Orthogonality of Dirichlet Characters

The following are all direct corollaries of the results given in orthogonality of group characters.

Theorem

Let nZ, then and χ a dirichlet character modulo n, then

0a<nχ(a)={φ(n)χ=χ00otherwise

Theorem

Let n,aZ, then

χ(modn)χ(a)={φ(n)a1(modn)0otherwise

where the sum is taken over all dirichlet characters modulo n.


Corollary

Let n,a,bZ with gcd(b,n)=1, then

χ(modn)χ(b)χ(a)={φ(n)ab(modn)0otherwise

where the sum is taken over all Dirichlet characters modulo n.