Dirichlet character

Definition

Let $n$ be a natural number. A Dirichlet character modulo $n$ is a function $\chi :\mathbb {Z} \to \mathbb {C}$ such that:

• $\chi (a)=\chi (a+n)$ for all $a\in \mathbb {Z}$,
• $\chi (m)=0$ whenever $m$ and $n$ are not relatively prime, and
• for any $a,b\in \mathbb {N}$:

$\chi (ab)=\chi (a)\chi (b)$.

In other words, it is a homomorphism from the multiplicative monoid of the ring of integers to the ring of complex numbers that descends to a homomorphism from the ring of integers modulo $n$.

If $n$ is the smallest period of $\chi$, and $\chi$ is nonzero for all integers relatively prime to $n$, then $\chi$ is termed a primitive character modulo $n$.

The all ones function is the trivial or principal character, and it is the only character with period $1$.

Facts

• Dirichlet characters are completely multiplicative: As part of the definition, any Dirichlet character is a completely multiplicative function. Thus, a Dirichlet character is completely determined by its value at prime numbers, and its Dirichlet series has an Euler product formula.