Riemann hypothesis

E47346

conjecture in number theory mathematical conjecture unsolved problem in mathematics

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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All labels observed (6)

Label	Occurrences
Riemann hypothesis canonical	24
Riemann Hypothesis	22
Hilbert's eighth problem	1
Hilbert’s eighth problem	1
Riemann Hypothesis is equivalent to Λ ≤ 0	1
Riemann hypothesis for the Riemann zeta function	1

How this entity was disambiguated

This entity first appeared as the object of triple T373776 — 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: Riemann hypothesis
Context triple: [Bernhard Riemann, knownFor, Riemann hypothesis]

A. Millennium Prize Problem
The Millennium Prize Problem is one of seven famous unsolved mathematical problems designated by the Clay Mathematics Institute, each carrying a $1 million reward for a correct solution.
B. Hilbert problems
The Hilbert problems are a famous list of 23 unsolved mathematical problems presented by David Hilbert in 1900 that profoundly influenced the development of 20th-century mathematics.
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. Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
E. 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.
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: Riemann hypothesis
Target entity description: 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.

A. Millennium Prize Problem
The Millennium Prize Problem is one of seven famous unsolved mathematical problems designated by the Clay Mathematics Institute, each carrying a $1 million reward for a correct solution.
B. Hilbert problems
The Hilbert problems are a famous list of 23 unsolved mathematical problems presented by David Hilbert in 1900 that profoundly influenced the development of 20th-century mathematics.
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. Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
E. 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.
F. None of above. chosen

Statements (52)

Predicate	Object
instanceOf	conjecture in number theory ⓘ mathematical conjecture ⓘ unsolved problem in mathematics ⓘ
asserts	all nontrivial zeros of the Riemann zeta function lie on the critical line Re(s) = 1/2 ⓘ
consideredOneOf	most important open problems in mathematics ⓘ
criticalLine	Re(s) = 1/2 ⓘ
criticalStrip	0 < Re(s) < 1 ⓘ
domainOfZetaFunction	complex plane ⓘ
equivalentTo	certain bounds on the error term in the prime number theorem ⓘ many statements about the distribution of primes ⓘ statements about the growth of the Mertens function ⓘ statements about the size of the Chebyshev functions ⓘ
excludes	trivial zeros of the Riemann zeta function ⓘ
field	analytic number theory ⓘ number theory ⓘ
hasConsequence	improved bounds in many problems of analytic number theory ⓘ results on error terms in various counting functions ⓘ results on the distribution of prime ideals in number fields ⓘ
hasGeneralization	extended Riemann hypothesis ⓘ generalized Riemann hypothesis ⓘ grand Riemann hypothesis ⓘ linked to: Selberg class
hasInfluenceOn	computational number theory ⓘ cryptography ⓘ mathematical physics ⓘ
hasNumericalEvidence	many zeros verified on the critical line ⓘ
hasPrize	Clay Millennium Prize of 1 million US dollars ⓘ linked to: Millennium Prize Problem
implies	best possible error term in the prime number theorem up to constants ⓘ bounds on the Chebyshev functions ⓘ results on gaps between primes ⓘ results on the Mertens function ⓘ results on the Möbius function ⓘ strong results on the distribution of prime numbers ⓘ
involves	Riemann zeta function ⓘ complex analysis ⓘ prime numbers ⓘ zeros of the Riemann zeta function ⓘ
listedAs	Hilbert's eighth problem ⓘ linked to: Riemann hypothesis
listedIn	Clay Millennium Prize Problems ⓘ linked to: Millennium Prize Problem Hilbert's problems ⓘ linked to: Hilbert problems
namedAfter	Bernhard Riemann ⓘ
relatedTo	Dirichlet L-functions ⓘ Hilbert–Pólya conjecture ⓘ L-functions ⓘ Montgomery's pair correlation conjecture ⓘ distribution of primes in short intervals ⓘ prime number theorem ⓘ random matrix theory ⓘ zero-free regions of the zeta function ⓘ
statedIn	Riemann's 1859 paper "Über die Anzahl der Primzahlen unter einer gegebenen Grösse" ⓘ linked to: Über die Anzahl der Primzahlen unter einer gegebenen Grösse
status	open problem ⓘ unproven ⓘ
yearProposed	1859 ⓘ

How these facts were elicited

Referenced by (50)

Full triples — surface form annotated when it differs from this entity's canonical label.

