Dirichlet Characters #
Let R
be a commutative monoid with zero. A Dirichlet character χ
of level n
over R
is a
multiplicative character from ZMod n
to R
sending non-units to 0. We then obtain some properties
of toUnitHom χ
, the restriction of χ
to a group homomorphism (ZMod n)ˣ →* Rˣ
.
Main definitions:
DirichletCharacter
: The type representing a Dirichlet character.changeLevel
: Extend the Dirichlet character χ of leveln
to levelm
, wheren
dividesm
.
TODO #
- properties of the conductor
Tags #
dirichlet character, multiplicative character
The type of Dirichlet characters of level n
.
Instances For
A function that modifies the level of a Dirichlet character to some multiple of its original level.
Instances For
χ
of level n
factors through a Dirichlet character χ₀
of level d
if d ∣ n
and
χ₀ = χ ∘ (ZMod n → ZMod d)
.
Instances For
The fact that d
divides n
when χ
factors through a Dirichlet character at level d
The Dirichlet character at level d
through which χ
factors
Instances For
The fact that χ
factors through χ₀
of level d
The set of natural numbers for which a Dirichlet character is periodic.
Instances For
The minimum natural number n
for which a Dirichlet character is periodic.
The Dirichlet character χ
can then alternatively be reformulated on ℤ/nℤ
.