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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Dirichlet convolution canonical | 6 |
| Cauchy product for Dirichlet series coefficients | 1 |
| Dirichlet characters form an abelian group under Dirichlet convolution | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4746242 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Dirichlet convolution Context triple: [Peter Gustav Lejeune Dirichlet, notableWork, Dirichlet convolution]
-
A.
Dirichlet series
A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
-
B.
Ramanujan’s sum
Ramanujan’s sum is a number-theoretic function introduced by Srinivasa Ramanujan, expressing certain periodic arithmetic functions as finite trigonometric sums over primitive roots of unity.
-
C.
Dirichlet L-functions
Dirichlet L-functions are complex analytic functions built from Dirichlet characters that generalize the Riemann zeta function and play a central role in number theory, particularly in the study of primes in arithmetic progressions.
-
D.
Dirichlet characters
Dirichlet characters are completely multiplicative periodic arithmetic functions modulo an integer, fundamental in analytic number theory for constructing Dirichlet L-functions and studying the distribution of primes in arithmetic progressions.
-
E.
Chebyshev functions
Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Dirichlet convolution Target entity description: 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.
-
A.
Dirichlet series
A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
-
B.
Ramanujan’s sum
Ramanujan’s sum is a number-theoretic function introduced by Srinivasa Ramanujan, expressing certain periodic arithmetic functions as finite trigonometric sums over primitive roots of unity.
-
C.
Dirichlet L-functions
Dirichlet L-functions are complex analytic functions built from Dirichlet characters that generalize the Riemann zeta function and play a central role in number theory, particularly in the study of primes in arithmetic progressions.
-
D.
Dirichlet characters
Dirichlet characters are completely multiplicative periodic arithmetic functions modulo an integer, fundamental in analytic number theory for constructing Dirichlet L-functions and studying the distribution of primes in arithmetic progressions.
-
E.
Chebyshev functions
Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
binary operation
ⓘ
operation on arithmetic functions ⓘ |
| appliesTo | Dirichlet characters ⓘ |
| 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
ⓘ
linked to:
Dirichlet convolution
|
| hasIdentityElement | true ⓘ |
| identityElement | delta function at 1 ⓘ |
| identityFunctionDefinition | δ(1)=1 and δ(n)=0 for n>1 ⓘ |
| identityFunctionName |
Dirichlet delta function
ⓘ
linked to:
Kronecker delta
|
| inverseExistenceCondition | every arithmetic function with f(1) ≠ 0 has a Dirichlet inverse ⓘ |
| inverseOperationName | Dirichlet inverse ⓘ |
| 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
ⓘ
linked to:
Möbius function
constant function 1 ⓘ identity function id(n)=n ⓘ |
| linearity | linear in each argument over pointwise addition of functions ⓘ |
| namedAfter | Peter Gustav Lejeune Dirichlet ⓘ |
| 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
ⓘ
linked to:
Dirichlet convolution
|
| relatedConcept |
convolution algebra
ⓘ
group of Dirichlet characters under convolution ⓘ |
| relatedTo |
Dirichlet series
ⓘ
Möbius inversion formula ⓘ 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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Dirichlet convolution Description of subject: 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.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Dirichlet convolution
Dirichlet convolution
→
propertyOnCharacters
→
Dirichlet characters form an abelian group under Dirichlet convolution
ⓘ
linked to: Dirichlet convolution