THIS IS DOCUMENTATION FOR AN OBSOLETE PRODUCT.
SEE THE DOCUMENTATION CENTER FOR THE LATEST INFORMATION.

# DirichletCharacter

 DirichletCharacter[k, j, n] 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 distinct Dirichlet characters for a given modulus k, as labeled by the index j. Different conventions can give different orderings for the possible characters.
A basic Dirichlet character:
All the modulo 7 characters:
Plot them:
A basic Dirichlet character:
 Out[1]=

All the modulo 7 characters:
 Out[1]=
Plot them:
 Out[2]=
 Scope   (3)
Evaluate for large arguments:
Compute the Dirichlet transform:
 Applications   (5)
Compute the number of primitive Dirichlet characters modulo k:
Plot of q[k]:
Define generalized Bernoulli numbers from DirichletCharacter:
Compute values at negative integers of DirichletL using generalized Bernoulli numbers:
The generalized Bernoulli number at 0 of a principal character is and zero otherwise:
Dirichlet characters modulo k form a group:
Zero element:
Inverse:
Operations on Dirichlet characters:
Gauss sums:
The product of and its character modulo k at values coprime to k gives Gauss sums:
For primitive characters modulo k, are zero at values not coprime to k:
For primitive characters modulo k, absolute values of Gauss sums are equal to :
Find conductors of Dirichlet characters modulo k with k an odd prime power:
DirichletCharacter[25, 11, n] has a conductor 5:
Verify:
DirichletCharacter is periodic:
DirichletCharacter is completely multiplicative:
Values of Dirichlet characters are equal to zero or roots of unity:
DirichletCharacter modulo k is nonzero at values coprime to k:
DirichletCharacter modulo k is zero at values not coprime to k:
The trivial character takes the value 1 for all integers:
A principal character modulo m has index 1 and gives 1 for values coprime to m; otherwise it gives 0:
Real Dirichlet characters modulo k have index 1 or :
JacobiSymbol[n, k] is a real Dirichlet character modulo k for odd integers k:
A real primitive character modulo k can be defined as JacobiSymbol[[-1]k, n]:
Nonprimitive real characters can be written in terms of JacobiSymbol at integers coprime to k:
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 k:
A character modulo k can be written as a product of characters modulo prime powers of k:
First find primitive roots of 32 and 5:
Lift the primitive roots:
Find exponents of 7 modulo 32 and 5:
Dirichlet characters are labeled in an increasing order of the number of factors:
Decomposition of the Dirichlet character modulo 32 5 with index 8:
Verify the decomposition formula for all integers coprime to 32 5:
Lift a Dirichlet character modulo 3 to a Dirichlet character modulo a power of 3:
Compute the index:
Results:
New in 7
 Site Index Choose Language