Number Theoretic Character

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 k is a complex function chi_k(n) for positive integer n such that


for all m,n, and


if (k,n)!=1. chi_k can only assume values which are phi(k) roots of unity, where phi is the totient function.

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

See also

Dirichlet L-Series, Multiplicative Character, Primitive Character

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

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Cite this as:

Sondow, Jonathan and Weisstein, Eric W. "Number Theoretic Character." From MathWorld--A Wolfram Web Resource.

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