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Character (Number Theory)

A number theoretic function $\chi_k(n)$ for Positive integral $n$ is a character modulo $k$ if

$\displaystyle \chi_k(1)$ $\textstyle =$ $\displaystyle 1$  
$\displaystyle \chi_k(n)$ $\textstyle =$ $\displaystyle \chi_k(n+k)$  
$\displaystyle \chi_k(m)\chi_k(n)$ $\textstyle =$ $\displaystyle \chi_k(mn)$  

for all $m, n$, and

\begin{displaymath} \chi_k(n)=0 \end{displaymath}

if $(k, n)\not=1$. $\chi_k$ can only assume values which are $\phi(k)$ Roots of Unity, where $\phi$ is the Totient Function.

See also Dirichlet L-Series



© 1996-9 Eric W. Weisstein
1999-05-26