gives the Dirichlet character with modulus k and index j.
- Integer mathematical function, suitable for both symbolic and numerical manipulation.
- DirichletCharacter[k,j,n] picks a particular ordering for possible Dirichlet characters modulo k.
- There are ϕ distinct Dirichlet characters for a given modulus k, as labeled by the index j. Different conventions can give different orderings for the possible characters.
- DirichletCharacter[k,j,n] is periodic in n with period k.
- DirichletCharacter[k,j,n] is zero when n is not coprime to k.
- DirichletCharacter[k,j,n] is a multiplicative function in n.
Examplesopen allclose all
DirichletCharacter threads element-wise over lists:
Define generalized Bernoulli numbers from DirichletCharacter:
Compute values at negative integers of DirichletL using generalized Bernoulli numbers:
DirichletCharacter[25,11,n] has a conductor 5:
Properties & Relations (11)
DirichletCharacter is periodic:
DirichletCharacter is completely multiplicative:
DirichletCharacter modulo is nonzero at values coprime to :
DirichletCharacter modulo is zero at values not coprime to :
JacobiSymbol[n,k] is a real Dirichlet character modulo k for odd integers k:
A real primitive character χ modulo k can be defined as JacobiSymbol[χ[-1]k,n]:
Nonprimitive real characters can be written in terms of JacobiSymbol at integers coprime to :
DirichletCharacter[k,j,n] gives at the primitive root n of k, when it exists:
Use the multiplicative property of DirichletCharacter to get values at integers coprime to :