Selberg zeta function

E865101

analytic function object in analytic number theory object of spectral theory zeta function

The Selberg zeta function is an analytic function associated with the lengths of closed geodesics on a Riemannian manifold, playing a central role in spectral theory and the study of automorphic forms.

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Statements (49)

Predicate	Object
instanceOf	analytic function ⓘ object in analytic number theory ⓘ object of spectral theory ⓘ zeta function ⓘ
appearsIn	Selberg’s work on harmonic analysis on locally symmetric spaces ⓘ theory of rank-one locally symmetric spaces ⓘ
associatedWith	Fuchsian groups NERFINISHED ⓘ Laplace–Beltrami operator NERFINISHED ⓘ Riemannian manifold NERFINISHED ⓘ closed geodesics ⓘ cocompact Fuchsian groups ⓘ cofinite Fuchsian groups ⓘ discrete groups of isometries ⓘ hyperbolic surfaces ⓘ spectrum of the Laplacian ⓘ
definedFor	Re(s) sufficiently large ⓘ
definedOn	compact hyperbolic surfaces ⓘ finite-area hyperbolic surfaces ⓘ quotients of hyperbolic plane by Fuchsian groups ⓘ
encodes	length spectrum of closed geodesics ⓘ multiplicities of closed geodesics ⓘ primitive closed geodesics ⓘ
field	analytic number theory ⓘ automorphic forms ⓘ differential geometry ⓘ spectral theory ⓘ
generalizationOf	Riemann zeta function in geometric setting ⓘ
hasDomain	complex plane ⓘ
hasEulerProduct	over primitive closed geodesics ⓘ
hasVariable	complex variable s ⓘ
namedAfter	Atle Selberg NERFINISHED ⓘ
property	admits meromorphic continuation to the complex plane ⓘ logarithmic derivative appears in Selberg trace formula ⓘ satisfies functional equation ⓘ zeros encode spectral data of Laplacian ⓘ zeros related to eigenvalues of Laplace–Beltrami operator ⓘ
relatedTo	Eisenstein series NERFINISHED ⓘ Maass forms NERFINISHED ⓘ Ruelle zeta function NERFINISHED ⓘ Selberg trace formula NERFINISHED ⓘ automorphic Laplacian ⓘ automorphic representations ⓘ dynamical zeta function ⓘ prime geodesic theorem ⓘ
usedIn	inverse spectral problems ⓘ proofs of prime geodesic theorem ⓘ quantum chaos on hyperbolic surfaces ⓘ study of automorphic spectra ⓘ study of resonances on hyperbolic manifolds ⓘ

Referenced by (2)

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

Selberg trace formula → relatedTo → Selberg zeta function ⓘ

L-functions → hasSpecialCase → Selberg zeta function ⓘ

subject surface form: L-function