Proceedings of the Japan Academy, Series A, Mathematical Sciences

Refinement of prime geodesic theorem

Shin-ya Koyama

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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

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

First available in Project Euclid: 21 July 2016

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Zentralblatt MATH identifier

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}

Prime geodesic theorem Selberg zeta functions arithmetic groups


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.

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