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
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“The actor should not play a part. Like the Aeolian harps that used to be hung in the trees to be played only by the breeze, the actor should be an instrument played upon by the character he depicts.”
—Alla Nazimova (1879–1945)