A number theoretic character, also called a Dirichlet character (because Dirichlet first introduced them in his famous proof that every arithmetic progression with relatively prime initial term and common difference contains infinitely many primes), modulo is a complex function for positive integer such that

			(1)
			(2)
			(3)

for all , and

(4)

if . can only assume values which are roots of unity, where is the totient function.

Number theoretic characters are implemented in Mathematica as DirichletCharacter[k, j, n], where is the modulus and is the index.

Portions of this entry contributed by Jonathan Sondow (author's link)

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Sondow, Jonathan and Weisstein, Eric W. "Number Theoretic Character." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NumberTheoreticCharacter.html