Prime geodesic theorem for higherdimensional hyperbolic manifold
Author:
Maki Nakasuji
Journal:
Trans. Amer. Math. Soc. 358 (2006), 32853303
MSC (2000):
Primary 11M36, 11F72
Published electronically:
March 24, 2006
MathSciNet review:
2218976
Fulltext PDF Free Access
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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 LaplaceBeltrami 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
(58 #31311)
 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
(82m:58049)
 3.
HarishChandra,
Spherical functions on a semisimple Lie group. I, Amer. J. Math.
80 (1958), 241–310. MR 0094407
(20 #925)
HarishChandra,
Spherical functions on a semisimple Lie group. II, Amer. J. Math.
80 (1958), 553–613. MR 0101279
(21 #92)
 4.
Dennis
A. Hejhal, The Selberg trace formula for
𝑃𝑆𝐿(2,𝑅). Vol. I, Lecture Notes in
Mathematics, Vol. 548, SpringerVerlag, Berlin, 1976. MR 0439755
(55 #12641)
 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
(91f:11064)
 6.
Maki
Nakasuji, Prime geodesic theorem via the explicit formula of Ψ
for hyperbolic 3manifolds, Proc. Japan Acad. Ser. A Math. Sci.
77 (2001), no. 7, 130–133. MR 1857290
(2002j:11099)
 7.
Maki
Nakasuji, Prime geodesic theorem for hyperbolic 3manifolds:
general cofinite cases, Forum Math. 16 (2004),
no. 3, 317–363. MR 2050187
(2005g:11085), http://dx.doi.org/10.1515/form.2004.015
 8.
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
(82f:40012b)
 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
(94d:11034), http://dx.doi.org/10.1007/BF01895514
 10.
Garth
Warner, Selberg’s trace formula for nonuniform lattices: the
𝑅rank one case, Studies in algebra and number theory, Adv.
in Math. Suppl. Stud., vol. 6, Academic Press, New York, 1979,
pp. 1–142. MR 535763
(81f:10044)
 1.
 R. Gangolli, The length spectra of some compact manifolds of negative curvature, J. Differential Geometry 12 (1977), pp. 403424. MR 0650997 (58:31311)
 2.
 R. Gangolli and G. Warner, Zeta functions of Selberg's type for some noncompact quotients of symmetric spaces of rank one, Nagoya Math. J. 78 (1980), pp. 144. MR 0571435 (82m:58049)
 3.
 HarishChandra, Spherical functions on a semisimple Lie group (I, II), Amer. J. Math. 80 (1958), pp. 241310, 533613. MR 0094407 (20:925); MR 0101279 (21:92)
 4.
 D. Hejhal, The Selberg trace formula for I, Lect. Notes Math. 548, Springer, Berlin Heidelberg, New York (1976). MR 0439755 (55:12641)
 5.
 A. E. Ingham, The distribution of prime numbers, Cambridge Univ. Press (1932). MR 1074573 (91f:11064) (Reprint)
 6.
 M. Nakasuji, Prime geodesic theorem via the explicit formula of for hyperbolic 3manifolds, Proceedings Japan academy 77A (2001), pp. 130133. MR 1857290 (2002j:11099)
 7.
 M. Nakasuji, Prime geodesic theorem for hyperbolic 3manifolds: general cofinite cases, Forum Math. 16 (2004), no. 3, pp. 317363. MR 2050187 (2005g:11085)
 8.
 A. Postnikov, Tauberian theory and its applications, Proc. Steklov Inst. Math. Issue 2 (1980). MR 0603991 (82f:40012b)
 9.
 A. Reznikov, Eisenstein matrix and existence of cusp forms in rank one symmetric spaces, Geom. Funct. Anal. 3 no. 1 (1993). MR 1204788 (94d:11034)
 10.
 G. Warner, Selberg's trace formula for nonuniform lattices: The rank one case, Advances in Math. Studies 6 (1979), pp. 1142. MR 0535763 (81f:10044)
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Additional Information
Maki Nakasuji
Affiliation:
Department of Mathematics, Keio University, 3141 Hiyoshi, 2238522, Japan
Email:
nakasuji@math.keio.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002994706041225
PII:
S 00029947(06)041225
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.
