Abstract
We derive approximate formulas for the logarithmic derivative of the Selberg and the Ruelle zeta functions over compact, even-dimensional, locally symmetric spaces of real rank one. The obtained formulas are given in terms of zeta singularities.
(Communicated by Filippo Nuccio)
References
[1] Avdispahić, M.— Gušić, Dž.: On the error term in the prime geodesic theorem, Bull. Korean Math. Soc. 49 (2012), 367–372.10.4134/BKMS.2012.49.2.367Search in Google Scholar
[2] Avdispahić, M.— Gušić, Dž.: Order of Selberg’s and Ruelle’s zeta functions for compact even-dimensional locally symmetric spaces, J. Math. Anal. Appl. 413 (2014), 525–531.10.1016/j.jmaa.2013.10.052Search in Google Scholar
[3] Bunke, U.—Olbrich, M.: Selberg Zeta and Theta Functions: A Differential Operator Approach, Akademie Verlag, Berlin, 1995.Search in Google Scholar
[4] Duistermaat, J. J.—Kolk, J. A. C.—Varadarajan, V. S.: Spectra of compact locally symmetric manifolds of negative curvature, Invent. Math. 52 (1979), 27–93.10.1007/BF01389856Search in Google Scholar
[5] Fried, D.: The zeta functions of Ruelle and Selberg. I, Ann. Sci. Ec. Norm. Sup. 19 (1986), 491–517.10.24033/asens.1515Search in Google Scholar
[6] Gangolli, R.: Zeta functions of Selberg’s type for compact space forms of symmetric spaces of rank one, Illinois J. Math. 21 (1977), 1–42.10.1215/ijm/1256049498Search in Google Scholar
[7] Gangolli, R.—Warner, G.: Zeta functions of Selberg’s type for some non-compact quotients of symmetric spaces of rank one, Nagoya Math. J. 78 (1980), 1–44.10.1017/S002776300001878XSearch in Google Scholar
[8] Hejhal, D.: The Selberg trace formula for Psl(2,ℝ), Vol. I. Lecture Notes in Math. 548, Springer-Verlag, Berlin-Heidelberg, 1976.10.1007/BFb0079608Search in Google Scholar
[9] Park, J.: Ruelle zeta function and prime geodesic theorem for hyperbolic manifolds with cusps. In: Casimir Force, Casimir Operators and Riemann Hypothesis (G. van Dijk, M. Wakayama, eds.), de Gruyter, Berlin, 2010, pp. 89–104.10.1515/9783110226133.89Search in Google Scholar
[10] Randol, B.: The Riemann hypothesis for Selberg’s zeta-function and the asymptotic behavior of eigenvalues of the Laplace operator, Trans. Amer. Math. Soc. 236 (1978), 209−223.10.1090/S0002-9947-1978-0472728-1Search in Google Scholar
[11] Titchmarsh, E. C.: The Theory of the Riemann Zeta-function, Clarendon Press, Oxford, 1986.Search in Google Scholar
© 2019 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
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Articles in the same Issue
- Regular papers
- RNDr. Miloslav Duchoň, DrSc. passed away ∗ aug. 21, 1932 – †dec. 31, 2018
- Prof. RNDr. Gejza Wimmer, DrSc. – A Septuagenerian?
- Two monads on the category of graphs
- Entropy of MV-algebraic dynamical systems: An example
- Going up and lying over in congruence-modular algebras
- The minimal arity of near unanimity polymorphisms
- On the logarithmic derivative of zeta functions for compact even-dimensional locally symmetric spaces of real rank one
- The generalized Fermat Conjecture
- Evaluation of sums involving products of Gaussian q-binomial coefficients with applications
- Integrals of logarithmic functions and alternating multiple zeta values
- An abelian subextension of the dyadic division field of a hyperelliptic Jacobian
- The class of all semigroups related to semihypergroups of order 2
- Criterion for the weak potency of certain HNN extensions
- Fekete Szegö theorem for a close-to-convex error function
- Bounds for distance between eigenvalues of boundary value problems with retarded argument
- Gradient estimates for a nonlinear heat equation under the Finsler-Ricci flow
- Weak module amenability of triangular banach algebras II
- Some consequences of quasicentral approximate units modulo Hilbert-Schmidt class
- Example of C-rigid polytopes which are not B-rigid
- Some comments on rU-spaces
- On the equivalence of various definitions of mixed poisson processes
- A note on discrete C-embedded subspaces