Riemann zeta function

E47609

Dirichlet series L-function complex-valued function meromorphic function special 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.

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All labels observed (6)

Label	Occurrences
Riemann zeta function canonical	50
Riemann zeta function ζ(s)	3
Riemann function	1
Riemann zeta function zeros	1
Riemann zeta function ζ(s)=L(s,χ₀) for trivial character χ₀ modulo 1	1
Riemann's Zeta Function	1

How this entity was disambiguated

This entity first appeared as the object of triple T373777 — 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 zeta function
Context triple: [Bernhard Riemann, knownFor, Riemann zeta function]

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. 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.
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. Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
E. Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
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 zeta function
Target entity description: 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.

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. 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.
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. Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
E. Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
F. None of above. chosen

Statements (51)

Predicate	Object
instanceOf	Dirichlet series ⓘ L-function ⓘ complex-valued function ⓘ meromorphic function ⓘ special function ⓘ
analyticContinuation	extends meromorphically to C except s = 1 ⓘ
approximateFunctionalEquation	ζ(s) expressed as finite sums plus error term ⓘ
BernoulliNumberRelation	ζ(2n) = (-1)^{n+1} (B_{2n} (2π)^{2n}) / (2 (2n)!) ⓘ
completedZetaDefinition	ξ(s) = ½ s(s-1) π^{-s/2} Γ(s/2) ζ(s) ⓘ
completedZetaProperty	ξ(s) is entire ⓘ
completedZetaSymmetry	ξ(s) = ξ(1-s) ⓘ
connectedTo	prime number theorem ⓘ
criticalLine	Re(s) = 1/2 ⓘ
definedOn	complex plane ⓘ
DirichletSeriesConvergenceRegion	Re(s) > 1 ⓘ
DirichletSeriesDefinition	ζ(s) = Σ_{n=1}^{∞} 1/n^s ⓘ
EulerProduct	ζ(s) = ∏_{p prime} (1 - p^{-s})^{-1} ⓘ
EulerProductConvergenceRegion	Re(s) > 1 ⓘ
field	analytic number theory ⓘ complex analysis ⓘ
functionalEquation	ζ(s) = 2^s π^{s-1} sin(πs/2) Γ(1-s) ζ(1-s) ⓘ
generalization	Dedekind zeta functions ⓘ Dirichlet L-functions ⓘ Hasse–Weil L-functions ⓘ linked to: Hasse–Weil zeta function
growthProperty	ζ(s) is of order 1 as an entire function of s after removing pole ⓘ
historicalPaper	Riemann 1859 memoir "On the Number of Primes Less Than a Given Magnitude" ⓘ linked to: Über die Anzahl der Primzahlen unter einer gegebenen Grösse
introducedBy	Bernhard Riemann ⓘ
momentStudy	moments of ζ(1/2+it) studied in random matrix theory ⓘ
nontrivialZerosRegion	0 < Re(s) < 1 ⓘ
pole	simple pole at s = 1 ⓘ
primeNumberTheoremRelation	nonvanishing of ζ(s) on Re(s) = 1 equivalent to prime number theorem ⓘ
relatedConjecture	Riemann hypothesis ⓘ
relatedTo	distribution of prime numbers ⓘ
residueAt1	1 ⓘ
RiemannHypothesisStatement	all nontrivial zeros lie on Re(s) = 1/2 ⓘ
specialValue	ζ(-1) = -1/12 ⓘ ζ(0) = -1/2 ⓘ ζ(1/2) is irrational (conjectured, not proved) ⓘ ζ(2) = π^2/6 ⓘ ζ(4) = π^4/90 ⓘ
symbol	ζ(s) ⓘ
trivialZero	s = -2 ⓘ s = -4 ⓘ s = -6 ⓘ
trivialZerosLocation	negative even integers ⓘ
universalityProperty	Voronin universality theorem ⓘ
valuesAtEvenIntegers	ζ(2n) expressed using Bernoulli numbers and powers of π ⓘ
variable	complex variable s ⓘ
VoroninUniversality	translates of ζ(s) approximate wide classes of analytic functions in critical strip ⓘ
zeroFreeRegion	Re(s) > 1 ⓘ there exists c > 0 such that ζ(s) ≠ 0 for Re(s) ≥ 1 - c/log(\|Im(s)\|+2) ⓘ

How these facts were elicited

Referenced by (57)

