Riemann zeta function
E47609
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Riemann zeta function canonical | 29 |
| Riemann zeta function ζ(s) | 3 |
| Riemann zeta function zeros | 1 |
| Riemann zeta function ζ(s)=L(s,χ₀) for trivial character χ₀ modulo 1 | 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.
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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.
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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 zeta function ⓘ
surface form:
Hasse–Weil L-functions
|
| growthProperty | ζ(s) is of order 1 as an entire function of s after removing pole ⓘ |
| historicalPaper |
Über die Anzahl der Primzahlen unter einer gegebenen Grösse
ⓘ
surface form:
Riemann 1859 memoir "On the Number of Primes Less Than a Given Magnitude"
|
| 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
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: Riemann zeta function Description of subject: 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.
Referenced by (34)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Georg Friedrich Bernhard Riemann
subject surface form:
Friedrich Bernhard Riemann
this entity surface form:
Riemann zeta function ζ(s)
subject surface form:
Riemann’s Zeta Function (book)
this entity surface form:
Riemann zeta function ζ(s)=L(s,χ₀) for trivial character χ₀ modulo 1
this entity surface form:
Riemann zeta function zeros
subject surface form:
Dedekind zeta function
this entity surface form:
Riemann zeta function ζ(s)
subject surface form:
Chebyshev function ψ(x)
this entity surface form:
Riemann zeta function ζ(s)
subject surface form:
L-function