# DirichletRing CoefΒΆ

DirichletRing is the ring of arithmetical functions with Dirichlet convolution as multiplication

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (%, Coef) -> % if Coef has CommutativeRing

from RightModule Coef

*: (Coef, %) -> % if Coef has CommutativeRing

from LeftModule Coef

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

=: (%, %) -> Boolean

from BasicType

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

`additive?(a, n)` returns `true` if the first `n` coefficients of a are additive

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %
associates?: (%, %) -> Boolean if Coef has CommutativeRing

from EntireRing

associator: (%, %, %) -> %
characteristic: () -> NonNegativeInteger
coerce: % -> % if Coef has CommutativeRing

from Algebra %

coerce: % -> OutputForm

coerce: % -> PositiveInteger -> Coef

coerce: % -> Stream Coef

coerce: (PositiveInteger -> Coef) -> %

coerce: Coef -> % if Coef has CommutativeRing

from Algebra Coef

coerce: Integer -> %

coerce: Stream Coef -> %

commutator: (%, %) -> %
elt: (%, PositiveInteger) -> Coef

from Eltable(PositiveInteger, Coef)

exquo: (%, %) -> Union(%, failed) if Coef has CommutativeRing

from EntireRing

latex: % -> String

from SetCategory

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

multiplicative?: (%, PositiveInteger) -> Boolean

`multiplicative?(a, n)` returns `true` if the first `n` coefficients of a are multiplicative

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

plenaryPower: (%, PositiveInteger) -> % if Coef has CommutativeRing
recip: % -> Union(%, failed)

from MagmaWithUnit

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

subtractIfCan: (%, %) -> Union(%, failed)
unit?: % -> Boolean if Coef has CommutativeRing

from EntireRing

unitCanonical: % -> % if Coef has CommutativeRing

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %) if Coef has CommutativeRing

from EntireRing

zero?: % -> Boolean

from AbelianMonoid

zeta: %

`zeta()` returns the function which is constantly one

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra % if Coef has CommutativeRing

Algebra Coef if Coef has CommutativeRing

BasicType

BiModule(%, %)

BiModule(Coef, Coef) if Coef has CommutativeRing

CancellationAbelianMonoid

CommutativeRing if Coef has CommutativeRing

CommutativeStar if Coef has CommutativeRing

Eltable(PositiveInteger, Coef)

EntireRing if Coef has CommutativeRing

IntegralDomain if Coef has CommutativeRing

LeftModule Coef if Coef has CommutativeRing

Magma

MagmaWithUnit

Module % if Coef has CommutativeRing

Module Coef if Coef has CommutativeRing

Monoid

NonAssociativeAlgebra % if Coef has CommutativeRing

NonAssociativeAlgebra Coef if Coef has CommutativeRing

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if Coef has CommutativeRing

RightModule Coef if Coef has CommutativeRing

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip if Coef has CommutativeRing

unitsKnown