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

Bernhard Riemann → knownFor → Riemann zeta function ⓘ

Riemann–Siegel formula → mainSubject → Riemann zeta function ⓘ

Riemann hypothesis → involves → Riemann zeta function ⓘ

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

Über die Anzahl der Primzahlen unter einer gegebenen Grösse → defines → Riemann zeta function ⓘ

Georg Friedrich Bernhard Riemann → notableWork → Riemann zeta function ⓘ

subject linked to: Georg

Friedrich Bernhard Riemann → notableWork → Riemann zeta function ⓘ

subject linked to: Friedrich

Euler product formula for the Riemann zeta function → relatesTo → Riemann zeta function ⓘ

Zeta → symbolFor → Riemann zeta function ⓘ

Bernoulli numbers → relatedTo → Riemann zeta function ⓘ

Multiplicative Number Theory → usesConcept → Riemann zeta function ⓘ

Hasse–Weil zeta function → generalizes → Riemann zeta function ⓘ

de Bruijn–Newman constant → relatedTo → Riemann zeta function ⓘ

Riemann–Siegel theta function → relatedTo → Riemann zeta function ⓘ

Hardy Z-function → dependsOn → Riemann zeta function ⓘ

Hardy Z-function → constructedFrom → Riemann zeta function ζ(s) ⓘ

linked to: Riemann zeta function

E. C. Titchmarsh, The Theory of the Riemann Zeta-Function → mainSubject → Riemann zeta function ⓘ

Riemann’s Zeta Function (book) → mainSubject → Riemann zeta function ⓘ

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

A. Ivić, The Riemann Zeta-Function → mainSubject → Riemann zeta function ⓘ

Selberg class → contains → Riemann zeta function ⓘ

Dirichlet L-functions → generalizes → Riemann zeta function ⓘ

Dirichlet L-functions → hasSpecialCase → Riemann zeta function ζ(s)=L(s,χ₀) for trivial character χ₀ modulo 1 ⓘ

linked to: Riemann zeta function

random matrix theory → relatedTo → Riemann zeta function zeros ⓘ

linked to: Riemann zeta function

Hilbert–Pólya conjecture → relatedTo → Riemann zeta function ⓘ

Montgomery's pair correlation conjecture → mainSubject → Riemann zeta function ⓘ

prime number theorem → relatedTo → Riemann zeta function ⓘ

Voronin universality theorem → mainObject → Riemann zeta function ⓘ

Voronin universality theorem → relatedTo → Riemann zeta function ⓘ

Dedekind zeta function → generalizes → Riemann zeta function ⓘ

subject linked to: Dedekind zeta functions

Jordan’s totient functions → relatedConcept → Riemann zeta function ζ(s) ⓘ

linked to: Riemann zeta function

Dirichlet series → specialCase → Riemann zeta function ⓘ

Chebyshev function ψ(x) → relatedTo → Riemann zeta function ζ(s) ⓘ

subject linked to: Chebyshev functions

linked to: Riemann zeta function

Mertens’ theorems → mainTopic → Riemann zeta function ⓘ

L-function → generalizes → Riemann zeta function ⓘ

subject linked to: L-functions

Du Bois-Reymond function → relatedTo → Riemann function ⓘ

linked to: Riemann zeta function

Dirichlet eta function → relatedTo → Riemann zeta function ⓘ

Lindelöf hypothesis → concerns → Riemann zeta function ⓘ

analytic number theory → centralObject → Riemann zeta function ⓘ

Prime Obsession → focusesOn → Riemann zeta function ⓘ

Liouville function → relatedConcept → Riemann zeta function ⓘ

Mellin transform → usedWith → Riemann zeta function ⓘ

subject linked to: Mellin transforms

Pólya’s conjecture → involvesConcept → Riemann zeta function ⓘ

Marcus du Sautoy → hasResearchInterest → Riemann zeta function ⓘ

Derrick Henry Lehmer → researchInterest → Riemann zeta function ⓘ

Riemann xi function → dependsOn → Riemann zeta function ⓘ

Riemann xi function → encodesZerosOf → Riemann zeta function ⓘ

Gram points → relatedTo → Riemann zeta function ⓘ

H. M. Edwards → notableWork → Riemann's Zeta Function ⓘ

linked to: Riemann zeta function

H. M. Edwards → hasAcademicSpecialization → Riemann zeta function ⓘ

Aleksandar Ivić → researchInterest → Riemann zeta function ⓘ

All labels observed (6)

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Statements (51)

How these facts were elicited Show

Referenced by (57)

How this entity was disambiguated

How these facts were elicited