Dirichlet character

English

Noun

Dirichlet character (plural Dirichlet characters)

1. (analytic number theory) A complex-valued arithmetic function $\chi :\mathbb {Z} \rightarrow \mathbb {C}$ that satisfies the following conditions for some positive integer $m$ and all integers $a$ and $b$:

   1)   $\chi (ab)=\chi (a)\chi (b);$  

   2)   $\chi (a){\begin{cases}=0&{\text{if }}\;\gcd(a,m)>1\\\neq 0&{\text{if }}\;\gcd(a,m)=1.\end{cases}}$ (gcd is the greatest common divisor)

   3)   $\chi (a+m)=\chi (a)$.