Hardy–Ramanujan asymptotic formula
E123877
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Hardy–Ramanujan asymptotic formula canonical | 3 |
| Hardy–Ramanujan theorem | 1 |
| Rademacher convergent series for p(n) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1060255 — 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–Ramanujan asymptotic formula Context triple: [G. H. Hardy, knownFor, Hardy–Ramanujan asymptotic formula]
-
A.
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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B.
Hardy–Littlewood circle method
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.
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C.
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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D.
Rogers–Ramanujan-type identities
Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
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E.
Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
- 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–Ramanujan asymptotic formula Target entity description: 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.
-
A.
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.
-
B.
Hardy–Littlewood circle method
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.
-
C.
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.
-
D.
Rogers–Ramanujan-type identities
Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
-
E.
Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
asymptotic formula
ⓘ
result in number theory ⓘ theorem in analytic number theory ⓘ |
| appliesTo |
integer partitions
ⓘ
partition function p(n) ⓘ |
| approximationQuality | asymptotically accurate as n \to \infty ⓘ |
| authors |
G. H. Hardy
ⓘ
Srinivasa Ramanujan ⓘ |
| classification | asymptotic expansion of arithmetic function ⓘ |
| describes | asymptotic growth of the partition function p(n) ⓘ |
| domainOfVariable | n \in \mathbb{N} ⓘ |
| field |
analytic number theory
ⓘ
number theory ⓘ |
| gives | approximate expression for p(n) ⓘ |
| givesLeadingTerm | \frac{1}{4n\sqrt{3}} e^{\pi \sqrt{2n/3}} ⓘ |
| growthType | subexponential in n but superpolynomial ⓘ |
| hasAsymptoticNotation | p(n) \sim f(n) as n \to \infty ⓘ |
| hasConstant | \pi\sqrt{2/3} ⓘ |
| hasDenominatorFactor | 4n\sqrt{3} ⓘ |
| hasErrorTerm | relative error tends to 0 as n \to \infty ⓘ |
| historicalSignificance | early major application of the circle method ⓘ |
| implies | log p(n) grows on the order of \sqrt{n} ⓘ |
| improvedBy |
Hardy–Ramanujan asymptotic formula
self-linksurface differs
ⓘ
surface form:
Rademacher convergent series for p(n)
|
| inspired | further work on partition asymptotics ⓘ |
| involvesConcept |
asymptotic analysis
ⓘ
circle method ⓘ integer partitions ⓘ |
| involvesFunction | partition function p(n) ⓘ |
| isLandmarkResultIn | partition theory ⓘ |
| mainExpression | p(n) \sim \frac{1}{4n\sqrt{3}} e^{\pi \sqrt{2n/3}} ⓘ |
| methodIntroducedWith | Hardy–Littlewood circle method ⓘ |
| namedAfter |
G. H. Hardy
ⓘ
Srinivasa Ramanujan ⓘ |
| publishedIn | Proceedings of the London Mathematical Society ⓘ |
| refinedBy | Rademacher exact formula for p(n) ⓘ |
| relatedTo |
generating function of p(n)
ⓘ
modular forms ⓘ q-series ⓘ |
| shows | p(n) grows rapidly with n ⓘ |
| showsBehavior | p(n) grows roughly like e^{C\sqrt{n}} for C = \pi\sqrt{2/3} ⓘ |
| topicOf |
expositions in analytic number theory textbooks
ⓘ
research in additive number theory ⓘ |
| usedFor | estimating p(n) for large n ⓘ |
| usedIn |
combinatorics
ⓘ
statistical mechanics models involving partitions ⓘ |
| usesGeneratingFunction | \sum_{n \ge 0} p(n) q^n = \prod_{m \ge 1} (1 - q^m)^{-1} ⓘ |
| yearProved | 1918 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Hardy–Ramanujan asymptotic formula Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
Hardy–Ramanujan asymptotic formula
→
improvedBy
→
Hardy–Ramanujan asymptotic formula
self-linksurface differs
ⓘ
this entity surface form:
Rademacher convergent series for p(n)
subject surface form:
G. H. Hardy
subject surface form:
G. H. Hardy
this entity surface form:
Hardy–Ramanujan theorem