Ramanujan–Petersson conjecture
E355436
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
All labels observed (6)
How this entity was disambiguated
This entity first appeared as the object of triple T3410519 — 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–Petersson conjecture Context triple: [Srinivasa Ramanujan, notableWork, Ramanujan–Petersson conjecture]
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A.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
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B.
Riemann hypothesis
The Riemann hypothesis is a famous unsolved conjecture in number theory asserting that all nontrivial zeros of the Riemann zeta function lie on a critical line in the complex plane, with deep implications for the distribution of prime numbers.
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C.
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.
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D.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
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E.
Hilbert–Pólya conjecture
The Hilbert–Pólya conjecture is an unproven idea in number theory suggesting that the nontrivial zeros of the Riemann zeta function correspond to eigenvalues of a suitable self-adjoint operator, offering a potential spectral approach to proving the Riemann hypothesis.
- 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–Petersson conjecture Target entity description: The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
-
A.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
-
B.
Riemann hypothesis
The Riemann hypothesis is a famous unsolved conjecture in number theory asserting that all nontrivial zeros of the Riemann zeta function lie on a critical line in the complex plane, with deep implications for the distribution of prime numbers.
-
C.
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.
-
D.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
-
E.
Hilbert–Pólya conjecture
The Hilbert–Pólya conjecture is an unproven idea in number theory suggesting that the nontrivial zeros of the Riemann zeta function correspond to eigenvalues of a suitable self-adjoint operator, offering a potential spectral approach to proving the Riemann hypothesis.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
conjecture in number theory
ⓘ
mathematical conjecture ⓘ |
| concerns |
Hecke eigenforms
ⓘ
holomorphic cusp forms of weight k for SL(2,Z) ⓘ |
| connectedTo |
eigenvalues of the Laplacian on modular curves
ⓘ
spectral theory of automorphic forms ⓘ |
| equivalentTo | temperedness of local components of automorphic representations of GL(2) ⓘ |
| extendedBy | Petersson’s work on Fourier coefficients of cusp forms ⓘ |
| field |
number theory
ⓘ
theory of modular forms ⓘ |
| formulatedInContextOf |
Hecke eigenforms with multiplicative Fourier coefficients
ⓘ
modular forms for SL(2,Z) ⓘ |
| generalizationOf |
Ramanujan–Petersson conjecture
self-linksurface differs
ⓘ
surface form:
Ramanujan conjecture for the tau function
|
| hasConsequence |
improved error terms in arithmetic counting problems
ⓘ
subconvexity bounds for certain L-functions ⓘ |
| hasLocalForm | bounds on Satake parameters ⓘ |
| historicalOrigin | Ramanujan’s 1916 conjectures on the tau function ⓘ |
| holdsFor | holomorphic cusp forms of any integral weight k ≥ 2 ⓘ |
| implies | Fourier coefficients of normalized Hecke eigenforms are bounded by n^{(k-1)/2+ε} ⓘ |
| inspired |
Ramanujan–Petersson conjecture
self-linksurface differs
ⓘ
surface form:
generalized Ramanujan–Petersson conjecture for GL(n)
|
| language |
algebraic geometry
ⓘ
complex analysis ⓘ representation theory ⓘ |
| motivation | understanding arithmetic properties of modular forms ⓘ |
| namedAfter |
Hans Petersson
ⓘ
Srinivasa Ramanujan ⓘ |
| openVariant | generalized Ramanujan conjecture for Maass forms ⓘ |
| predicts |
Deligne bound for Fourier coefficients of modular forms
ⓘ
growth conditions on Fourier coefficients ⓘ strong bounds on Fourier coefficients of cusp forms ⓘ |
| proofUses |
Weil conjectures
ⓘ
étale cohomology ⓘ |
| proofYear | 1974 ⓘ |
| provedBy | Pierre Deligne ⓘ |
| relatedProblem | Selberg eigenvalue conjecture ⓘ |
| relatedTo |
Hecke operators
ⓘ
Langlands program ⓘ Ramanujan tau function ⓘ automorphic forms ⓘ Ramanujan–Petersson conjecture self-linksurface differs ⓘ
surface form:
generalized Ramanujan conjecture
|
| status | proved for holomorphic modular forms of integral weight ⓘ |
| subject |
Fourier coefficients of cusp forms
ⓘ
Fourier coefficients of modular forms ⓘ |
| type | growth conjecture ⓘ |
| usedIn |
analytic number theory
ⓘ
bounds for exponential sums ⓘ estimates for L-functions ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Ramanujan–Petersson conjecture Description of subject: The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Ramanujan–Petersson conjecture (proved by Deligne)
this entity surface form:
Ramanujan conjectures
Ramanujan–Petersson conjecture
→
relatedTo
→
Ramanujan–Petersson conjecture
self-linksurface differs
ⓘ
this entity surface form:
generalized Ramanujan conjecture
Ramanujan–Petersson conjecture
→
generalizationOf
→
Ramanujan–Petersson conjecture
self-linksurface differs
ⓘ
this entity surface form:
Ramanujan conjecture for the tau function
Ramanujan–Petersson conjecture
→
inspired
→
Ramanujan–Petersson conjecture
self-linksurface differs
ⓘ
this entity surface form:
generalized Ramanujan–Petersson conjecture for GL(n)