The Selberg trace formula and the Riemann zeta function

Hejhal, Dennis A.

doi:10.1215/s0012-7094-76-04338-6

Cited by 155 publications

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“…The restriction G Φ SU(2n 9 1) comes in for a purely technical reason. In studying the analytic continuation of z Γ9 one writes it as a sum Σ 7 i=1 Ai(s) and studies A^s) separately.…”

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confidence: 99%

Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one

Gangolli

Warner

1980

Nagoya Mathematical Journal

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IntroductionIn a previous paper [5], one of the present authors has worked out a theory of zeta functions of Selberg's type for compact quotients of symmetric spaces of rank one. In the present paper, we consider the analogues of those results when G/K is a noncompact symmetric space of rank one and Γ is a discrete subgroup of G such that G/Γ is not compact but such that vol(GjΓXoo. Thus, Γ is a non-uniform lattice. Certain mild restrictions, which are fulfilled in many arithmetic cases, will be put on Γ, and we shall consider how one can define a zeta function Z Γ of Selberg's type attached to the data (G, K, Γ).This will be attempted as in [5], by first defining, via the trace formula, the logarithmic derivative z Γ of Z Γ . The main reason why we get somewhere is the fact that in the case rank (GjK) = 1, the parabolic terms in the trace formula can be fully evaluated, for a spherical admissible function /, in terms of the spherical Fourier transform /. The computations necessary for this were performed by the present authors, and were reported in [21].For technical reasons, presently to be explained, we shall eventually exclude the case G = SU(2n, 1). Except for this case, our results for Z Γ are pretty satisfactory. We shall see that Z Γ is a meromorphic function. The location and orders of its zeroes and poles will turn out to contain topological and/or spectral information. It will also be seen that Z Γ enjoys a functional equation which involves not only the Harish-Chandra c-function (which determines the Plancherel measure of GjK) but also the determinant Ψ(s) of the intertwining operator M(s) which occurs in the

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“…The restriction G Φ SU(2n 9 1) comes in for a purely technical reason. In studying the analytic continuation of z Γ9 one writes it as a sum Σ 7 i=1 Ai(s) and studies A^s) separately.…”

mentioning

confidence: 99%

Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one

Gangolli

Warner

1980

Nagoya Mathematical Journal

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“…the last expression can be transformed to (38) which proves the functional equation (37). From the functional equation (37) is follows that ζ(s) has 'trivial' zeros at negative even integers (except zero) s = −2, −4, .…”

Section: Functional Equationmentioning

confidence: 82%

“…. which appear from sin(πs/2) in (38). All other non-trivial zeros, ζ(s n ) = 0, are situated in the socalled critical strip 0 < Re s < 1.…”

Section: Functional Equationmentioning

confidence: 97%

Quantum and Arithmetical Chaos

Bogomolny¹

Frontiers in Number Theory, Physics, and Geometry I

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“…n j L'intérêt de cette écriture est de faire apparaître Zo(t) dont une expression exacte est donnée dans le cas X = F\H par la formule des traces de Selberg [15] (et aussi [21] dans le cas où la courbure de Xo n'est pas constante). Pour déterminer le support singulier de Z^(Q, à défaut d'une formule de traces exactes, qui existe peut-être dans le cas X = r\H, on s'inspire des méthodes de la théorie des équations hyperboliques et du calcul des fronts d'ondes, comme dans le cas d'une variété compacte.…”

Section: Za(0 = S Cos Tr^ + ^ Cos Tr(a) = Zo(r) + W(t)unclassified

Pseudo-laplaciens II

Verdìère¹

1983

Annales De L’institut Fourier

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The Selberg trace formula and the Riemann zeta function

Cited by 155 publications

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Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one

Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one

Quantum and Arithmetical Chaos

Pseudo-laplaciens II

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