Dirichlet convolution
E466252
Dirichlet convolution is a binary operation on arithmetic functions that combines them via summation over divisors and plays a central role in multiplicative number theory and Dirichlet series.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
binary operation
ⓘ
operation on arithmetic functions ⓘ |
| appliesTo | Dirichlet characters NERFINISHED ⓘ |
| definedOn | arithmetic functions ⓘ |
| definition | (f * g)(n) = ∑_{d | n} f(d) g(n/d) ⓘ |
| domain | positive integers ⓘ |
| field |
analytic number theory
ⓘ
multiplicative number theory ⓘ number theory ⓘ |
| formsAlgebraicStructure |
commutative monoid of arithmetic functions
ⓘ
commutative ring of arithmetic functions ⓘ |
| generalizationOf | Cauchy product for Dirichlet series coefficients NERFINISHED ⓘ |
| hasIdentityElement | true ⓘ |
| identityElement | delta function at 1 ⓘ |
| identityFunctionDefinition | δ(1)=1 and δ(n)=0 for n>1 ⓘ |
| identityFunctionName | Dirichlet delta function NERFINISHED ⓘ |
| inverseExistenceCondition | every arithmetic function with f(1) ≠ 0 has a Dirichlet inverse ⓘ |
| inverseOperationName | Dirichlet inverse NERFINISHED ⓘ |
| inverseRecurrence | f^{-1}(1)=1/f(1) and f^{-1}(n) = -(1/f(1)) ∑_{d|n, d<n} f(d) f^{-1}(n/d) ⓘ |
| isAssociative | true ⓘ |
| isCommutative | true ⓘ |
| isDistributiveOverAddition | true ⓘ |
| keyFunction |
Möbius function μ
NERFINISHED
ⓘ
constant function 1 ⓘ identity function id(n)=n ⓘ |
| linearity | linear in each argument over pointwise addition of functions ⓘ |
| namedAfter | Peter Gustav Lejeune Dirichlet NERFINISHED ⓘ |
| preservesCompleteMultiplicativity | convolution of completely multiplicative functions need not be completely multiplicative ⓘ |
| preservesMultiplicativity | convolution of multiplicative functions is multiplicative ⓘ |
| property |
(Df)(s) (Dg)(s) = D(f * g)(s) for Dirichlet series D
ⓘ
Dirichlet series turn Dirichlet convolution into ordinary multiplication ⓘ |
| propertyOnCharacters | Dirichlet characters form an abelian group under Dirichlet convolution NERFINISHED ⓘ |
| relatedConcept |
convolution algebra
ⓘ
group of Dirichlet characters under convolution ⓘ |
| relatedTo |
Dirichlet series
NERFINISHED
ⓘ
Möbius inversion formula NERFINISHED ⓘ multiplicative functions ⓘ |
| relation |
(1 * id)(n) = σ_1(n) (sum of divisors function)
ⓘ
1 * 1 = d(n) (divisor-counting function) ⓘ 1 * μ = δ (Dirichlet delta) ⓘ |
| ringType | ring with identity under Dirichlet convolution and pointwise addition ⓘ |
| symbol |
*
ⓘ
⋆ ⓘ |
| unitCondition | an arithmetic function is a unit iff f(1) ≠ 0 ⓘ |
| usedFor |
Möbius inversion in combinatorial number theory
ⓘ
expressing arithmetic functions via divisor sums ⓘ proving identities between multiplicative functions ⓘ |
| zeroElement | zero arithmetic function ⓘ |
| zeroElementProperty | f * 0 = 0 for all arithmetic functions f ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.