On the asymptotic distribution of closed geodesics on compact Riemann surfaces

Randol, Burton

doi:10.1090/S0002-9947-1977-0482582-9

On the asymptotic distribution of closed geodesics on compact Riemann surfaces
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by Burton Randol

Trans. Amer. Math. Soc. 233 (1977), 241-247

DOI: https://doi.org/10.1090/S0002-9947-1977-0482582-9

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Abstract:

The set of lengths of closed geodesics on a compact Riemann surface is related to the Selberg zeta function in a manner which is evocative of the relationship between the rational primes and the Riemann zeta function. In this paper, this connection is developed to derive results about the asymptotic distribution of these lengths.

References

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

The general theory of Dirichlet’s series

Heinz Huber, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen. II, Math. Ann. 142 (1960/61), 385–398 (German). MR 126549, DOI 10.1007/BF01451031
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

The asymptotic behavior of the zeros of zeta functions of Selberg’s type

David Ruelle, Zeta-functions for expanding maps and Anosov flows, Invent. Math. 34 (1976), no. 3, 231–242. MR 420720, DOI 10.1007/BF01403069
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