In analytic number theory and related branches of mathematics , Dirichlet characters are certain complex-valued arithmetic functions . Specifically, given a positive integer
m
{\displaystyle m}
χ
:
Z
→
C
{\displaystyle \chi :\mathbb {Z} \rightarrow \mathbb {C} }
Dirichlet character of modulus
m
{\displaystyle m}
if for all integers
a
{\displaystyle a}
b
{\displaystyle b}
1)
χ
(
a
b
)
=
χ
(
a
)
χ
(
b
)
;
{\displaystyle \chi (ab)=\chi (a)\chi (b);}
χ
{\displaystyle \chi }
completely multiplicative . 2)
χ
(
a
)
{
=
0
if
gcd
(
a
,
m
)
>
1
≠
0
if
gcd
(
a
,
m
)
=
1.
{\displaystyle \chi (a){\begin{cases}=0&{\text{if }}\;\gcd(a,m)>1\\\neq 0&{\text{if }}\;\gcd(a,m)=1.\end{cases}}}
3)
χ
(
a
+
m
)
=
χ
(
a
)
{\displaystyle \chi (a+m)=\chi (a)}
χ
{\displaystyle \chi }
m
{\displaystyle m}
The simplest possible character, called the principal character (usually denoted
χ
0
{\displaystyle \chi _{0}}
Notation below) exists for all moduli:
χ
0
(
a
)
=
{
0
if
gcd
(
a
,
m
)
>
1
1
if
gcd
(
a
,
m
)
=
1.
{\displaystyle \chi _{0}(a)={\begin{cases}0&{\text{if }}\;\gcd(a,m)>1\\1&{\text{if }}\;\gcd(a,m)=1.\end{cases}}}
Dirichlet introduced these functions in his 1837 paper on primes in arithmetic progressions .
ϕ
(
n
)
{\displaystyle \phi (n)}
Euler totient function .
E
(
x
)
:=
e
2
π
i
x
.
{\displaystyle \operatorname {E} (x):=e^{2\pi ix}.}
E
(
x
)
E
(
y
)
=
E
(
x
+
y
)
.
{\displaystyle \operatorname {E} (x)\operatorname {E} (y)=\operatorname {E} (x+y).}
ζ
n
=
E
(
1
n
)
{\displaystyle \zeta _{n}=\operatorname {E} \left({\frac {1}{n}}\right)}
n-th root of unity :
ζ
n
n
=
1
,
{\displaystyle \zeta _{n}^{n}=1,}
ζ
n
≠
1
,
ζ
n
2
≠
1
,
.
.
.
ζ
n
n
−
1
≠
1.
{\displaystyle \zeta _{n}\neq 1,\zeta _{n}^{2}\neq 1,...\zeta _{n}^{n-1}\neq 1.}
(
Z
/
q
Z
)
×
{\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{\times }}
group of units mod
q
{\displaystyle q}
. It has order
ϕ
(
q
)
.
{\displaystyle \phi (q).}
χ
(
a
)
{\displaystyle \chi (a)}
χ
′
(
a
)
{\displaystyle \chi '(a)}
χ
r
(
a
)
{\displaystyle \chi _{r}(a)}
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
m
{\displaystyle m}
χ
m
,
t
(
a
)
{\displaystyle \chi _{m,\;t}(a)}
t
{\displaystyle t}
m
{\displaystyle m}
Explicit construction below. Note that the principal character for modulus
m
{\displaystyle m}
χ
m
,
1
(
a
)
{\displaystyle \chi _{m,\;1}(a)}
4) Since
gcd
(
1
,
m
)
=
1
,
{\displaystyle \gcd(1,m)=1,}
χ
(
1
)
≠
0
{\displaystyle \;\chi (1)\neq 0}
χ
(
1
)
χ
(
1
)
=
χ
(
1
×
1
)
=
χ
(
1
)
{\displaystyle \chi (1)\chi (1)=\chi (1\times 1)=\chi (1)}
χ
(
1
)
=
1.
{\displaystyle \chi (1)=1.}
5) Property 3) is equivalent to
if
a
≡
b
(
mod
m
)
{\displaystyle a\equiv b{\pmod {m}}}
χ
(
a
)
=
χ
(
b
)
.
{\displaystyle \chi (a)=\chi (b).}
6) Property 1) implies that, for any positive integer
n
{\displaystyle n}
χ
(
a
n
)
=
χ
(
a
)
n
.
