Let \(\{0=\lambda_0<\lambda_1\leq\dots \}\) be the spectrum of the scalar Laplacian \(\Delta=\delta d\) on a compact Riemannian manifold \(M\). Let \(\zeta(s):=\sum_{\nu\geq 1}\lambda_\nu^{-s}\) and \(h(t):=\sum_{\nu\geq 0}e^{-t\lambda_\nu}\) be the associated zeta and heat functions. The series defining the zeta function \(\zeta(s)\) converges for \(\operatorname{Re}(s)\gg 0\) and has a meromorphic extension to the complex plane with isolated simple poles on the real axis; the series defining the heat function \(h(t)\) converges for \(t>0\) and has an asymptotic expansion as \(t\downarrow 0\). The asymptotic coefficients in the expansion of \(h\) determine the pole structure of the zeta function.
The author uses group theoretic methods to study the case when \(M=U/K\) is a rank \(1\) symmetric space. He determines the multiplicities of the Laplacian using a relation between these multiplicities and the spherical Plancherel measure - this method gives a unified formula for these cases and provides another derivation of a result of R. S. Cahn and J. A. Wolf [Comment. Math. Helv. 51, 1-21 (1976; Zbl 0327.43013)]. The author also discusses the analytic continuation in the simply connected case and gives a relationship between the compact zeta function and the noncompact (local) one \(\zeta_{G/K}\).