Proceedings of the Japan Academy, Series A, Mathematical Sciences

Refinement of prime geodesic theorem

Shin-ya Koyama

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Abstract

We prove existence of a set $E$ of positive real numbers, which is relatively small in the sense that its logarithmic measure is finite, such that we can improve the error term of the prime geodesic theorem as $x\to\infty$ $(x\notin E)$. The result holds for any compact hyperbolic surfaces, and it would also be true for generic hyperbolic surfaces of finite volume according to the philosophy of Phillips and Sarnak.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 92, Number 7 (2016), 77-81.

Dates
First available in Project Euclid: 21 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.pja/1469104539

Digital Object Identifier
doi:10.3792/pjaa.92.77

Mathematical Reviews number (MathSciNet)
MR3529088

Zentralblatt MATH identifier
06673656

Subjects
Primary: 11F72: Spectral theory; Selberg trace formula
Secondary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}

Keywords
Prime geodesic theorem Selberg zeta functions arithmetic groups

Citation

Koyama, Shin-ya. Refinement of prime geodesic theorem. Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 7, 77--81. doi:10.3792/pjaa.92.77. https://projecteuclid.org/euclid.pja/1469104539


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References

  • P. X. Gallagher, A large sieve density estimate near $\sigma = 1$, Invent. Math. 11 (1970), 329–339.
  • P. X. Gallagher, Some consequences of the Riemann hypothesis, Acta Arith. 37 (1980), 339–343.
  • D. A. Hejhal, The Selberg trace formula for $\mathit{PSL}(2,\mathbf{R})$. Vol. I, Lecture Notes in Mathematics, Vol. 548, Springer, Berlin, 1976.
  • D. A. Hejhal, The Selberg trace formula for $\mathit{PSL}(2,\mathbf{R})$. Vol. II, Lecture Notes in Mathematics, Vol. 1001, Springer, Berlin, 1983.
  • H. Iwaniec, Prime geodesic theorem, J. Reine Angew. Math. 349 (1984), 136–159.
  • S. Koyama, Prime geodesic theorem for arithmetic compact surfaces, Internat. Math. Res. Notices 1998, no. 8, 383–388.
  • W. Luo, Z. Rudnick and P. Sarnak, On Selberg's eigenvalue conjecture, Geom. Funct. Anal. 5 (1995), no. 2, 387–401.
  • W. Z. Luo and P. Sarnak, Quantum ergodicity of eigenfunctions on $\mathit{PSL}_{2}(\mathbf{Z})\backslash \mathbf{H}^{2}$, Publ. Math. Inst. Hautes Études Sci. 81 (1995), 207–237.
  • R. S. Phillips and P. Sarnak, On cusp forms for co-finite subgroups of $\mathit{PSL}(2,\mathbf{R})$, Invent. Math. 80 (1985), no. 2, 339–364.