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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Observed surface forms (3)

Surface form	Occurrences
Riemann Hypothesis	6
Hilbert's eighth problem	1
Hilbert’s eighth problem	1

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 ⓘ Selberg class ⓘ surface form: grand Riemann hypothesis
hasInfluenceOn	computational number theory ⓘ cryptography ⓘ mathematical physics ⓘ
hasNumericalEvidence	many zeros verified on the critical line ⓘ
hasPrize	Millennium Prize Problem ⓘ surface form: Clay Millennium Prize of 1 million US dollars
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	Riemann hypothesis self-linksurface differs ⓘ surface form: Hilbert's eighth problem
listedIn	Millennium Prize Problem ⓘ surface form: Clay Millennium Prize Problems Hilbert problems ⓘ surface form: Hilbert's 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	Über die Anzahl der Primzahlen unter einer gegebenen Grösse ⓘ surface form: Riemann's 1859 paper "Über die Anzahl der Primzahlen unter einer gegebenen Grösse"
status	open problem ⓘ unproven ⓘ
yearProposed	1859 ⓘ

Referenced by (14)

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

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

this entity surface form: Riemann Hypothesis

Hilbert problems → hasPart → Riemann hypothesis ⓘ

this entity surface form: Hilbert’s eighth problem

Millennium Prize Problem → hasProblem → Riemann hypothesis ⓘ

this entity surface form: Riemann Hypothesis

Alain Connes → hasResearchInterest → Riemann hypothesis ⓘ

Millennium Prize Problem → includes → Riemann hypothesis ⓘ

this entity surface form: Riemann Hypothesis

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

this entity surface form: Riemann Hypothesis

Bernhard Riemann → knownFor → Riemann hypothesis ⓘ

Riemann hypothesis → listedAs → Riemann hypothesis self-linksurface differs ⓘ

this entity surface form: Hilbert's eighth problem

Hilbert problems → notableProblem → Riemann hypothesis ⓘ

Friedrich → notableWork → Riemann hypothesis ⓘ

subject surface form: Friedrich Bernhard Riemann

Georg → notableWork → Riemann hypothesis ⓘ

subject surface form: Georg Friedrich Bernhard Riemann

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

this entity surface form: Riemann Hypothesis

Riemann zeta function → relatedConjecture → Riemann hypothesis ⓘ

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

this entity surface form: Riemann Hypothesis