Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON THE ERROR TERM IN THE PRIME GEODESIC THEOREM

  • Received : 2010.11.05
  • Published : 2012.03.31
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Abstract

Taking the integrated Chebyshev-type counting function of the appropriate order, we improve the error term in Park's prime geodesic theorem for hyperbolic manifolds with cusps. The obtained estimate coincides with the best known result in the Riemann surfaces case.

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References

  1. M. Avdispahic and L. Smajlovic, On the prime number theorem for a compact Riem- mann surface, Rocky Mountain J. Math. 39 (2009), no. 6, 1837-1845. https://doi.org/10.1216/RMJ-2009-39-6-1837
  2. P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Progress in Mathemat- ics, Vol. 106 Birkhauser, Boston-Basel-Berlin, 1992.
  3. D. L. DeGeorge, Length spectrum for compact locally symmetric spaces of strictly neg- ative curvature, Ann. Sci. Ecole Norm. Sup. (4) 10 (1977), no. 2, 133-152. https://doi.org/10.24033/asens.1323
  4. D. Fried, The zeta functions of Ruelle and Selberg, Ann. Sci. Ecole Norm. Sup. (4) 19 (1986), no. 4, 491-517. https://doi.org/10.24033/asens.1515
  5. R. Gangolli, The length spectrum of some compact manifolds of negative curvature, J. Differential Geom. 12 (1977), no. 3, 403-424. https://doi.org/10.4310/jdg/1214434092
  6. R. Gangolli and G. Warner, Zeta functions of Selberg's type for some noncompact quo- tients of symmetric spaces of rank one, Nagoya Math. J. 78 (1980), 1-44. https://doi.org/10.1017/S002776300001878X
  7. Y. Gon and J. Park, The zeta functions of Ruelle and Selberg for hyperbolic manifolds with cusps, Math. Ann. 346 (2010), no. 3, 719-767. https://doi.org/10.1007/s00208-009-0408-7
  8. D. Hejhal, The Selberg Trace Formula for PSL (2;R). Vol. I, Lecture Notes in Mathe- matics 548. Springer-Verlag, Berlin-Heidelberg, 1973.
  9. D. Hejhal, The Selberg Trace Formula for PSL (2;R). Vol. II, Lecture Notes in Mathematics 1001. Springer-Verlag, Berlin-Heidelberg, 1983.
  10. H. Huber, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgrupen II, Math. Ann. 142 (1961), 385-398. https://doi.org/10.1007/BF01451031
  11. H. Huber, Nachtrag zu [10], Math. Ann. 143 (1961), 463-464. https://doi.org/10.1007/BF01470758
  12. J. Park, Ruelle zeta function and prime geodesic theorem for hyperbolic manifolds with cusps, in: G. van Dijk, M. Wakayama (eds.), Casimir Force, Casimir Operators and the Riemann Hypothesis, 9-13 November 2009, Kyushu University, Fukuoka, Japan, Walter de Gruyter, 2010.
  13. W. Parry and M. Pollicott, An analogue of the prime number theorem for closed orbits of Axiom A flows, Ann. of Math. (2) 118 (1983), no. 3, 573-591. https://doi.org/10.2307/2006982
  14. B. Randol, On the asymptotic distribution of closed geodesics on compact Riemann surfaces, Trans. Amer. Math. Soc. 233 (1977), 241-247. https://doi.org/10.1090/S0002-9947-1977-0482582-9

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