{\displaystyle \chi (a^{n})=\chi (a)^{n}.}
7) Euler's theorem states that if
gcd
(
a
,
m
)
=
1
{\displaystyle \gcd(a,m)=1}
a
ϕ
(
m
)
≡
1
(
mod
m
)
.
{\displaystyle a^{\phi (m)}\equiv 1{\pmod {m}}.}
χ
(
a
)
ϕ
(
m
)
=
χ
(
a
ϕ
(
m
)
)
=
χ
(
1
)
=
1.
{\displaystyle \chi (a)^{\phi (m)}=\chi (a^{\phi (m)})=\chi (1)=1.}
That is, the nonzero values of
χ
(
a
)
{\displaystyle \chi (a)}
ϕ
(
m
)
{\displaystyle \phi (m)}
roots of unity :
χ
(
a
)
=
{
0
if
gcd
(
a
,
m
)
>
1
e
2
π
i
r
ϕ
(
m
)
=
E
(
r
ϕ
(
m
)
)
if
gcd
(
a
,
m
)
=
1
{\displaystyle \chi (a)={\begin{cases}0&{\text{if }}\;\gcd(a,m)>1\\e^{2\pi i{\frac {r}{\phi (m)}}}=\operatorname {E} \left({\frac {r}{\phi (m)}}\right)&{\text{if }}\;\gcd(a,m)=1\end{cases}}}
for some integer
r
{\displaystyle r}
χ
{\displaystyle \chi }
a
{\displaystyle a}
8) If
χ
{\displaystyle \chi }
χ
′
{\displaystyle \chi '}
χ
χ
′
,
{\displaystyle \chi \chi ',}
χ
χ
′
(
a
)
=
χ
(
a
)
χ
′
(
a
)
,
{\displaystyle \chi \chi '(a)=\chi (a)\chi '(a),}
χ
χ
′
{\displaystyle \chi \chi '}
The principal character is an identity:
χ
χ
0
(
a
)
=
χ
(
a
)
χ
0
(
a
)
=
{
0
×
0
=
χ
(
a
)
if
gcd
(
a
,
m
)
>
1
χ
(
a
)
×
1
=
χ
(
a
)
if
gcd
(
a
,
m
)
=
1.
{\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}}}
9) The complex conjugate of a root of unity is its inverse (see here for details):
χ
(
a
)
χ
¯
(
a
)
=
{
0
if
gcd
(
a
,
m
)
>
1
1
if
gcd
(
a
,
m
)
=
1.
{\displaystyle \chi (a){\overline {\chi }}(a)={\begin{cases}0&{\text{if }}\;\gcd(a,m)>1\\1&{\text{if }}\;\gcd(a,m)=1.\end{cases}}}
In other words
χ
χ
¯
=
χ
0
{\displaystyle \chi {\overline {\chi }}=\chi _{0}}
Note that this implies for
gcd
(
a
,
m
)
=
1
,
χ
(
a
−
1
)
=
χ
(
a
)
−
1
,
{\displaystyle \gcd(a,m)=1,\;\chi (a^{-1})=\chi (a)^{-1},}
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
edit
There are three cases to consider: powers of odd primes, powers of 2, and products of prime powers.
Powers of odd primes
edit
If
q
=
p
k
{\displaystyle q=p^{k}}
(
Z
/
q
Z
)
×
{\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{\times }}
ϕ
(
q
)
{\displaystyle \phi (q)}
primitive root .
Let
g
q
{\displaystyle g_{q}}
q
{\displaystyle q}
ν
q
(
a
)
{\displaystyle \nu _{q}(a)}
a
,
gcd
(
a
,
q
)
=
1
{\displaystyle a,\;\gcd(a,q)=1}
a
≡
g
q
ν
q
(
a
)
(
mod
q
)
,
{\displaystyle a\equiv g_{q}^{\nu _{q}(a)}{\pmod {q}},}
0
≤
ν
q
(
a
)
<
ϕ
(
q
)
.
{\displaystyle 0\leq \nu _{q}(a)<\phi (q).}
For
a
,
gcd
(
a
,
q
)
=
1
{\displaystyle a,\;\gcd(a,q)=1}
χ
(
a
)
{\displaystyle \chi (a)}
χ
(
g
q
)
.
{\displaystyle \chi (g_{q}).}
ω
q
=
ζ
ϕ
(
q
)
{\displaystyle \omega _{q}=\zeta _{\phi (q)}}
ϕ
(
q
)
{\displaystyle \phi (q)}
χ
(
g
q
)
{\displaystyle \chi (g_{q})}
ω
q
,
ω
q
2
,
.
