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 the Wolfram Language as DirichletCharacter[k,
j, n], where 
is the modulus and 
is the index.
