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 ${\displaystyle b}$:

The simplest possible character, called the principal character and usually denoted ${\displaystyle \chi _{0}}$, exists for all moduli:

Dirichlet characters were named after German mathematician Peter Gustav Lejeune Dirichlet, who introduced these functions in his 1837 paper on primes in arithmetic progressions.

${\displaystyle \phi (n)}$ is Euler's totient function.

${\displaystyle \zeta _{n}}$ is a complex primitive n-th root of unity:

${\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }}$ is the group of units mod ${\displaystyle m}$. It has order ${\displaystyle \phi (m).}$

${\displaystyle {\widehat {(\mathbb {Z} /m\mathbb {Z} )^{\times }}}}$ is the group of Dirichlet characters mod ${\displaystyle m}$.

${\displaystyle p,p_{k},}$ etc. are prime numbers.