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DirichletCharacter

DirichletCharacter
gives the Dirichlet character with modulus k and index j.
  • Integer mathematical function, suitable for both symbolic and numerical manipulation.
  • DirichletCharacter 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:
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All the modulo 7 characters:
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Plot them:
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Evaluate for large arguments:
Compute the Dirichlet transform:
DirichletCharacter threads element-wise over lists:
Compute the number of primitive Dirichlet characters modulo :
Plot of :
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 form a group:
Addition:
Zero element:
Inverse:
Operations on Dirichlet characters:
Gauss sums:
The product of and its character modulo at values coprime to gives Gauss sums:
For primitive characters modulo , are zero at values not coprime to :
For primitive characters modulo , absolute values of Gauss sums are equal to :
Find conductors of Dirichlet characters modulo with an odd prime power:
DirichletCharacter has a conductor :
Verify:
DirichletCharacter is periodic:
DirichletCharacter is completely multiplicative:
Values of Dirichlet characters are equal to zero or roots of unity:
DirichletCharacter modulo is nonzero at values coprime to :
DirichletCharacter modulo is zero at values not coprime to :
The trivial character takes the value for all integers:
A principal character modulo has index and gives for values coprime to ; otherwise it gives :
Real Dirichlet characters modulo have index or :
JacobiSymbol is a real Dirichlet character modulo k for odd integers k:
A real primitive character modulo k can be defined as JacobiSymbol:
Nonprimitive real characters can be written in terms of JacobiSymbol at integers coprime to :
DirichletCharacter gives at the primitive root n of k, when it exists:
Use the multiplicative property of DirichletCharacter to get values at integers coprime to :
A character modulo can be written as a product of characters modulo prime powers of :
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