AUTHORS: Dzenan Gusic
Download as PDF
ABSTRACT: Applying a suitably derived, Titchmarsh-Landau style approximate formula for the logarithmic derivative of the Ruelle zeta function, we obtain an another proof of the recently improved variant of the prime geodesic theorem for compact, even-dimensional, locally symmetric spaces of real rank one
KEYWORDS: Prime geodesic theorem, Selberg zeta function, Ruelle zeta function
REFERENCES:
[1] M. Avdispahic and D ´ z. Gu ˇ siˇ c, On the error term ´ in the prime geodesic theorem, Bull. Korean Math. Soc. 49, 2012, pp. 367–372.
[2] M. Avdispahic and D ´ z. Gu ˇ siˇ c, Order of Selberg’s ´ and Ruelle’s zeta functions for compact evendimensional locally symmetric spaces, J. Math. Anal. Appl. 413, 2014, pp. 525–531.
[3] M. Avdispahic and D ´ z. Gu ˇ siˇ c, On the length ´ spectrum for compact locally symmetric spaces of real rank one, WSEAS Trans. on Math. 16, 2017, pp. 303-321.
[4] U. Bunke and M. Olbrich, Selberg zeta and theta functions. A Differential Operator Approach, Akademie–Verlag, Berlin 1995
[5] U. Bunke and M. Olbrich, Theta and zeta functions for locally symmetric spaces of rank one, available at https://arxiv.org/abs/dg-ga/9407013
[6] Y. Cai, Prime geodesic theorem, J. The´or. Nombres. Bordeaux 14, 2002, pp. 59–72.
[7] D.–L. DeGeorge, Length spectrum for compact locally symmetric spaces of strictly negative curvature, Ann. Sci. Ec. Norm. Sup. 10, 1977, pp. 133–152.
[8] D. Fried, The zeta functions of Ruelle and Selberg. I, Ann. Sci. Ec. Norm. Sup. 19, 1986, pp. 491–517.
[9] R. Gangolli, The length spectrum of some compact manifolds of negative curvature, J. Diff. Geom. 12, 1977, pp. 403–426.
[10] Y. Gon and J. Park, The zeta functions of Ruelle and Selberg for hyperbolic manifolds with cusps, Math. Ann. 346, 2010, pp. 719–767.
[11] D. Hejhal, The Selberg trace formula for PSL (2, R), Vol. I. Lecture Notes in Mathematics 548, Springer–Verlag, Berlin–Heidelberg 1976
[12] D. Hejhal, The Selberg trace formula for PSL (2, R), Vol. II. Lecture Notes in Mathematics 1001, Springer–Verlag, Berlin 1983
[13] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Acad.–Press 1978
[14] M.–D. Horton, H.–M. Stark and A.–A. Terras, What are zeta functions of graphs and what are they good for?, in: G. Berkolaiko, R. Carlson, S. A. Fulling and P. Kuchment (eds.), Quantum graphs and their applications, Contemp. Math., 2006, pp. 173–190.
[15] H. Iwaniec, Prime geodesic theorem, J. Reine Angew. Math. 349, 1984, pp. 136–159.
[16] N. Katz and P. Sarnak Zeros of zeta functions and symmetry, Bull. Amer. Math. Soc. 36, 1999, pp. 1–26.
[17] S. Lang, Algebraic Number Theory, Adison– Wesley, Mass. 1968
[18] W. Luo and P. Sarnak, Quantum ergodicity of eigenfunctions on PSL2 (R) \ H2 , Inst. Hautes E´tudes Sci. Publ. Math. 81, 1995, pp. 207–237.
[19] 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 Riemann hypothesis, de Gruyter, Berlin 2010, pp. 89–104.
[20] M. Pavey, Class Numbers of Orders in Quartic Fields, Ph.D. dissertation, University of T¨ubingen, 2006.
[21] B. Randol, On the asymptotic distribution of closed geodesics on compact Riemann surfaces, Trans. Amer. Math. Soc. 233, 1977, pp. 241–247.
[22] D. Ruelle, Dynamical zeta functions and transfer operators, Notices Amer. Math. Soc. 49, 2002, pp. 887–895.
[23] K. Soundararajan and M.–P. Young, The prime geodesic theorem, J. Reine Angew. Math. 676, 2013, pp. 105–120.
[24] E.–C. Titchmarsh, The Theory of the Riemann Zeta-function, Claredon–Press, Oxford 1986