Carletti, E.; Monti Bragadin, G. On Minakshisundaram-Pleijel zeta functions of spheres. (English) Zbl 0838.11057 Proc. Am. Math. Soc. 122, No. 4, 993-1001 (1994). 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) Cited in 1 ReviewCited in 7 Documents 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 Keywords:spectral zeta-functions; eigenvalues of the Laplace operators; Hurwitz zeta-functions PDFBibTeX XMLCite \textit{E. Carletti} and \textit{G. Monti Bragadin}, Proc. Am. Math. Soc. 122, No. 4, 993--1001 (1994; Zbl 0838.11057) Full Text: DOI References: [1] Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). · Zbl 0223.53034 [2] E. Carletti and G. Monti Bragadin, On Dirichlet series associated with polynomials, Proc. Amer. Math. Soc. 121 (1994), no. 1, 33 – 37. · Zbl 0804.11051 [3] Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Mathematics Lecture Series, vol. 11, Publish or Perish, Inc., Wilmington, DE, 1984. · Zbl 0565.58035 [4] Charles Jordan, Calculus of finite differences, Third Edition. Introduction by Harry C. Carver, Chelsea Publishing Co., New York, 1965. · Zbl 0060.12309 [5] S. Minakshisundaram and Å. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canadian J. Math. 1 (1949), 242 – 256. · Zbl 0041.42701 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.