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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 18, 2019

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 18, 2019



On the Logarithmic Derivative of Zeta Functions for Compact, Odd-dimensional Hyperbolic Spaces

AUTHORS: Dzenan Gusic

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In this paper we investigate the Selberg zeta functions and the Ruelle zeta functions associated with locally homogeneous bundles over compact locally symmetric spaces of rank one. Our basic object will be a compact locally symmetric Riemannian manifold with negative sectional curvature. In particular, our research will be restricted to compact, odd-dimensional, real hyperbolic spaces. For this class of spaces, the Titchmarsh-Landau style approximate formulas for the logarithmic derivative of the aforementioned zeta functions are derived. As expected in this setting, the obtained formulas are given in terms of zeros of the attached Selberg zeta functions. Our results follow from the fact that these zeta functions can be represented as quotients of two entire functions of order not larger than the dimension of the underlying compact, odd-dimensional, locally symmetric space, and the application of suitably chosen Weyl asymptotic law. The obtained formulas can be further applied in the proof of the corresponding prime geodesic theorem.

KEYWORDS: Zeta functions of Selberg and Ruelle, logarithmic derivative, locally symmetric spaces, approximate formulas

REFERENCES:

[ 1] 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.

[2] M. Avdispahic and D ´ z. Gu ˇ siˇ c, On the loga- ´ rithmic derivative of zeta functions for compact even-dimensional locally symmetric spaces of real rank one, Mathematica Slovaca 69, 2019, to appear.

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

[4] 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

[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] 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.

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

[9] 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.

[10] B. Randol, The Riemann hypothesis for Selberg’s zeta-function and the asymptotic behavior of eigenvalues of the Laplace operator, Trans. Amer. Math. Soc. 236, 1978, pp. 209–233.

[11] E.–C. Titchmarsh, The Theory of the Riemann zeta-function, Clarendon Press, Oxford 1986

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 18, 2019, Art. #25, pp. 176-184


Copyright Β© 2018 Author(s) retain the copyright of this article. This article is published under the terms of the Creative Commons Attribution License 4.0

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