Hardy–Littlewood circle method
E120394
The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hardy–Littlewood circle method canonical | 8 |
| Vinogradov's method | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1060254 — 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: Hardy–Littlewood circle method Context triple: [G. H. Hardy, knownFor, Hardy–Littlewood circle method]
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A.
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.
-
B.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
C.
Ramanujan partition congruences
Ramanujan partition congruences are remarkable number-theoretic results discovered by Srinivasa Ramanujan that describe surprising modular patterns in the partition function, such as specific arithmetic progressions where the number of integer partitions of an integer is divisible by a given prime.
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D.
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.
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E.
Statistical Independence in Probability, Analysis and Number Theory
"Statistical Independence in Probability, Analysis and Number Theory" is a mathematical monograph by Mark Kac that explores the concept of independence across probability theory, real analysis, and number theory.
- 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: Hardy–Littlewood circle method Target entity description: The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
-
A.
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.
-
B.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
C.
Ramanujan partition congruences
Ramanujan partition congruences are remarkable number-theoretic results discovered by Srinivasa Ramanujan that describe surprising modular patterns in the partition function, such as specific arithmetic progressions where the number of integer partitions of an integer is divisible by a given prime.
-
D.
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.
-
E.
Statistical Independence in Probability, Analysis and Number Theory
"Statistical Independence in Probability, Analysis and Number Theory" is a mathematical monograph by Mark Kac that explores the concept of independence across probability theory, real analysis, and number theory.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
analytic number theory method
ⓘ
circle method ⓘ mathematical method ⓘ |
| appliedTo |
Goldbach conjecture
ⓘ
Waring's problem ⓘ additive prime number theory ⓘ representation of integers as sums of powers ⓘ representation of integers as sums of primes ⓘ |
| approach |
approximation of generating functions near rational points
ⓘ
bounding exponential sums on minor arcs ⓘ decomposition of the unit circle into major and minor arcs ⓘ |
| basedOn | integration over the unit circle in the complex plane ⓘ |
| coreConcept |
exponential sums
ⓘ
generating functions ⓘ major arcs ⓘ minor arcs ⓘ singular integral ⓘ singular series ⓘ |
| domain | problems about representations of integers ⓘ |
| field | analytic number theory ⓘ |
| formalSetting | analysis on the torus ⓘ |
| goal |
asymptotic formulas for representation functions
ⓘ
estimation of the number of representations of integers ⓘ |
| historicalPeriod | early 20th century ⓘ |
| influenced |
development of sieve methods
ⓘ
modern additive combinatorics ⓘ |
| introducedBy |
G. H. Hardy
ⓘ
J. E. Littlewood ⓘ
linked to:
John Edensor Littlewood
|
| namedAfter |
G. H. Hardy
ⓘ
J. E. Littlewood ⓘ
linked to:
John Edensor Littlewood
|
| notableApplication |
Vinogradov's three-primes theorem
ⓘ
asymptotic formula in Waring's problem ⓘ |
| relatedTo |
Hardy–Littlewood conjectures
ⓘ
Vinogradov's method ⓘ
linked to:
Hardy–Littlewood circle method
|
| requires |
Diophantine approximation
ⓘ
estimates for exponential sums ⓘ |
| studies | additive problems in number theory ⓘ |
| techniqueType |
Fourier-analytic method
ⓘ
sieve-related method ⓘ |
| typicalOutput |
asymptotic formulas with singular series factors
ⓘ
density results for representable integers ⓘ |
| usedFor |
proofs of asymptotic versions of additive conjectures
ⓘ
quantitative results in additive number theory ⓘ |
| uses |
Fourier analysis on the unit circle
ⓘ
Fourier series ⓘ
linked to:
Fourier analysis
complex analysis ⓘ |
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: Hardy–Littlewood circle method Description of subject: The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
Referenced by (9)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Hardy–Littlewood circle method
subject linked to:
Godfrey
subject linked to:
Littlewood
van der Corput method for estimating exponential sums
→
relatedTo
→
Hardy–Littlewood circle method
ⓘ