Dirichlet character

In analytic number theory and related branches of mathematics, a complex-valued arithmetic function ${\displaystyle \chi :\mathbb {Z} \rightarrow \mathbb {C} }$ is a Dirichlet character of modulus ${\displaystyle m}$ (where ${\displaystyle m}$ is a positive integer) if for all integers ${\displaystyle a}$ and ${\displaystyle b}$:^[1]

1. ${\displaystyle \chi (ab)=\chi (a)\chi (b);}$ that is, ${\displaystyle \chi }$ is completely multiplicative.
2. ${\displaystyle \chi (a){\begin{cases}=0&{\text{if }}\gcd(a,m)>1\\\neq 0&{\text{if }}\gcd(a,m)=1.\end{cases}}}$ (gcd is the greatest common divisor)
3. ${\displaystyle \chi (a+m)=\chi (a)}$; that is, ${\displaystyle \chi }$ is periodic with period ${\displaystyle m}$.

The simplest possible character, called the principal character, usually denoted ${\displaystyle \chi _{0}}$, (see Notation below) exists for all moduli:^[2]

${\displaystyle \chi _{0}(a)={\begin{cases}0&{\text{if }}\gcd(a,m)>1\\1&{\text{if }}\gcd(a,m)=1.\end{cases}}}$

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]