DirichletRing Coef

dirichlet.spad line 35 [edit on github]

• Coef: Ring

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?: (%, PositiveInteger) -> Boolean

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

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

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

from EntireRing

associator: (%, %, %) -> %

from NonAssociativeRng

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

coerce: % -> % if Coef has CommutativeRing

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: % -> PositiveInteger -> Coef

coerce: % -> Stream Coef

coerce: (PositiveInteger -> Coef) -> %

coerce: Coef -> % if Coef has CommutativeRing

from Algebra Coef

coerce: Integer -> %

from NonAssociativeRing

coerce: Stream Coef -> %

commutator: (%, %) -> %

from NonAssociativeRng

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

from NonAssociativeAlgebra %

recip: % -> Union(%, failed)

from MagmaWithUnit

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

subtractIfCan: (%, %) -> Union(%, failed)

from CancellationAbelianMonoid

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

CoercibleTo OutputForm

CommutativeRing if Coef has CommutativeRing

CommutativeStar if Coef has CommutativeRing

Eltable(PositiveInteger, Coef)

EntireRing if Coef has CommutativeRing

IntegralDomain if Coef has CommutativeRing

LeftModule %

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 %

RightModule Coef if Coef has CommutativeRing

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip if Coef has CommutativeRing

unitsKnown