AUTHORS: Dzenan Gusic
Download as PDF
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
REFERENCES:
[
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.