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.
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“Gross and obscure natures, however decorated, seem impure shambles; but character gives splendor to youth, and awe to wrinkled skin and gray hairs.”
—Ralph Waldo Emerson (1803–1882)