Selberg class

E246701

class of Dirichlet series concept in analytic number theory

The Selberg class is a collection of Dirichlet series with specific analytic properties introduced to generalize and axiomatize L-functions in number theory.

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All labels observed (3)

Label	Occurrences
Selberg class canonical	2
Selberg orthonormality conjecture	1
grand Riemann hypothesis	1

Statements (48)

Predicate	Object
instanceOf	class of Dirichlet series ⓘ concept in analytic number theory ⓘ
appearsIn	literature on Selberg’s conjectures on L-functions ⓘ literature on generalized Riemann hypothesis ⓘ
contains	Dirichlet L-functions ⓘ L-functions ⓘ surface form: Hecke L-functions Riemann zeta function ⓘ automorphic L-functions under suitable conditions ⓘ
context	axiomatic theory of L-functions ⓘ
definedAs	collection of Dirichlet series satisfying specific axioms ⓘ
field	number theory ⓘ
generalizes	Dedekind zeta functions ⓘ classical Dirichlet L-functions ⓘ
hasAxiom	Dirichlet series representation ⓘ Euler product formula for the Riemann zeta function ⓘ surface form: Euler product Ramanujan hypothesis type growth condition ⓘ analytic continuation ⓘ functional equation ⓘ
hasConjecture	Selberg class self-linksurface differs ⓘ surface form: Selberg orthonormality conjecture degree conjecture for elements of the Selberg class ⓘ
hasInvariant	conductor of an L-function ⓘ degree of an L-function ⓘ
hasProperty	Euler product has local factors of polynomial type in p^{-s} ⓘ closed under multiplication of L-functions ⓘ closed under taking Dirichlet series quotients in some formulations ⓘ coefficients satisfy polynomial growth conditions ⓘ each element admits meromorphic continuation to the complex plane ⓘ each element has an Euler product expansion ⓘ each element is a Dirichlet series absolutely convergent in some right half-plane ⓘ each element satisfies a functional equation relating s and 1−s ⓘ elements satisfy certain growth bounds in vertical strips ⓘ
introducedBy	Atle Selberg ⓘ
introducedInContextOf	L-functions ⓘ
namedAfter	Atle Selberg ⓘ
purpose	to axiomatize L-functions ⓘ to generalize classical L-functions ⓘ
relatedTo	Dirichlet characters ⓘ Euler products ⓘ automorphic representations ⓘ functional equations of L-functions ⓘ modular forms ⓘ
studiedFor	distribution of zeros of L-functions ⓘ value distribution of L-functions ⓘ
subfield	analytic number theory ⓘ
topicOf	research in analytic number theory ⓘ
usedFor	formulating generalized Riemann hypothesis ⓘ studying zeros of L-functions ⓘ unifying different types of L-functions ⓘ

Referenced by (4)

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

Atle Selberg → knownFor → Selberg class ⓘ

Atle Selberg → notableWork → Selberg class ⓘ

Riemann hypothesis → hasGeneralization → Selberg class ⓘ

this entity surface form: grand Riemann hypothesis

Selberg class → hasConjecture → Selberg class self-linksurface differs ⓘ

this entity surface form: Selberg orthonormality conjecture