## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Refinement of prime geodesic theorem

Shin-ya Koyama

#### 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

https://projecteuclid.org/euclid.pja/1469104539

Digital Object Identifier
doi:10.3792/pjaa.92.77

Mathematical Reviews number (MathSciNet)
MR3529088

Zentralblatt MATH identifier
06673656

#### 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

#### 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.