Lindelöf hypothesis
E518478
The Lindelöf hypothesis is an unproven conjecture in analytic number theory about the growth rate of the Riemann zeta function along the critical line, with deep implications for the distribution of prime numbers.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Lindelöf hypothesis canonical | 3 |
| Lindelöf hypothesis (background and consequences) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5425834 — 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: Lindelöf hypothesis Context triple: [Ernst Lindelöf, notableFor, Lindelöf hypothesis]
-
A.
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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B.
Ramanujan–Petersson conjecture
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.
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C.
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.
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D.
generalized Riemann hypothesis
The generalized Riemann hypothesis is a major unproven conjecture in number theory asserting that the nontrivial zeros of all Dirichlet L-functions lie on a critical line in the complex plane, extending the classical Riemann hypothesis.
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E.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
- 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: Lindelöf hypothesis Target entity description: The Lindelöf hypothesis is an unproven conjecture in analytic number theory about the growth rate of the Riemann zeta function along the critical line, with deep implications for the distribution of prime numbers.
-
A.
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.
-
B.
Ramanujan–Petersson conjecture
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.
-
C.
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.
-
D.
generalized Riemann hypothesis
The generalized Riemann hypothesis is a major unproven conjecture in number theory asserting that the nontrivial zeros of all Dirichlet L-functions lie on a critical line in the complex plane, extending the classical Riemann hypothesis.
-
E.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
conjecture in analytic number theory
ⓘ
mathematical conjecture ⓘ unproven conjecture ⓘ |
| concerns |
Riemann zeta function
NERFINISHED
ⓘ
asymptotic behavior of the Riemann zeta function ⓘ growth of the Riemann zeta function on the critical line ⓘ |
| concernsProperty | order of growth of |ζ(1/2+it)| ⓘ |
| domain | critical line Re(s) = 1/2 of the complex plane ⓘ |
| field |
analytic number theory
ⓘ
number theory ⓘ |
| hasConsequence |
bounds for error terms in prime-counting functions
ⓘ
results on the distribution of primes in arithmetic progressions ⓘ results on the distribution of primes in short intervals ⓘ |
| hasGeneralization |
Lindelöf hypothesis for Dirichlet L-functions
ⓘ
Lindelöf hypothesis for automorphic L-functions NERFINISHED ⓘ |
| hasHeuristicSupportFrom | random matrix theory models of the zeta function ⓘ |
| hasImplicationFor |
distribution of zeros of the Riemann zeta function
ⓘ
error term in the prime number theorem ⓘ size of gaps between primes ⓘ |
| hasOpenStatusIn | Clay Millennium Problem context (via relation to Riemann Hypothesis) ⓘ |
| implies |
strong bounds on the growth of the Riemann zeta function on the critical line
ⓘ
subpolynomial growth of the Riemann zeta function on the critical line ⓘ upper bounds for moments of the Riemann zeta function on the critical line ⓘ |
| isConnectedTo |
moment conjectures for the Riemann zeta function
ⓘ
subconvexity problems for L-functions ⓘ |
| isDiscussedIn |
analytic number theory textbooks
ⓘ
research literature on the Riemann zeta function ⓘ |
| isNotKnownToBeEquivalentTo | Riemann Hypothesis NERFINISHED ⓘ |
| isPartOf | classical problems about the Riemann zeta function ⓘ |
| isStrongerThan | trivial bounds for |ζ(1/2+it)| ⓘ |
| isSubjectOf | ongoing research in analytic number theory ⓘ |
| isWeakerThan | Riemann Hypothesis NERFINISHED ⓘ |
| mathematicalObject | statement about complex-valued function growth ⓘ |
| motivatedBy | understanding fine distribution of primes ⓘ |
| namedAfter | Ernst Leonard Lindelöf NERFINISHED ⓘ |
| relatedTo |
Dirichlet L-functions
NERFINISHED
ⓘ
Riemann Hypothesis NERFINISHED ⓘ distribution of prime numbers ⓘ generalized Lindelöf hypothesis ⓘ |
| statedIn | early 20th century ⓘ |
| status | open problem ⓘ |
| typeOfBound | upper bound on |ζ(1/2+it)| as t → ∞ ⓘ |
| usedIn |
analytic estimates in prime number theory
ⓘ
study of zeros of L-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: Lindelöf hypothesis Description of subject: The Lindelöf hypothesis is an unproven conjecture in analytic number theory about the growth rate of the Riemann zeta function along the critical line, with deep implications for the distribution of prime numbers.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Lindelöf hypothesis (background and consequences)