8524b1e81d46c8f54e6adf7e3f811)
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
![{\displaystyle 5^{1}\equiv 5,\;5^{2}\equiv 5^{0}\equiv 1{\pmod {8}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb8b5de7b7b6c368ab152894600834801336239)
![{\displaystyle 5^{1}\equiv 5,\;5^{2}\equiv 9,\;5^{3}\equiv 13,\;5^{4}\equiv 5^{0}\equiv 1{\pmod {16}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0b86a03322753d2504189598702d0729cfb8b01)
![{\displaystyle 5^{1}\equiv 5,\;5^{2}\equiv 25,\;5^{3}\equiv 29,\;5^{4}\equiv 17,\;5^{5}\equiv 21,\;5^{6}\equiv 9,\;5^{7}\equiv 13,\;5^{8}\equiv 5^{0}\equiv 1{\pmod {32}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c8ccc294f100f9f55c67a329819e9c89c40a6e8)
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
![{\displaystyle a\equiv (-1)^{\nu _{0}(a)}5^{\nu _{q}(a)}{\pmod {q}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb47af0f5a89641941e8057b827437717db322c1)
![{\displaystyle 0\leq \nu _{0}<2,\;\;0\leq \nu _{q}<{\frac {\phi (q)}{2}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4375733a17ebff215745394f3f71a45b2f34c9ee)
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
![{\displaystyle \chi _{q,r}(a)={\begin{cases}0&{\text{if }}a{\text{ is even}}\\(-1)^{\nu _{0}(r)\nu _{0}(a)}\omega _{q}^{\nu _{q}(r)\nu _{q}(a)}&{\text{if }}a{\text{ is odd}}.\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bada750fd2a48c59eba69def94e1c990e0a6acea)
Then for odd
and
and all
and
showing that
is a character and
showing that ![{\displaystyle {\widehat {(\mathbb {Z} /2^{k}\mathbb {Z} )^{\times }}}\cong (\mathbb {Z} /2^{k}\mathbb {Z} )^{\times }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2384c5971b6e9a4289eb65a4e93a08b0f9c3392c)
Examples m = 2, 4, 8, 16
The only character mod 2 is the principal character
.
−1 is a primitive root mod 4 (
)
![{\displaystyle {\begin{array}{|||}a&1&3\\\hline \nu _{0}(a)&0&1\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8584965621d2c16eafc9a36ce3c10de9a7c6047c)
The nonzero values of the characters mod 4 are
![{\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}&1&3\\\hline \chi _{4,1}&1&1\\\chi _{4,3}&1&-1\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b4f680747f7909f6665bab9202ba9f0cf812e77)
−1 is and 5 generate the units mod 8 (
)
.
The nonzero values of the characters mod 8 are
![{\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}&1&3&5&7\\\hline \chi _{8,1}&1&1&1&1\\\chi _{8,3}&1&1&-1&-1\\\chi _{8,5}&1&-1&-1&1\\\chi _{8,7}&1&-1&1&-1\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4afc55284feed6cd699b47df1e632ebd5677dfe)
−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]
![{\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }\cong (\mathbb {Z} /q_{1}\mathbb {Z} )^{\times }\times (\mathbb {Z} /q_{2}\mathbb {Z} )^{\times }\times \dots \times (\mathbb {Z} /q_{k}\mathbb {Z} )^{\times }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9178e3647207010fcca5b129c8c5007c84122e91)
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 ![{\displaystyle c_{i}\equiv a_{i}b_{i}{\pmod {q_{i}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0204acc60c7008cfe6008a320973be08b8020e24)
The Chinese remainder theorem (CRT) implies that the
are simply
There are subgroups
such that [17]
and ![{\displaystyle G_{i}\equiv {\begin{cases}(\mathbb {Z} /q_{i}\mathbb {Z} )^{\times }&\mod q_{i}\\\{1\}&\mod q_{j},j\neq i.\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9e0c07ccaecd020593b9a699ab152f79b5cc1b1)
Then
and every
corresponds to a
-tuple
where
and
Every
can be uniquely factored as
[18] [19]
If
is a character mod
on the subgroup
it must be identical to some
mod
Then
![{\displaystyle \chi _{m,\_}(a)=\chi _{m,\_}(a_{1}a_{2}...)=\chi _{m,\_}(a_{1})\chi _{m,\_}(a_{2})...=\chi _{q_{1},\_}(a_{1})\chi _{a_{2},\_}(a_{2})...,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba56585b181dd5acd10382095c373fbd2e1b4736)
showing that every character mod
is the product of characters mod the
.