.
.
ω
q
ϕ
(
q
)
=
1.
{\displaystyle \omega _{q},\omega _{q}^{2},...\omega _{q}^{\phi (q)}=1.}
ϕ
(
q
)
{\displaystyle \phi (q)}
q
.
{\displaystyle q.}
r
,
gcd
(
q
,
r
)
=
1
{\displaystyle r,\;\gcd(q,r)=1}
χ
q
,
r
(
a
)
{\displaystyle \chi _{q,\;r}(a)}
χ
q
,
r
(
a
)
=
{
0
if
gcd
(
a
,
q
)
>
1
ω
q
ν
q
(
r
)
ν
q
(
a
)
if
gcd
(
a
,
q
)
=
1.
{\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}}}
Then for
r
,
s
{\displaystyle r,s}
q
{\displaystyle q}
gcd
(
r
,
q
)
=
gcd
(
s
,
q
)
=
1
{\displaystyle \gcd(r,q)=\gcd(s,q)=1}
χ
q
,
r
(
a
)
χ
q
,
r
(
b
)
=
χ
q
,
r
(
a
b
)
,
{\displaystyle \chi _{q,\;r}(a)\chi _{q,\;r}(b)=\chi _{q,\;r}(ab),}
and
χ
q
,
r
(
a
)
χ
q
,
s
(
a
)
=
χ
q
,
r
s
(
a
)
{\displaystyle \chi _{q,\;r}(a)\chi _{q,\;s}(a)=\chi _{q,\;rs}(a)}
where the latter formula shows an explicit isomorphism between the group of characters mod
q
{\displaystyle q}
(
Z
/
q
Z
)
×
.
{\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{\times }.}
ϕ
(
9
)
=
6
{\displaystyle \phi (9)=6}
2
≡
2
,
2
2
≡
4
,
2
3
≡
8
,
2
4
≡
7
,
2
5
≡
5
,
2
6
≡
2
0
≡
1
(
mod
9
)
,
{\displaystyle 2\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}},}
so the values of
ν
9
{\displaystyle \nu _{9}}
a
1
2
4
5
7
8
ν
9
(
a
)
0
1
2
5
4
3
{\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}a&1&2&4&5&7&8\\\hline \nu _{9}(a)&0&1&2&5&4&3\\\end{array}}}
The characters mod 9 are (
ω
=
ζ
6
{\displaystyle \omega =\zeta _{6}}
a
1
2
4
5
7
8
χ
9
,
1
(
a
)
1
1
1
1
1
1
χ
9
,
2
(
a
)
1
ω
ω
2
−
ω
2
−
ω
−
1
χ
9
,
4
(
a
)
1
ω
2
−
ω
−
ω
ω
2
1
χ
9
,
5
(
a
)
1
−
ω
2
−
ω
ω
ω
2
−
1
χ
9
,
7
(
a
)
1
−
ω
ω
2
ω
2
−
ω
1
χ
9
,
8
(
a
)
1
−
1
1
−
1
1
−
1
{\displaystyle {\begin{array}{|c|c|c|c|c|c|c|}a&1&2&4&5&7&8\\\hline \chi _{9,\;1}(a)&1&1&1&1&1&1\\\chi _{9,\;2}(a)&1&\omega &\omega ^{2}&-\omega ^{2}&-\omega &-1\\\chi _{9,\;4}(a)&1&\omega ^{2}&-\omega &-\omega &\omega ^{2}&1\\\chi _{9,\;5}(a)&1&-\omega ^{2}&-\omega &\omega &\omega ^{2}&-1\\\chi _{9,\;7}(a)&1&-\omega &\omega ^{2}&\omega ^{2}&-\omega &1\\\chi _{9,\;8}(a)&1&-1&1&-1&1&-1\\\end{array}}}
(
Z
/
2
Z
)
×
{\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{\times }}
(
Z
/
4
Z
)
×
{\displaystyle (\mathbb {Z} /4\mathbb {Z} )^{\times }}
≡
1
(
mod
4
)
{\displaystyle \equiv 1{\pmod {4}}}
≡
3
(
mod
4
)
.
