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On the Prime Geodesic Theorem for Non-Compact Riemann Surfaces
Department of Mathematics, Faculty of Sciences and Mathematics, University of Sarajevo, Sarajevo, Bosnia and Herzegovina
Department of Mathematics, Faculty of Sciences and Mathematics, University of Sarajevo, Sarajevo, Bosnia and Herzegovina
- 1 Department of Mathematics, Faculty of Sciences and Mathematics, University of Sarajevo, Sarajevo, Bosnia and Herzegovina
- 2 Department of Mathematics, Faculty of Sciences and Mathematics, University of Sarajevo, Sarajevo, Bosnia and Herzegovina
Advances in Pure Mathematics·Volume 06 (2016)·Pages 903–914·Published 8 November 2016·DOI10.4236/apm.2016.612068
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Abstract
We use B. Randol’s method to improve the error term in the prime geodesic theorem for a noncompact Riemann surface having at least one cusp. The case considered is a general one, corresponding to a Fuchsian group of the first kind and a multiplier system with a weight on it.
KeywordsSelberg Trace FormulaSelberg Zeta FunctionPrime Geodesic Theorem
- Selberg, A. (1956) Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces with Applications to Dirichlet Series. Journal of the Indian Mathematical Society, 20, 47-87.
- Huber, H. (1961) Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgrupen II. Mathematische Annalen, 142, 385-398. https://doi.org/10.1007/BF01451031
- Huber, H. (1961) Nachtrag zu. Mathematische Annalen, 143, 463-464. https://doi.org/10.1007/BF01470758
- Hejhal, D. (1973) The Selberg Trace Formula for , Vol. I. Lecture Notes in Mathematics, Volume 548. Springer-Verlag, Berlin-Heidelberg.
- Hejhal, D. (1983) The Selberg Trace Formula for , Vol. II. Lecture Notes in Mathematics, Volume 1001. Springer-Verlag, Berlin-Heidelberg.
- Randol, B. (1977) On the Asymptotic Distribution of Closed Geodesics on Compact Riemann Surfaces. Transactions of the American Mathematical Society, 233, 241-247. https://doi.org/10.1090/S0002-9947-1977-0482582-9
- Buser, P. (1992) Geometry and Spectra of Compact Riemann Surfaces, Progress in Mathematics, Vol. 106. Birkhäuser, Boston-Basel-Berlin.
- Avdispahić, M. and Smajlović, L. (2009) On the Prime Number Theorem for a Compact Riemmann Surface. Rocky Mountain Journal of Mathematics, 39, 1837-1845. https://doi.org/10.1216/RMJ-2009-39-6-1837
- Avdispahić, M. and Smajlović, L. (2006) An explicit Formula and Its Application to the Selberg Trace Formula. Monatshefte für Mathematik, 147, 183-198. https://doi.org/10.1007/s00605-005-0317-0
- Avdispahić, M. and Smajlović, L. (2008) Euler Constants for a Fuchsian Group of the First Kind. Acta Arithmetica, 131, 125-143. https://doi.org/10.4064/aa131-2-2
- Avdispahić, M. and Smajlović, L. (2016) Selberg Trace Formula as an Explicit Formula and the Prime Geodesic Theorem. (Submitted)
- Fischer, J. (19760 An Approach to the Selberg Trace Formula via Selberg Zeta-Function. Lecture Notes in Mathematics, Volume 1253. Springer-Verlag, Berlin-Heidelberg.
- Hardy, G.H. and Riesz. M. (1915) The General Theory of Dirichlet’s Series. Cambridge University Press, Cambridge.
- Jameson, G.J. (2003) The Prime Number Theorem. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9781139164986
- Park, J. (2010) Ruelle Zeta Function and Prime Geodesic Theorem for Hyperbolic Manifolds with Cusps. In: van Dijk, G. and Wakayama, M., Eds., Casimir Force, Casimir Operators and Riemann Hypothesis, de Gruyter, Berlin, 89-104.