On Koyama’s refinement of the prime geodesic theorem. (English) Zbl 1422.11185

Summary: We give a new proof of the best presently-known error term in the prime geodesic theorem for compact hyperbolic surfaces, without the assumption of excluding a set of finite logarithmic measure. Stronger implications of the Gallagher-Koyama approach are derived, yielding to a further reduction of the error term outside a set of finite logarithmic measure.


11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
Full Text: DOI arXiv Euclid


[1] M. Avdispahić and Dž. Gušić, On the error term in the prime geodesic theorem, Bull. Korean Math. Soc. 49 (2012), no. 2, 367-372. · Zbl 1333.11087
[2] M. Avdispahić and L. Smajlović, An explicit formula and its application to the Selberg trace formula, Monatsh. Math. 147 (2006), no. 3, 183-198. · Zbl 1092.11026
[3] P. Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, 106, Birkhäuser Boston, Inc., Boston, MA, 1992. · Zbl 0770.53001
[4] P. X. Gallagher, Some consequences of the Riemann hypothesis, Acta Arith. 37 (1980), 339-343. · Zbl 0444.10034
[5] D. A. Hejhal, The Selberg trace formula for \(\text{PSL}(2,\mathbf{R})\). Vol. I, Lecture Notes in Mathematics, 548, Springer-Verlag, Berlin, 1976.
[6] S. Koyama, Refinement of prime geodesic theorem, Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 7, 77-81. · Zbl 1417.11092
[7] B. Randol, On the asymptotic distribution of closed geodesics on compact Riemann surfaces, Trans. Amer. Math. Soc. 233 (1977), 241-247.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.