Journal of the Mathematical Society of Japan

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

Yasufumi HASHIMOTO

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 13 October 2006

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1160745823

Digital Object Identifier
doi:10.2969/jmsj/1160745823

Mathematical Reviews number (MathSciNet)
MR2114730

Zentralblatt MATH identifier
1084.11052

Subjects
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

Keywords
spectral zeta function Selberg's zeta function first eigenvalue

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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