For
define[20]
![{\displaystyle \chi _{m,t}=\chi _{q_{1},t}\chi _{q_{2},t}...}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99b1d7707f8fb6653e23d26d6da711475caf58e3)
Then for
and all
and
[21]
showing that
is a character and
showing an isomorphism ![{\displaystyle {\widehat {(\mathbb {Z} /m\mathbb {Z} )^{\times }}}\cong (\mathbb {Z} /m\mathbb {Z} )^{\times }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a401d0c12fd0b7b98a139f40ff36fe9469b9a3e)
Examples m = 15, 24, 40
The factorization of the characters mod 15 is
![{\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}&\chi _{5,1}&\chi _{5,2}&\chi _{5,3}&\chi _{5,4}\\\hline \chi _{3,1}&\chi _{15,1}&\chi _{15,7}&\chi _{15,13}&\chi _{15,4}\\\chi _{3,2}&\chi _{15,11}&\chi _{15,2}&\chi _{15,8}&\chi _{15,14}\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bdcac6a0f3a405a8a042c42e3d60267baf636d0)
The nonzero values of the characters mod 15 are
.
The factorization of the characters mod 24 is
![{\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}&\chi _{8,1}&\chi _{8,3}&\chi _{8,5}&\chi _{8,7}\\\hline \chi _{3,1}&\chi _{24,1}&\chi _{24,19}&\chi _{24,13}&\chi _{24,7}\\\chi _{3,2}&\chi _{24,17}&\chi _{24,11}&\chi _{24,5}&\chi _{24,23}\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54c82001feea9cb1135c07f68dd96d69c911b12d)
The nonzero values of the characters mod 24 are
.
The factorization of the characters mod 40 is
![{\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}&\chi _{8,1}&\chi _{8,3}&\chi _{8,5}&\chi _{8,7}\\\hline \chi _{5,1}&\chi _{40,1}&\chi _{40,11}&\chi _{40,21}&\chi _{40,31}\\\chi _{5,2}&\chi _{40,17}&\chi _{40,27}&\chi _{40,37}&\chi _{40,7}\\\chi _{5,3}&\chi _{40,33}&\chi _{40,3}&\chi _{40,13}&\chi _{40,23}\\\chi _{5,4}&\chi _{40,9}&\chi _{40,19}&\chi _{40,29}&\chi _{40,39}\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d89e0a610e8e9dffbca4e1a0be62df9243cd4213)
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
[22]
Each character mod
has a unique factorization as the product of characters mod the prime powers dividing
:
![{\displaystyle \chi _{m,r}=\chi _{q_{1},r}\chi _{q_{2},r}...}](https://wikimedia.org/api/rest_v1/media/math/render/svg/960afb38ca662eae58fcb28b6ee8924dac2c8fb7)
If
the product
is a character
where
is given by
and
Also,[23][24]
Orthogonality
The two orthogonality relations are[25]
and ![{\displaystyle \sum _{\chi \in {\widehat {(\mathbb {Z} /m\mathbb {Z} )^{\times }}}}\chi (a)={\begin{cases}\phi (m)&{\text{ if }}\;a\equiv 1{\pmod {m}}\\0&{\text{ if }}\;a\not \equiv 1{\pmod {m}}.\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09593e9ab746f62290527867ac2dac4ca2014185)
The relations can be written in the symmetric form
and ![{\displaystyle \sum _{r\in (\mathbb {Z} /m\mathbb {Z} )^{\times }}\chi _{m,r}(a)={\begin{cases}\phi (m)&{\text{ if }}\;a\equiv 1\\0&{\text{ if }}\;a\not \equiv 1.\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c9ee71cf7dc17cc139da8ee8b7142ab05d8de13)
The first relation is easy to prove: If
there are
non-zero summands each equal to 1. If
there is[26] some
Then
[27] 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 lemma[28]
- Given
there is a ![{\displaystyle \chi ^{*},\;\chi ^{*}(a)\neq 1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6beca16fb02d153cc66638c1fb8cc6aac047e20c)
The second relation has an important corollary: if
define the function
Then ![{\displaystyle f_{a}(n)={\frac {1}{\phi (m)}}\sum _{\chi }\chi (a^{-1})\chi (n)={\frac {1}{\phi (m)}}\sum _{\chi }\chi (a^{-1}n)={\begin{cases}1,&n\equiv a{\pmod {m}}\\0,&n\not \equiv a{\pmod {m}},\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23daa93e23d1f6d59d72ab1da5cf4fe35d273744)
