## Journal of the Mathematical Society of Japan

### Special values of the spectral zeta functions for locally symmetric Riemannian manifolds

Yasufumi HASHIMOTO

#### Abstract

In this paper, we establish the formulas expressing the special values of the spectral zeta function $\zeta_{\rm\Delta}(n)$ of the Laplacian $\rm\Delta$ on some locally symmetric Riemannian manifold $\Gamma\backslash G/K$ in terms of the coefficients of the Laurent expansion of the corresponding Selberg zeta function. As an application, we give a numerical estimation of the first eigenvalue of $\rm\Delta$ by computing the values $\zeta_{\rm\Delta}(n)$ numerically, when $\Gamma\backslash G/K$ is a Riemann surface with $\Gamma$ being the quaternion group.

#### Article information

Source
J. Math. Soc. Japan, Volume 57, Number 1 (2005), 217-232.

Dates
First available in Project Euclid: 13 October 2006

https://projecteuclid.org/euclid.jmsj/1160745823

Digital Object Identifier
doi:10.2969/jmsj/1160745823

Mathematical Reviews number (MathSciNet)
MR2114730

Zentralblatt MATH identifier
1084.11052

#### Citation

HASHIMOTO, Yasufumi. Special values of the spectral zeta functions for locally symmetric Riemannian manifolds. J. Math. Soc. Japan 57 (2005), no. 1, 217--232. doi:10.2969/jmsj/1160745823. https://projecteuclid.org/euclid.jmsj/1160745823

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