Proceedings of the Japan Academy, Series A, Mathematical Sciences

Euler products beyond the boundary for Selberg zeta functions

Shin-ya Koyama and Fumika Suzuki

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Convergence of Euler products in the critical strip is directly related to a proof of the generalized Riemann hypothesis. Moreover its behavior on the critical line is called the deep Riemann hypothesis (DRH). Kimura-Koyama-Kurokawa recently proved DRH over function fields in case the $L$-function is regular at $s=1$ [3]. In this paper we generalize their results to Selberg zeta functions. Our results imply the DRH for principal congruence groups over function fields.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 90, Number 8 (2014), 101-106.

First available in Project Euclid: 3 October 2014

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Zentralblatt MATH identifier

Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$

Selberg zeta functions Riemann hypothesis Euler products


Koyama, Shin-ya; Suzuki, Fumika. Euler products beyond the boundary for Selberg zeta functions. Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), no. 8, 101--106. doi:10.3792/pjaa.90.101.

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