The Riemann hypothesis for Selberg’s zeta-function and the asymptotic behavior of eigenvalues of the Laplace operator

Randol, Burton

doi:10.1090/S0002-9947-1978-0472728-1

The Riemann hypothesis for Selberg’s zeta-function and the asymptotic behavior of eigenvalues of the Laplace operator
HTML articles powered by AMS MathViewer

by Burton Randol

Trans. Amer. Math. Soc. 236 (1978), 209-223

DOI: https://doi.org/10.1090/S0002-9947-1978-0472728-1

PDF | Request permission

Abstract:

Much of that part of the theory of the Riemann zeta-function based on the Riemann hypothesis carries over to zeta-functions of Selberg’s type, and in this way one can get asymptotic information about various eigenvalue problems. The methods are illustrated in the case of a compact Riemann surface.

References

Sur la fonction spectrale du Laplacien d’une variété riemannienne compacte sans point conjugués

283

K. Chandrasekharan, Arithmetical functions, Die Grundlehren der mathematischen Wissenschaften, Band 167, Springer-Verlag, New York-Berlin, 1970. MR 0277490, DOI 10.1007/978-3-642-50026-8
J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), no. 1, 39–79. MR 405514, DOI 10.1007/BF01405172
Ramesh Gangolli, Zeta functions of Selberg’s type for compact space forms of symmetric spaces of rank one, Illinois J. Math. 21 (1977), no. 1, 1–41. MR 485702
Heinz Huber, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen. II, Math. Ann. 142 (1960/61), 385–398 (German). MR 126549, DOI 10.1007/BF01451031

Lecture notes on the representation theory of

H. P. McKean, Selberg’s trace formula as applied to a compact Riemann surface, Comm. Pure Appl. Math. 25 (1972), 225–246. MR 473166, DOI 10.1002/cpa.3160250302
Burton Randol, Small eigenvalues of the Laplace operator on compact Riemann surfaces, Bull. Amer. Math. Soc. 80 (1974), 996–1000. MR 400316, DOI 10.1090/S0002-9904-1974-13609-8
Burton Randol, On the asymptotic distribution of closed geodesics on compact Riemann surfaces, Trans. Amer. Math. Soc. 233 (1977), 241–247. MR 482582, DOI 10.1090/S0002-9947-1977-0482582-9
A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. MR 88511
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon Press, 1951. MR 0046485

The theory of functions