BUILT-IN MATHEMATICA SYMBOL

DirichletCharacter

gives the Dirichlet character

with modulus k and index j.

MORE INFORMATION

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.

DirichletCharacter is periodic in n with period k.

DirichletCharacter is zero when n is not coprime to k.

DirichletCharacter 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:

In[1]:=

Out[1]=

Plot them:

In[2]:=

Out[2]=

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

:

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:

Properties & Relations (11)

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: