Jordan’s totient functions
E300758
Jordan’s totient functions are a family of arithmetic functions in number theory that generalize Euler’s totient function to count k-tuples of integers modulo n with certain coprimality conditions.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Jordan’s totient functions canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2815407 — 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: Jordan’s totient functions Context triple: [Euler’s totient function φ(n), relatedFunction, Jordan’s totient functions]
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A.
Euler’s totient function φ(n)
Euler’s totient function φ(n) is a fundamental arithmetic function in number theory that counts the positive integers up to n that are relatively prime to n and plays a key role in topics such as modular arithmetic and cryptography.
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B.
Multiplicative Number Theory
Multiplicative Number Theory is a branch of analytic number theory that studies arithmetic functions and prime number distributions through their multiplicative properties and associated Dirichlet series.
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C.
Jacobi’s four-square theorem
Jacobi’s four-square theorem is a fundamental result in number theory that gives a precise formula for the number of ways an integer can be expressed as a sum of four squares.
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D.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
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E.
Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
- 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: Jordan’s totient functions Target entity description: Jordan’s totient functions are a family of arithmetic functions in number theory that generalize Euler’s totient function to count k-tuples of integers modulo n with certain coprimality conditions.
-
A.
Euler’s totient function φ(n)
Euler’s totient function φ(n) is a fundamental arithmetic function in number theory that counts the positive integers up to n that are relatively prime to n and plays a key role in topics such as modular arithmetic and cryptography.
-
B.
Multiplicative Number Theory
Multiplicative Number Theory is a branch of analytic number theory that studies arithmetic functions and prime number distributions through their multiplicative properties and associated Dirichlet series.
-
C.
Jacobi’s four-square theorem
Jacobi’s four-square theorem is a fundamental result in number theory that gives a precise formula for the number of ways an integer can be expressed as a sum of four squares.
-
D.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
-
E.
Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
family of arithmetic functions
ⓘ
number-theoretic functions ⓘ |
| averageOrder | ∑_{n≤x} J_k(n) ~ x^{k+1}/((k+1)ζ(k+1)) ⓘ |
| codomain | nonnegative integers ⓘ |
| completelyMultiplicative | no ⓘ |
| convolutionExplanation | J_k(n) = ∑_{d|n} μ(d)(n/d)^k ⓘ |
| convolutionRelation | J_k = μ * id_k ⓘ |
| coprimalityCondition | J_k(n) counts k-tuples whose gcd with n is 1 in a suitable sense ⓘ |
| definitionInformal |
J_k(n) counts ordered k-tuples of integers modulo n that form a generating set for the additive group Z/nZ
ⓘ
J_k(n) counts ordered k-tuples of integers modulo n whose greatest common divisor with n is 1 ⓘ |
| DirichletSeries | ∑_{n≥1} J_k(n) n^{-s} = ζ(s-k)/ζ(s) ⓘ |
| domain | positive integers ⓘ |
| EulerProduct | ∑_{n≥1} J_k(n) n^{-s} = ∏_{p}(1 - p^{k-s})/(1 - p^{-s}) ⓘ |
| field | number theory ⓘ |
| generalizes | Euler’s totient function ⓘ |
| generatingFunctionType | Dirichlet generating function ⓘ |
| growthRate | J_k(n) is of order n^k ⓘ |
| introducedBy | Camille Jordan ⓘ |
| inverseConvolution | id_k = 1 * J_k ⓘ |
| multiplicative | yes ⓘ |
| multiplicativeDefinition | J_k(mn) = J_k(m)J_k(n) if gcd(m,n)=1 ⓘ |
| multiplicativeFactor | J_k(n)/n^k = ∏_{p|n}(1 - p^{-k}) ⓘ |
| multiplicativeGroupInterpretation | J_k(n) counts k-tuples that generate (Z/nZ,+) ⓘ |
| multiplicativeOnCoprimeArguments | yes ⓘ |
| namedAfter | Camille Jordan ⓘ |
| primePowerFactorization | J_k(n) = n^k ∏_{p|n}(1 - p^{-k}) ⓘ |
| primePowerFormula | J_k(p^a) = p^{ak} - p^{(a-1)k} ⓘ |
| relatedConcept |
Euler product
ⓘ
Möbius function ⓘ
surface form:
Möbius function μ(n)
Riemann zeta function ⓘ
surface form:
Riemann zeta function ζ(s)
identity function id_k(n) = n^k ⓘ |
| relationToDivisorFunction | J_k is related to the k-th power identity function id_k via Möbius inversion ⓘ |
| relationToEulerPhi | J_1(n) = φ(n) ⓘ |
| specialCase | J_1(n) = φ(n) ⓘ |
| subfield | multiplicative number theory ⓘ |
| sumOverDivisorsIdentity | ∑_{d|n} J_k(d) = n^k ⓘ |
| symbol | J_k(n) ⓘ |
| topic |
Dirichlet series of multiplicative functions
ⓘ
arithmetic functions ⓘ coprimality in residue classes ⓘ |
| usedIn |
analysis of random k-tuples modulo n
ⓘ
generalizations of Euler’s theorem ⓘ probabilistic number theory ⓘ study of finite abelian groups ⓘ zeta-function identities ⓘ |
| valueAtOne | J_k(1) = 1 ⓘ |
| valueAtPrime | J_k(p) = p^k - 1 ⓘ |
| valueAtPrimeSquare | J_k(p^2) = p^{2k} - p^k ⓘ |
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
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Input
Subject: Jordan’s totient functions Description of subject: Jordan’s totient functions are a family of arithmetic functions in number theory that generalize Euler’s totient function to count k-tuples of integers modulo n with certain coprimality conditions.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.