Dirichlet character
التعريفات والمعاني
== English ==
=== Noun ===
Dirichlet character (plural Dirichlet characters)
(analytic number theory) A complex-valued arithmetic function
χ
:
Z
→
C
{\displaystyle \chi :\mathbb {Z} \rightarrow \mathbb {C} }
that satisfies the following conditions for some positive integer
m
{\displaystyle m}
and all integers
a
{\displaystyle a}
and
b
{\displaystyle b}
:
1)
χ
(
a
b
)
=
χ
(
a
)
χ
(
b
)
;
{\displaystyle \chi (ab)=\chi (a)\chi (b);}
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}}}
(gcd is the greatest common divisor)
3)
χ
(
a
+
m
)
=
χ
(
a
)
{\displaystyle \chi (a+m)=\chi (a)}
.