Character Orthogonality
The orthogonality relations for characters of a finite group transfer to Dirichlet characters. If we fix a character χ modulo n then the sum
unless χ is principal, in which case the sum is φ(n). Similarly, if we fix a residue class a modulo n and sum over all characters we have
unless a=1 in which case the sum is φ(n). We deduce that any periodic function with period n supported on the residue classes prime to n is a linear combination of Dirichlet characters.
Read more about this topic: Dirichlet Character
Famous quotes containing the word character:
“Pity the man who has a character to support—it is worse than a large family—he is silent poor indeed.”
—Henry David Thoreau (1817–1862)