Prime geodesic theorem for higher-dimensional hyperbolic manifold

Author:
Maki Nakasuji

Journal:
Trans. Amer. Math. Soc. **358** (2006), 3285-3303

MSC (2000):
Primary 11M36, 11F72

DOI:
https://doi.org/10.1090/S0002-9947-06-04122-5

Published electronically:
March 24, 2006

MathSciNet review:
2218976

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Abstract | References | Similar Articles | Additional Information

Abstract: For a -dimensional hyperbolic manifold , we consider an estimate of the error term of the prime geodesic theorem. Put the fundamental group of to be a discrete subgroup of with cofinite volume. When the contribution of the discrete spectrum of the Laplace-Beltrami operator is larger than that of the continuous spectrum in Weyl's law, we obtained a lower estimate as goes to .

**1.**Ramesh Gangolli,*The length spectra of some compact manifolds of negative curvature*, J. Differential Geom.**12**(1977), no. 3, 403–424. MR**0650997****2.**Ramesh Gangolli and Garth Warner,*Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one*, Nagoya Math. J.**78**(1980), 1–44. MR**571435****3.**Harish-Chandra,*Spherical functions on a semisimple Lie group. I*, Amer. J. Math.**80**(1958), 241–310. MR**0094407**, https://doi.org/10.2307/2372786

Harish-Chandra,*Spherical functions on a semisimple Lie group. II*, Amer. J. Math.**80**(1958), 553–613. MR**0101279**, https://doi.org/10.2307/2372772**4.**Dennis A. Hejhal,*The Selberg trace formula for 𝑃𝑆𝐿(2,𝑅). Vol. I*, Lecture Notes in Mathematics, Vol. 548, Springer-Verlag, Berlin-New York, 1976. MR**0439755****5.**A. E. Ingham,*The distribution of prime numbers*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. Reprint of the 1932 original; With a foreword by R. C. Vaughan. MR**1074573****6.**Maki Nakasuji,*Prime geodesic theorem via the explicit formula of Ψ for hyperbolic 3-manifolds*, Proc. Japan Acad. Ser. A Math. Sci.**77**(2001), no. 7, 130–133. MR**1857290****7.**Maki Nakasuji,*Prime geodesic theorem for hyperbolic 3-manifolds: general cofinite cases*, Forum Math.**16**(2004), no. 3, 317–363. MR**2050187**, https://doi.org/10.1515/form.2004.015**8.**A. G. Postnikov,*Tauberian theory and its applications*, Trudy Mat. Inst. Steklov.**144**(1979), 147 (Russian). MR**543488**

A. G. Postnikov,*Tauberian theory and its applications*, Proc. Steklov Inst. Math.**2**(1980), v+138. A translation of Trudy Mat. Inst. Steklov. 144 (1979). MR**603991****9.**Andrei Reznikov,*Eisenstein matrix and existence of cusp forms in rank one symmetric spaces*, Geom. Funct. Anal.**3**(1993), no. 1, 79–105. MR**1204788**, https://doi.org/10.1007/BF01895514**10.**Garth Warner,*Selberg’s trace formula for nonuniform lattices: the 𝑅-rank one case*, Studies in algebra and number theory, Adv. in Math. Suppl. Stud., vol. 6, Academic Press, New York-London, 1979, pp. 1–142. MR**535763**

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Additional Information

**Maki Nakasuji**

Affiliation:
Department of Mathematics, Keio University, 3-14-1 Hiyoshi, 223-8522, Japan

Email:
nakasuji@math.keio.ac.jp

DOI:
https://doi.org/10.1090/S0002-9947-06-04122-5

Keywords:
Prime geodesic theorem,
Selberg zeta function,
Weyl's law

Received by editor(s):
April 22, 2003

Published electronically:
March 24, 2006

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.