DirichletRing(Coef)

dirichlet.spad line 34 [edit on github]

• Coef : Ring

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

* : (%, %) -> %

from Magma

* : (%, 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

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

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 : Coef -> % if Coef has CommutativeRing

from Algebra(Coef)

coerce : Integer -> %

from NonAssociativeRing

coerce : Mapping(Coef, PositiveInteger) -> %

coerce : Stream(Coef) -> %

coerce : % -> Mapping(Coef, PositiveInteger)

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

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(Coef)

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

~= : (%, %) -> Boolean

from BasicType

IntegralDomain

Module(Coef)

noZeroDivisors

LeftModule(Coef)

Algebra(Coef)

Monoid

Algebra(%)

AbelianMonoid

CancellationAbelianMonoid

MagmaWithUnit

NonAssociativeRing

AbelianGroup

CommutativeStar

Eltable(PositiveInteger, Coef)

LeftModule(%)

Module(%)

SetCategory

Rng

CommutativeRing

TwoSidedRecip

Magma

SemiGroup

RightModule(Coef)

BiModule(%, %)

unitsKnown

CoercibleTo(OutputForm)

AbelianSemiGroup

NonAssociativeSemiRing

RightModule(%)

NonAssociativeAlgebra(%)

BiModule(Coef, Coef)

NonAssociativeRng

Ring

SemiRng

EntireRing

NonAssociativeSemiRng

BasicType

NonAssociativeAlgebra(Coef)

SemiRing