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]
![{\displaystyle {\begin{array}{|||}&1&3&5&7&9&11&13&15\\\hline \chi _{16,3}&1&-i&-i&1&-1&i&i&-1\\\chi _{16,9}&1&-1&-1&1&1&-1&-1&1\\\chi _{16,15}&1&-1&1&-1&1&-1&1&-1\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3e6b3ee1c677e59c8861b3581172f2a31975fc3)
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 ![{\displaystyle \chi _{m,\_}=\chi _{q,\_}\chi _{r,1},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/369adb52e882a1efae12aab7577e61b6e4420c80)
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.[34]
The character
is primitive if and only if each of the factors is primitive.[35]
Primitive characters often simplify (or make possible) formulas in the theories of L-functions[36] 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 (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]
Real characters are Kronecker symbols;[40] for example, the principal character can be written[41]
.
The real characters in the examples are:
Principal
If
the principal character is[42]
Primitive
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]
Imprimitive
Applications
L-functions
The Dirichlet L-series for a character
is
![{\displaystyle L(s,\chi )=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b0973bd5e53e9b36d4bbda44aa7d08ee11f677e)
This series only converges for
; it can be analytically continued to a meromorphic function
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[44]
Let
and let
be primitive.
If
[45]
define
,[46]
Then
. If
is a cusp form so is ![{\displaystyle f_{\chi _{1}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79148ea2a1987649b71c52ed4ed2121e17880b5e)
See theta series of a Dirichlet character for another example.
Gauss sum
The Gauss sum of a Dirichlet character modulo N is
![{\displaystyle G(\chi )=\sum _{a=1}^{N}\chi (a)e^{\frac {2\pi ia}{N}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a207ea407b73419862d8778ea965b8839a436b5a)
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
![{\displaystyle J(\chi ,\psi )=\sum _{a=2}^{p-1}\chi (a)\psi (1-a).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f6048e0011a3a1c8091efed2901f448dba1deb4)
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[47]
![{\displaystyle K(a,b,\chi )=\sum _{r\in (\mathbb {Z} /q\mathbb {Z} )^{\times }}\chi (r)\zeta ^{ar+{\frac {b}{r}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51637b5248f9b31603989906dfe8256239656fd1)
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
- 1)
![{\displaystyle \mathrm {X} (ab)=\mathrm {X} (a)\mathrm {X} (b),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc942f04698377f472a43324b55823624a64f181)
- 2)
, - 3) If
then
, but - 4)
is not always 0,
then
is one of the
characters mod
[48]
Sárközy's Condition
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]
See also
Notes
- ^ This is the standard definition; e.g. Davenport p.27; Landau p. 109; Ireland and Rosen p. 253
- ^ Note the special case of modulus 1: the unique character mod 1 is the constant 1; all other characters are 0 at 0
- ^ Davenport p. 1
- ^ An English translation is in External Links
- ^ Used in Davenport, Landau, Ireland and Rosen
- ^
is equivalent to
- ^ See Multiplicative character
- ^ Ireland and Rosen p. 253-254
- ^ See Character group#Orthogonality of characters
- ^ Davenport p. 27
- ^ These properties are derived in all introductions to the subject, e.g. Davenport p. 27, Landau p. 109.
