Gallagherian Prime Geodesic Theorem in Higher Dimensions

  • Muharem Avdispahić
  • Zenan ŠabanacEmail author


Using the Gallagher–Koyama approach, we reduce the exponent in the error term of the prime geodesic theorem for real hyperbolic manifolds with cusps.


Hyperbolic manifolds Prime geodesic theorem Selberg and Ruelle zeta functions 

Mathematics Subject Classification

11M36 11F72 58J50 



We would like to thank the referee for suggestions that resulted in adding the remark on lower dimensions (and related references) to the initial version of the paper.


  1. 1.
    Avdispahić, M.: On Koyama’s refinement of the prime geodesic theorem. Proc. Japan Acad. Ser. A 94(3), 21–24 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Avdispahić, M.: Gallagherian \(PGT\) on \(PSL(2, {\mathbb{Z}}) \). Funct. Approximatio. Comment. Math. 58(2), 207–213 (2018)Google Scholar
  3. 3.
    Avdispahić, M.: Errata and addendum to On the prime geodesic theorem for hyperbolic 3-manifolds Math. Nachr. 291 (2018), no. 14–15, 2160–2167, Math. Nachr. 292(4), 691–693 (2019)Google Scholar
  4. 4.
    Avdispahić, M.: Prime geodesic theorem of Gallagher type for Riemann surfaces, Anal. Math. (to appear)Google Scholar
  5. 5.
    Avdispahić, M.: Prime geodesic theorem for the modular surface. Hacet. J. Math. Stat. (to appear).
  6. 6.
    Avdispahić, M., Gušić, Dž: On the error term in the prime geodesic theorem. Bull. Korean Math. Soc. 49(2), 367–372 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Balkanova, O., Chatzakos, D., Cherubini, G., Frolenkov, D., Laaksonen, N.: Prime geodesic theorem in the 3-dimensional hyperbolic space. Trans. Am. Math. Soc. 372(8), 5355–5374 (2019)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Balkanova, O., Frolenkov, D.: Sums of Kloosterman sums in the prime geodesic theorem. Q. J. Math. 70(2), 649–674 (2019)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Balkanova, O., Frolenkov, D.: Prime geodesic theorem for the Picard manifold, arXiv:1804.00275v2
  10. 10.
    Balog, A., Biró, A., Harcos, G., Maga, P.: The prime geodesic theorem in square mean. J. Number Theory 198, 239–249 (2019)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chatzakos, D., Cherubini, G., Laaksonen, N.: Second moment of the prime geodesic theorem for \(PSL(2, {\mathbb{Z}}[i])\), arXiv:1812.11916
  12. 12.
    Cherubini, G., Guerreiro, J.: Mean square in the prime geodesic theorem. Algebra Number Theory 12(3), 571–597 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    DeGeorge, D.L.: Length spectrum for compact locally symmetric spaces of strictly negative curvature. Ann. Sci. Ecole Norm. Sup. 10, 133–152 (1977)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gallagher, P.X.: A large sieve density estimate near \(\sigma =1\). Invent. Math. 11, 329–339 (1970)Google Scholar
  15. 15.
    Gallagher, P.X.: Some consequences of the Riemann hypothesis. Acta Arith. 37, 339–343 (1980)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gangolli, R.: The length spectra of some compact manifolds of negative curvature. J. Differ. Geom. 12, 403–424 (1977)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gangolli, R., Warner, G.: Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one. Nagoya Math. J. 78, 1–44 (1980)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kaneko, I.: Second moment of the prime geodesic theorem for \(PSL(2, {\mathbb{Z}}[i])\) and bounds on a spectral exponential sum, arXiv:1903.05111
  19. 19.
    Koyama, S.: Refinement of prime geodesic theorem. Proc. Japan Acad. Ser. A 92(7), 77–81 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Park, J.: Ruelle zeta function and prime geodesic theorem for hyperbolic manifolds with cusps, In: van Dijk, G., Wakayama, M. (eds.) Casimir Force, Casimir Operators and the Riemann Hypothesis, 9–13 November 2009, Kyushu University, Fukuoka, Japan, pp. 89–104, Walter de Gruyter (2010)Google Scholar
  21. 21.
    Randol, B.: On the asymptotic distribution of closed geodesics on compact Riemann surfaces. Trans. Am. Math. Soc. 233, 241–247 (1977)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Sarnak, P.: The arithmetic and geometry of some hyperbolic three-manifolds. Acta Math. 151, 253–295 (1983)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Soundararajan, K., Young, M.P.: The prime geodesic theorem. J. Reine Angew. Math. 676, 105–120 (2013)MathSciNetzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of SarajevoSarajevoBosnia and Herzegovina

Personalised recommendations