§27.8 Dirichlet Characters

If $k$ $(>1)$ is a given integer, then a function $\chi \left(n\right)$ is called a Dirichlet character (mod $k$) if it is completely multiplicative, periodic with period $k$, and vanishes when $\left(n,k\right)>1$. In other words, Dirichlet characters (mod $k$) satisfy the four conditions:

An example is the principal character (mod $k$):

For any character $\chi (modk)$, $\chi \left(n\right)\ne 0$ if and only if $\left(n,k\right)=1$, in which case the Euler–Fermat theorem (27.2.8) implies ${\left(\chi \left(n\right)\right)}^{\varphi \left(k\right)}=1$. There are exactly $\varphi \left(k\right)$ different characters (mod $k$), which can be labeled as ${\chi }_1,\mathrm{\dots },{\chi }_{\varphi \left(k\right)}$. If $\chi$ is a character (mod $k$), so is its complex conjugate $\overline{\chi }$. If $\left(n,k\right)=1$, then the characters satisfy the orthogonality relation

A Dirichlet character $\chi (modk)$ is called primitive (mod $k$) if for every proper divisor $d$ of $k$ (that is, a divisor ), there exists an integer $a\equiv 1(modd)$, with $\left(a,k\right)=1$ and $\chi \left(a\right)\ne 1$. If $k$ is prime, then every nonprincipal character $\chi (modk)$ is primitive. A divisor $d$ of $k$ is called an induced modulus for $\chi$ if

Every Dirichlet character $\chi$ (mod $k$) is a product

where ${\chi }_0$ is a character (mod $d$) for some induced modulus $d$ for $\chi$, and ${\chi }_1$ is the principal character (mod $k$). A character is real if all its values are real. If $k$ is odd, then the real characters (mod $k$) are the principal character and the quadratic characters described in the next section.