Bernhard Riemann → knownFor → Riemann hypothesis ⓘ

Millennium Prize Problem → hasProblem → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

Millennium Prize Problem → includes → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

Hilbert problems → hasPart → Hilbert’s eighth problem ⓘ

linked to: Riemann hypothesis

Hilbert problems → notableProblem → Riemann hypothesis ⓘ

Riemann hypothesis → listedAs → Hilbert's eighth problem ⓘ

linked to: Riemann hypothesis

Über die Anzahl der Primzahlen unter einer gegebenen Grösse → topic → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

Über die Anzahl der Primzahlen unter einer gegebenen Grösse → introduces → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

Über die Anzahl der Primzahlen unter einer gegebenen Grösse → associatedConjecture → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

Riemann zeta function → relatedConjecture → Riemann hypothesis ⓘ

Georg Friedrich Bernhard Riemann → notableWork → Riemann hypothesis ⓘ

subject linked to: Georg

Alain Connes → hasResearchInterest → Riemann hypothesis ⓘ

Friedrich Bernhard Riemann → notableWork → Riemann hypothesis ⓘ

subject linked to: Friedrich

Euler product formula for the Riemann zeta function → relatedConcept → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

Hardy–Littlewood conjectures → relatedTo → Riemann hypothesis ⓘ

Multiplicative Number Theory → relatedTo → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

de Bruijn–Newman constant → relatedTo → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

de Bruijn–Newman constant → conjecturedRelationToRiemannHypothesis → Riemann Hypothesis is equivalent to Λ ≤ 0 ⓘ

linked to: Riemann hypothesis

Riemann–Siegel theta function → relatedTo → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

Hardy Z-function → relatedTo → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

E. C. Titchmarsh, The Theory of the Riemann Zeta-Function → topic → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

Riemann’s Zeta Function (book) → mainSubject → Riemann Hypothesis ⓘ

subject linked to: H. M. Edwards, Riemann’s Zeta Function

linked to: Riemann hypothesis

Weil conjectures → inspiredBy → Riemann hypothesis ⓘ

Weil conjectures → analogousTo → Riemann hypothesis for the Riemann zeta function ⓘ

linked to: Riemann hypothesis

generalized Riemann hypothesis → extends → Riemann hypothesis ⓘ

generalized Riemann hypothesis → relatedTo → Riemann hypothesis ⓘ

Hilbert–Pólya conjecture → relatedTo → Riemann hypothesis ⓘ

Hilbert–Pólya conjecture → implies → Riemann hypothesis ⓘ

Montgomery's pair correlation conjecture → relatedTo → Riemann hypothesis ⓘ

prime number theorem → relatedTo → Riemann hypothesis ⓘ

Mertens’ theorems → relatedTo → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

Goldbach conjecture → relatedTo → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

Lindelöf hypothesis → relatedTo → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

Lindelöf hypothesis → isWeakerThan → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

Lindelöf hypothesis → isNotKnownToBeEquivalentTo → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

analytic number theory → centralConjecture → Riemann hypothesis ⓘ

Prime Obsession → subject → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

Liouville function → relatedConjecture → Riemann hypothesis ⓘ

Pólya’s conjecture → involvesConcept → Riemann hypothesis ⓘ

Farey sequence → relatedTo → Riemann hypothesis ⓘ

Legendre’s conjecture on primes between consecutive squares → relatedTo → Riemann hypothesis ⓘ

Miller primality test → basedOn → Riemann hypothesis ⓘ

Λ → associatedWith → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

Harold M. Edwards → hasWrittenOn → Riemann hypothesis ⓘ

H. M. Edwards → hasWrittenOn → Riemann hypothesis ⓘ

GRH → relatedTo → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

GRH → generalizes → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

GRH → isStrongerThan → Riemann Hypothesis ⓘ

linked to: Riemann hypothesis

grand Riemann hypothesis → generalizes → Riemann hypothesis ⓘ

grand Riemann hypothesis → implies → Riemann hypothesis ⓘ

All labels observed (6)

How this entity was disambiguated Show

Statements (52)

How these facts were elicited Show

Referenced by (50)

How this entity was disambiguated

How these facts were elicited