{\displaystyle \equiv 3{\pmod {4}}.}
For example
5
≡
5
,
5
2
≡
5
0
≡
1
(
mod
8
)
{\displaystyle 5\equiv 5,\;5^{2}\equiv 5^{0}\equiv 1{\pmod {8}}}
5
≡
5
,
5
2
≡
9
,
5
3
≡
13
,
5
4
≡
5
0
≡
1
(
mod
16
)
{\displaystyle 5\equiv 5,\;5^{2}\equiv 9,\;5^{3}\equiv 13,\;5^{4}\equiv 5^{0}\equiv 1{\pmod {16}}}
5
≡
5
,
5
2
≡
25
,
5
3
≡
29
,
5
4
≡
17
,
5
5
≡
21
,
5
6
≡
9
,
5
7
≡
13
,
5
8
≡
5
0
≡
1
(
mod
32
)
.
{\displaystyle 5\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}}.}
Let
q
=
2
k
,
k
≥
3
{\displaystyle q=2^{k},\;\;k\geq 3}
(
Z
/
q
Z
)
×
{\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{\times }}
ϕ
(
q
)
2
{\displaystyle {\frac {\phi (q)}{2}}}
a
{\displaystyle a}
ν
0
{\displaystyle \nu _{0}}
ν
2
{\displaystyle \nu _{2}}
a
≡
(
−
1
)
ν
0
(
a
)
5
ν
2
(
a
)
(
mod
q
)
,
{\displaystyle a\equiv (-1)^{\nu _{0}(a)}5^{\nu _{2}(a)}{\pmod {q}},}
0
≤
ν
0
<
2
,
0
≤
ν
2
<
ϕ
(
q
)
2
.
{\displaystyle 0\leq \nu _{0}<2,\;\;0\leq \nu _{2}<{\frac {\phi (q)}{2}}.}
For odd
a
{\displaystyle a}
χ
(
a
)
{\displaystyle \chi (a)}
χ
(
−
1
)
{\displaystyle \chi (-1)}
χ
(
5
)
.
{\displaystyle \chi (5).}
ω
2
=
ζ
ϕ
(
q
)
2
{\displaystyle \omega _{2}=\zeta _{\frac {\phi (q)}{2}}}
ϕ
(
q
)
2
{\displaystyle {\frac {\phi (q)}{2}}}
χ
(
(
−
1
)
ν
0
(
a
)
5
ν
2
(
a
)
)
{\displaystyle \chi ((-1)^{\nu _{0}(a)}5^{\nu _{2}(a)})}
±
ω
2
,
±
ω
2
2
,
.
.
.
±
ω
2
ϕ
(
q
)
2
=
±
1.
{\displaystyle \pm \omega _{2},\pm \omega _{2}^{2},...\pm \omega _{2}^{\frac {\phi (q)}{2}}=\pm 1.}
ϕ
(
q
)
{\displaystyle \phi (q)}
q
.
{\displaystyle q.}
r
{\displaystyle r}
χ
q
,
r
(
a
)
{\displaystyle \chi _{q,\;r}(a)}
χ
q
,
r
(
a
)
=
{
0
if
a
is even
(
−
1
)
ν
0
(
r
)
ν
0
(
a
)
ω
2
ν
2
(
r
)
ν
2
(
a
)
if
a
is odd
.
{\displaystyle \chi _{q,\;r}(a)={\begin{cases}0&{\text{if }}\;a{\text{ is even}}\\(-1)^{\nu _{0}(r)\nu _{0}(a)}\omega _{2}^{\nu _{2}(r)\nu _{2}(a)}&{\text{if }}\;a{\text{ is odd}}.\end{cases}}}
Then for odd
r
,
s
{\displaystyle r,s}
χ
q
,
r
(
a
)
χ
q
,
r
(
b
)
=
χ
q
,
r
(
a
b
)
{\displaystyle \chi _{q,\;r}(a)\chi _{q,\;r}(b)=\chi _{q,\;r}(ab)}
and
χ
q
,
r
(
a
)
χ
q
,
s
(
a
)
=
χ
q
,
r
s
(
a
)
{\displaystyle \chi _{q,\;r}(a)\chi _{q,\;s}(a)=\chi _{q,\;rs}(a)}
and the later formula is an isomorphism between the group of characters mod
2
k
{\displaystyle 2^{k}}
(
Z
/
2
k
Z
)
×
.
