Voronin universality theorem
E262116
The Voronin universality theorem is a result in analytic number theory stating that, in a precise sense, the Riemann zeta function can approximate any non-vanishing analytic function arbitrarily well on certain regions of the complex plane.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Riemann zeta function is universal for analytic functions in the critical strip | 1 |
| Voronin universality theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2394171 — 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: Voronin universality theorem Context triple: [Riemann zeta function, universalityProperty, Voronin universality theorem]
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A.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
B.
Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
-
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 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.
-
E.
Euler product formula for the Riemann zeta function
The Euler product formula for the Riemann zeta function is a fundamental identity in analytic number theory that expresses the zeta function as an infinite product over all prime numbers, revealing a deep connection between primes and the distribution of integers.
- 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: Voronin universality theorem Target entity description: The Voronin universality theorem is a result in analytic number theory stating that, in a precise sense, the Riemann zeta function can approximate any non-vanishing analytic function arbitrarily well on certain regions of the complex plane.
-
A.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
B.
Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
-
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 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.
-
E.
Euler product formula for the Riemann zeta function
The Euler product formula for the Riemann zeta function is a fundamental identity in analytic number theory that expresses the zeta function as an infinite product over all prime numbers, revealing a deep connection between primes and the distribution of integers.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in analytic number theory ⓘ |
| appliesTo |
compact subsets of the critical strip
ⓘ
non-vanishing analytic functions ⓘ |
| approximationRegion | compact subsets of {s : 1/2 < Re(s) < 1} ⓘ |
| approximationType | uniform approximation ⓘ |
| approximationVariable | vertical shift parameter t in zeta(s+it) ⓘ |
| assumption |
target function is analytic on an open set containing the compact set
ⓘ
target function is non-zero on the compact set ⓘ |
| citedIn |
monographs on the Riemann zeta function
ⓘ
surveys on universality of L-functions ⓘ |
| concerns |
approximation of analytic functions
ⓘ
universality of the Riemann zeta function ⓘ vertical shifts of the Riemann zeta function ⓘ |
| domain | complex analysis ⓘ |
| field |
analytic number theory
ⓘ
number theory ⓘ |
| generalizedBy |
universality theorems for Dirichlet L-functions
ⓘ
universality theorems for automorphic L-functions ⓘ |
| hasConsequence |
existence of many zeros of differences between zeta and given analytic functions
ⓘ
topological richness of the orbit {zeta(s+it)} under vertical shifts ⓘ |
| implies |
Riemann zeta function has dense set of values in complex plane on certain regions
ⓘ
Voronin universality theorem self-linksurface differs ⓘ
surface form:
Riemann zeta function is universal for analytic functions in the critical strip
|
| influenced |
probabilistic models of the Riemann zeta function
ⓘ
research on universality phenomena in dynamical systems ⓘ |
| isStrongerThan | value-distribution results for the Riemann zeta function ⓘ |
| languageOfOriginalPublication | Russian ⓘ |
| mainObject | Riemann zeta function ⓘ |
| namedAfter | Sergei Voronin ⓘ |
| relatedTo |
Bohr–Courant theorem
ⓘ
Riemann zeta function ⓘ critical strip of the Riemann zeta function ⓘ universality theorems for L-functions ⓘ |
| requires |
compact set contained in the strip 1/2 < Re(s) < 1
ⓘ
compact set with connected complement ⓘ |
| statedBy | Sergei Voronin ⓘ |
| statementFeature |
arbitrarily small error in approximation
ⓘ
existence of vertical shifts of zeta ⓘ uniform approximation on compact sets ⓘ |
| typicalFormulation | for any non-vanishing analytic function on a suitable compact set there exist vertical shifts of zeta approximating it uniformly ⓘ |
| usedIn |
study of random behavior of zeta and L-functions
ⓘ
value-distribution theory of L-functions ⓘ |
| yearProved | 1975 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Voronin universality theorem Description of subject: The Voronin universality theorem is a result in analytic number theory stating that, in a precise sense, the Riemann zeta function can approximate any non-vanishing analytic function arbitrarily well on certain regions of the complex plane.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Riemann zeta function is universal for analytic functions in the critical strip