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On Minakshisundaram-Pleijel zeta functions of spheres. (English) Zbl 0838.11057

Let \(0 = \lambda_0 < \lambda_1 \leq \lambda_2 \leq \cdots\) be the sequence of the eigenvalues of the Laplace operator acting on \(L^2 (M)\) for a compact Riemannian manifold \(M\), and let \(\zeta_M (s) = \sum^\infty_{i = 0} \lambda_i^{-s}\) for \(\text{Re} s>{1\over 2}\dim M\). The function \(s \mapsto \zeta_M (s)\) is known to have a meromorphic continuation to the whole complex plane. For \(M = S^k\) and for \(M = \mathbb{P}^k (\mathbb{R})\), the authors express \(\zeta_M (s)\) in terms of Hurwitz zeta-functions \(\zeta (s, \alpha) = \sum^\infty_{n = 0} (n + \alpha)^{-s}\) and calculate the residues of \(\zeta_M (s)\) at \(s = k/2 - m\), \(m \in \mathbb{Z}\), \(m \geq 0\).
Reviewer: B.Z.Moroz (Bonn)

MSC:

11M41 Other Dirichlet series and zeta functions
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
11B73 Bell and Stirling numbers
11M35 Hurwitz and Lerch zeta functions
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References:

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