- ^ In general, the product of a character mod
and a character mod
is a character mod
- ^ Except for the use of the modified Conrie labeling, this section follows Davenport pp. 1-3, 27-30
- ^ There is a primitive root mod
which is a primitive root mod
and all higher powers of
. See, e.g., Landau p. 106 - ^ Landau pp. 107-108
- ^ See group of units for details
- ^ To construct the
for each
use the CRT to find
where ![{\displaystyle a_{i}\equiv {\begin{cases}a&\mod q_{i}\\1&\mod q_{j},j\neq i.\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ac081dddb97c90c65463f69713f6d68ab2a07d8)
- ^ 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 ![{\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}&1&9&17&33\\\hline 1&1&9&17&33\\11&11&19&27&3\\21&21&29&37&13\\31&31&39&7&23\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f36229eff94e32ff19ffb07b56579b0b6fe7f0d)
- ^ See Conrey labeling.
- ^ Because these formulas are true for each factor.
- ^ This is true for all finite abelian groups:
; See Ireland & Rosen pp. 253-254 - ^ 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.
- ^ See #Relation to group characters above.
- ^ by the definition of
- ^ 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]
- ^ Davenport chs. 1, 4; Landau p. 114
- ^ Note that if
is any function
; see Fourier transform on finite groups#Fourier transform for finite abelian groups - ^ This section follows Davenport pp. 35-36,
- ^ Platt, Dave. "Dirichlet characters Def. 11.10" (PDF). Retrieved April 5, 2024.
- ^ "Conductor of a Dirichlet character (reviewed)". LMFDB. Retrieved April 5, 2024.
- ^ Davenport classifies it as neither primitive nor imprimitive; the LMFDB induces it from
- ^ a b Note that if
is two times an odd number,
, all characters mod
are imprimitive because
- ^ For example the functional equation of
is only valid for primitive
. See Davenport, p. 85 - ^ In fact, for prime modulus
is the Legendre symbol:
Sketch of proof:
is even (odd) if a is a quadratic residue (nonresidue) - ^ Davenport, chs. 1, 4.
- ^ Ireland and Rosen's proof, valid for all moduli, also has these two cases. pp. 259 ff
- ^ Davenport p. 40
- ^ The notation
is a shorter way of writing
- ^ The product of primes ensures it is zero if
; the squares ensure its only nonzero value is 1. - ^ Davenport pp. 38-40
- ^ Koblittz, prop. 17b p. 127
- ^
means 1)
where
and
and 2)
where
and
See Koblitz Ch. III. - ^ the twist of
by
- ^ LMFDB definition of Kloosterman sum
- ^ Davenport p. 30
- ^ Sarkozy
- ^ Chudakov
- ^ Klurman
References
- Chudakov, N.G. "Theory of the characters of number semigroups". J. Indian Math. Soc. 20: 11–15.
- Davenport, Harold (1967). Multiplicative number theory. Lectures in advanced mathematics. Vol. 1. Chicago: Markham. Zbl 0159.06303.
- Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer, ISBN 0-387-97329-X
- Klurman, Oleksiy; Mangerel, Alexander P. (2017). "Rigidity Theorems for Multiplicative Functions". Math. Ann. 372 (1): 651–697. arXiv:1707.07817. Bibcode:2017arXiv170707817K. doi:10.1007/s00208-018-1724-6. S2CID 119597384.
- Koblitz, Neal (1993). Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics. Vol. 97 (2nd revised ed.). Springer-Verlag. ISBN 0-387-97966-2.
- Landau, Edmund (1966), Elementary Number Theory, New York: Chelsea
- Sarkozy, Andras. "On multiplicative arithmetic functions satisfying a linear recursion". Studia Sci. Math. Hung. 13 (1–2): 79–104.
External links
- English translation of Dirichlet's 1837 paper on primes in arithmetic progressions
- LMFDB Lists 30,397,486 Dirichlet characters of modulus up to 10,000 and their L-functions