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 (1)
| Label | Occurrences |
|---|---|
| Dirichlet convolution canonical | 2 |
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.
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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.
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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.
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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.
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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 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 ⓘ |
How these facts were elicited
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Instruction
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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 (2)
Full triples — surface form annotated when it differs from this entity's canonical label.