a8b6b4b6-38eb-403e-8350-1e7a2d4c62e520210318052618079wseamdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON SYSTEMS AND CONTROL1991-876310.37394/23203http://wseas.org/wseas/cms.action?id=4073220202022020201510.37394/23203.2020.15http://wseas.org/wseas/cms.action?id=23195On Some Higher Order Counting Functions for PSL(2,R)DzenanGusicUniversity of Sarajevo, Faculty of Sciences and Mathematics, Department of Mathematics, Sarajevo, BOSNIA AND HERZEGOVINAThis paper is devoted to some counting functions of level one and level three in the case of quotient space generated by some strictly hyperbolic Fuchsian group and the upper half-plane. Each of the functions is represented as a sum of some explicit part plus the error term. The explicit part is indexed over singularities of the corresponding Selberg zeta function. In particular, the obtained error term is not larger than O ( x 3/4) . The method applied in this paper follows traditional approach for achieving the error terms in the case of locally symmetric spaces of real rank one. In order to establish an analogy with the classical case, we consider the counting functions divided by x and x3, respectively.320202032020207380https://www.wseas.org/multimedia/journals/control/2020/a185103-058.pdf10.37394/23203.2020.15.9http://www.wseas.org/multimedia/journals/control/2020/a185103-058.pdfM. Avdispahic and Dz. Gusic, A Weighted Prime Geodesic Theorem, Mathematica Balkanica 25, 2011, pp. 463–474.10.4134/bkms.2012.49.2.367M. Avdispahic and Dz. Gusic, On the error termin the prime geodesic theorem, Bull. Korean Math. Soc.49, 2012, pp. 367–372.M. Avdispahic and Dz. Gusic, On the length spectrum for compact locally symmetric spaces of real rank one, WSEAS Trans. on Math.16, 2017, pp. 303–321.R. P. Boas, Entire Functions, Academic Press Inc., Publishers, New York 1954J. B. Conway,Functions of One Complex Variable, Springer–Verlag, New York–Heidelberg–Berlin 1978Dz. Gusic, Prime geodesic theorem for com-pact even - dimensional locally symmetric Riemannian manifolds of strictly negative sectional curvature, WSEAS Trans. on Math.17, 2018, pp. 188–196.Dz. Gusic, A Weighted Generalized Prime Geodesic Theorem, WSEAS Trans. on Math.17,2018, pp. 237–251.Dz. Gusic, Prime Geodesic Theorem for Com-pact Riemann Surfaces,Int. J. on Circuits, Systems and Signal Processing13, 2019, pp. 747–753.10.1007/bfb0079608D. Hejhal,The Selberg trace formula for PSL(2,R). Vol. I. Lecture Notes in Mathematics 548, Springer–Verlag, Berlin–Heidelberg 1976.10.1007/bf01451031H. Huber, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen. II, Math. Annalen142, 1961, pp. 385–398.10.1007/bf01470758H. Huber, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungs gruppen. II (Nachtrag zu Math. Annalen 142, 385-398,1961),Math. Annalen143, 1961, pp. 463–464.A. E. Ingham,The Distribution of Prime Numbers,Cambridge Mathematical Library, 199010.1002/cpa.3160250302H. P. McKean, Selberg’s trace formula as applied to a compact Riemann surface,Comm.Pure Appl. Math.25, 1972, pp. 225–246.10.1515/9783110226133.89J. Park, Ruelle zeta function and prime geodesic theorem for hyperbolic manifolds with cusps,in: G. van Dijk, M. Wakayama (eds.), Casimirforce, Casimir operators and Riemann hypothesis, de Gruyter, Berlin 2010, pp. 89–104.10.1090/s0002-9947-1977-0482582-9B. Randol, On the asymptotic distributon ofclosed geodesics on compact Riemann surfaces,Trans. Amer. Math. Soc.233, 1977, pp. 241–247.A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannians paces with applications to Diriclet series,J. In-dian Math.20, 1956, pp. 47–87.C. L. Siegel,Topics in Complex Function Theory,Wiley–Interscience, 1971J. J. Stocker, Differential Geometry,Wiley–Interscience, 196910.1007/s11401-008-0172-0H. Tang, The Generalized Prime Number Theorem for Automorphic L-Functions,Chin. Ann.Math.3, 2009, pp. 251–260.