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Dzenan Gusic

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Dzenan Gusic

WSEAS Transactions on Mathematics

Print ISSN: 1109-2769
E-ISSN: 2224-2880

Volume 17, 2018

Notice: As of 2014 and for the forthcoming years, the publication frequency/periodicity of WSEAS Journals is adapted to the 'continuously updated' model. What this means is that instead of being separated into issues, new papers will be added on a continuous basis, allowing a more regular flow and shorter publication times. The papers will appear in reverse order, therefore the most recent one will be on top.

Volume 17, 2018

A Weighted Generalized Prime Geodesic Theorem

AUTHORS: Dzenan Gusic

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ABSTRACT: Prime geodesic theorem gives an asymptotic estimate for the number of prime geodesics over underlying symmetric space counted by their lengths. In any setting, the search for the optimal error term is widely open. Our objective is to derive a weighted, generalized form of the prime geodesic theorem for compact, even-dimensional, locally symmetric Riemannian manifolds of strictly negative sectional curvature. We base our methodology on an application of the integrated, Chebyshev-type counting function of appropriate order. The obtained error term improves the corresponding, and best known one in the case of classical prime geodesic theorem. Our conclusion in the case at hand is that a weighted sense yields a better result.

KEYWORDS: Weighted prime geodesic theorem, counting functions, zeta functions, topological singularities, spectral singularities


[ 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, On the length ´ spectrum for compact locally symmetric spaces of real rank one, WSEAS Trans. on Math. 16, 2017, pp. 303–321.

[3] U. Bunke and M. Olbrich, Selberg zeta and theta functions. A Differential Operator Approach, Akademie–Verlag, Berlin 1995

[4] D.–L. DeGeorge, Length spectrum for compact locally symmetric spaces of strictly negative curvature, Ann. Sci. Ec. Norm. Sup. 10, 1977, pp. 133–152.

[5] J.–J. Duistermaat, J.–A.–C. Kolk and V.–S. Varadarajan, Spectra of compact locally symmetric manifolds of negative curvature, Invent. Math. 52, 1979, pp. 27–93.

[6] D. Fried, The zeta functions of Ruelle and Selberg. I, Ann. Sci. Ec. Norm. Sup. 19, 1986, pp. 491–517.

[7] R. Gangolli, The length spectrum of some compact manifolds of negative curvature, J. Diff. Geom. 12, 1977, pp. 403–426.

[8] Dz. Gu ˇ siˇ c, 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.

[9] D. Hejhal, The Selberg trace formula for PSL (2, R), Vol. I. Lecture Notes in Mathematics 548, Springer–Verlag, Berlin–Heidelberg 1976

[10] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Acad.–Press 1978

[11] A.–E. Ingham, The distribution of prime numbers, Cambridge University Press 1990

[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 Riemann hypothesis, de Gruyter, Berlin 2010, pp. 89–104.

[13] B. Randol, On the asymptotic distribution of closed geodesics on compact Riemann surfaces, Trans. Amer. Math. Soc. 233, 1977, pp. 241–247.

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 17, 2018, Art. #30, pp. 237-251

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