Built-in Mathematica Symbol

DirichletCharacter

DirichletCharacter[k, j, n]
gives the Dirichlet character $chi_({k,j})(n)$ with modulus k and index j.

MORE INFORMATION

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.

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.

EXAMPLES

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Basic Examples (2)

A basic Dirichlet character:

All the modulo 7 characters:

Plot them:

A basic Dirichlet character:

In[1]:=

Out[1]=

All the modulo 7 characters:

Plot them:

Scope (3)

Evaluate for large arguments:

Compute the Dirichlet transform:

DirichletCharacter threads element-wise over lists:

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:

Addition:

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:

Properties & Relations (11)

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: