Journal of the Mathematical Society of Japan

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


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In this paper, we establish the formulas expressing the special values of the spectral zeta function ζ Δ ( n ) of the Laplacian Δ on some locally symmetric Riemannian manifold Γ \ 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 Δ by computing the values ζ Δ ( n ) numerically, when Γ \ G / K is a Riemann surface with Γ being the quaternion group.

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

First available in Project Euclid: 13 October 2006

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Zentralblatt MATH identifier

Primary: 11F72: Spectral theory; Selberg trace formula
Secondary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas 35P15: Estimation of eigenvalues, upper and lower bounds

spectral zeta function Selberg's zeta function first eigenvalue


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.

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