Voronin universality theorem

E262116

mathematical theorem theorem in analytic number theory

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.

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All labels observed (2)

Label	Occurrences
Riemann zeta function is universal for analytic functions in the critical strip	1
Voronin universality theorem canonical	1

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 ⓘ

Referenced by (2)

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

Riemann zeta function → universalityProperty → Voronin universality theorem ⓘ

Voronin universality theorem → implies → Voronin universality theorem self-linksurface differs ⓘ

this entity surface form: Riemann zeta function is universal for analytic functions in the critical strip