{\displaystyle (\mathbb {Z} /2^{k}\mathbb {Z} )^{\times }.}
For example, mod 16 (
ϕ
(
16
)
=
8
{\displaystyle \phi (16)=8}
a
1
3
5
7
9
11
13
15
ν
0
(
a
)
0
1
0
1
0
1
0
1
ν
2
(
a
)
0
3
1
2
2
1
3
0
{\displaystyle {\begin{array}{|||}a&1&3&5&7&9&11&13&15\\\hline \nu _{0}(a)&0&1&0&1&0&1&0&1\\\nu _{2}(a)&0&3&1&2&2&1&3&0\\\end{array}}}
The characters mod 16 are (
i
=
ζ
4
{\displaystyle i=\zeta _{4}}
a
1
3
5
7
9
11
13
15
χ
16
,
1
(
a
)
1
1
1
1
1
1
1
1
χ
16
,
3
(
a
)
1
−
i
−
i
1
−
1
i
i
−
1
χ
16
,
5
(
a
)
1
−
i
i
−
1
−
1
i
−
i
1
χ
16
,
7
(
a
)
1
1
−
1
−
1
1
1
−
1
−
1
χ
16
,
9
(
a
)
1
−
1
−
1
1
1
−
1
−
1
1
χ
16
,
11
(
a
)
1
i
i
1
−
1
−
i
−
i
−
1
χ
16
,
13
(
a
)
1
i
−
i
−
1
−
1
−
i
i
1
χ
16
,
15
(
a
)
1
−
1
1
−
1
1
−
1
1
−
1
{\displaystyle {\begin{array}{|||}a&1&3&5&7&9&11&13&15\\\hline \chi _{16,\;1}(a)&1&1&1&1&1&1&1&1\\\chi _{16,\;3}(a)&1&-i&-i&1&-1&i&i&-1\\\chi _{16,\;5}(a)&1&-i&i&-1&-1&i&-i&1\\\chi _{16,\;7}(a)&1&1&-1&-1&1&1&-1&-1\\\chi _{16,\;9}(a)&1&-1&-1&1&1&-1&-1&1\\\chi _{16,\;11}(a)&1&i&i&1&-1&-i&-i&-1\\\chi _{16,\;13}(a)&1&i&-i&-1&-1&-i&i&1\\\chi _{16,\;15}(a)&1&-1&1&-1&1&-1&1&-1\\\end{array}}}
Products of prime powers
edit
Let
m
=
p
1
k
1
p
2
k
2
.
.
.
p
r
k
r
{\displaystyle m=p_{1}^{k_{1}}p_{2}^{k_{2}}...p_{r}^{k_{r}}}
m
{\displaystyle m}
here
(
Z
/
m
Z
)
×
≅
(
Z
/
p
1
k
1
Z
)
×
×
(
Z
/
p
2
k
2
Z
)
×
.
.
.
×
(
Z
/
p
r
k
r
Z
)
×
.
{\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }\cong (\mathbb {Z} /p_{1}^{k_{1}}\mathbb {Z} )^{\times }\times (\mathbb {Z} /p_{2}^{k_{2}}\mathbb {Z} )^{\times }...\times (\mathbb {Z} /p_{r}^{k_{r}}\mathbb {Z} )^{\times }.}
Summary and consequences
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The group of Dirichlet characters mod
q
{\displaystyle q}
(
Z
/
q
Z
)
×
{\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{\times }}
q
{\displaystyle q}
Unique factorization
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If
m
=
p
1
k
1
p
2
k
2
.
.
.
p
r
k
r
{\displaystyle m=p_{1}^{k_{1}}p_{2}^{k_{2}}...p_{r}^{k_{r}}}
q
1
=
p
1
k
1
,
.
.
.
q
r
=
p
r
k
r
.
{\displaystyle q_{1}=p_{1}^{k_{1}},...q_{r}=p_{r}^{k_{r}}.}
gcd
(
m
,
t
)
=
1
{\displaystyle \gcd(m,t)=1}
χ
m
,
t
=
χ
q
1
,
t
χ
q
2
,
t
.
.
.
χ
q
r
,
t
.
{\displaystyle \chi _{m,\;t}=\chi _{q_{1},\;t}\chi _{q_{2},\;t}...\chi _{q_{r},\;t}.}
∑
a
∈
A
χ
(
a
)
=
{
|
A
|
if
χ
=
χ
0
0
if
χ
≠
χ
0
{\displaystyle \sum _{a\in A}\chi (a)={\begin{cases}|A|&{\text{ if }}\;\chi =\chi _{0}\\0&{\text{ if }}\;\chi \neq \chi _{0}\end{cases}}}
∑
χ
∈
A
^
χ
(
a
)
=
{
|
A
|
if
a
=
e
0
if
a
≠
e
{\displaystyle \sum _{\chi \in {\hat {A}}}\chi (a)={\begin{cases}|A|&{\text{ if }}\;a=e\\0&{\text{ if }}\;a\neq e\end{cases}}}
Classification of characters
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