DirichletRing CoefΒΆ

dirichlet.spad line 35

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

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
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
=: (%, %) -> 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: Coef -> % if Coef has CommutativeRing
from Algebra Coef
coerce: Integer -> %
from NonAssociativeRing

coerce: PositiveInteger -> Coef -> %

coerce: Stream Coef -> %

commutator: (%, %) -> %
from NonAssociativeRng
elt: (%, PositiveInteger) -> Coef
from Eltable(PositiveInteger, Coef)
exquo: (%, %) -> Union(%, failed) if Coef has CommutativeRing
from EntireRing
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
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
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

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if Coef has CommutativeRing

RightModule %

RightModule Coef if Coef has CommutativeRing

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown