Ramanujan’s sum
E355439
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Ramanujan expansions | 1 |
| Ramanujan sum | 1 |
| Ramanujan sums | 1 |
| Ramanujan’s sum canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3410526 — 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: Ramanujan’s sum Context triple: [Srinivasa Ramanujan, notableWork, Ramanujan’s sum]
-
A.
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.
-
B.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
C.
Poisson summation formula
The Poisson summation formula is a fundamental result in harmonic analysis that links sums of a function over the integers to sums of its Fourier transform, with deep applications in number theory, signal processing, and physics.
-
D.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
E.
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.
- 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: Ramanujan’s sum Target entity description: 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.
-
A.
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.
-
B.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
C.
Poisson summation formula
The Poisson summation formula is a fundamental result in harmonic analysis that links sums of a function over the integers to sums of its Fourier transform, with deep applications in number theory, signal processing, and physics.
-
D.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
E.
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.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
arithmetical function
ⓘ
multiplicative function ⓘ number-theoretic function ⓘ |
| alternativeForm | c_q(n) = sum_{a mod q, (a,q)=1} e^{2πi a n / q} ⓘ |
| appearsIn | papers of Srinivasa Ramanujan on highly composite numbers and related topics ⓘ |
| closedForm |
c_q(n) = sum_{d | gcd(n,q)} μ(q/d) d
ⓘ
c_q(n) = μ(q/(q,n)) φ((q,n)) / φ(q/(q,n)) ⓘ |
| definition | c_q(n) = sum_{1 ≤ a ≤ q, gcd(a,q)=1} exp(2πi a n / q) ⓘ |
| dependsOn |
integer n
ⓘ
integer q ⓘ |
| domain |
n ∈ ℤ
ⓘ
q ∈ ℕ, q ≥ 1 ⓘ |
| field | number theory ⓘ |
| generalizationOf | finite Fourier sums over primitive roots of unity ⓘ |
| hasSeriesUse | basis for expansions of arithmetic functions with period q ⓘ |
| hasVariable |
argument n
ⓘ
modulus q ⓘ |
| introducedBy | Srinivasa Ramanujan ⓘ |
| isPeriodicIn | n modulo q ⓘ |
| isRealValued | true ⓘ |
| namedAfter | Srinivasa Ramanujan ⓘ |
| orthogonalityRelation |
sum_{n mod q} c_q(n) = 0 for q > 1
ⓘ
sum_{q ≥ 1} c_q(n) c_q(m) / φ(q) converges to a function of gcd(m,n) ⓘ |
| property |
c_1(n) = 1 for all integers n
ⓘ
c_q(0) = φ(q) ⓘ c_q(n) depends only on gcd(n,q) ⓘ c_q(n) is an integer for all integers n and q ≥ 1 ⓘ c_q(n) is bounded in absolute value by φ(q) ⓘ multiplicative in q for fixed n ⓘ |
| relatedConcept |
Fourier analysis on finite abelian groups
ⓘ
Ramanujan expansion of the divisor function ⓘ Ramanujan expansion of the von Mangoldt function ⓘ |
| relatedTo |
Dirichlet characters
ⓘ
Euler’s totient function ⓘ
linked to:
Euler’s totient function φ(n)
Fourier series on arithmetic progressions ⓘ Möbius function ⓘ primitive roots of unity ⓘ |
| specialCase |
c_p(n) = -1 if p is prime and p ∤ n
ⓘ
c_p(n) = p-1 if p is prime and p | n ⓘ c_q(1) = μ(q) ⓘ |
| symbol | c_q(n) ⓘ |
| takesValuesIn | integers ⓘ |
| usedFor |
Ramanujan expansions
ⓘ
linked to:
Ramanujan’s sum
expansion of periodic arithmetic functions ⓘ expressing arithmetic functions as trigonometric sums ⓘ |
| usedIn |
Waring’s problem and related additive problems
ⓘ
analytic number theory ⓘ circle method ⓘ study of multiplicative functions ⓘ |
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: Ramanujan’s sum Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Ramanujan’s sum
linked to: Ramanujan’s sum