# §27.8 Dirichlet Characters

If is a given integer, then a function is called a Dirichlet character (mod ) if it is completely multiplicative, periodic with period , and vanishes when . In other words, Dirichlet characters (mod ) satisfy the four conditions:

An example is the principal character (mod ):

For any character , if and only if , in which case the Euler–Fermat theorem (27.2.8) implies . There are exactly different characters (mod ), which can be labeled as . If is a character (mod ), so is its complex conjugate . If , then the characters satisfy the orthogonality relation

A Dirichlet character is called primitive (mod ) if for every proper divisor of (that is, a divisor ), there exists an integer , with and . If is prime, then every nonprincipal character is primitive. A divisor of is called an induced modulus for if

Every Dirichlet character (mod ) is a product

where is a character (mod ) for some induced modulus for , and is the principal character (mod ). A character is real if all its values are real. If is odd, then the real characters (mod ) are the principal character and the quadratic characters described in the next section.