On the Analytic Continuation of the Minakshisundaram]Pleijel Zeta Function for Compact Symmetric Spaces of Rank One

We give two equivalent analytic continuations of the Minakshisundaram]Pleijel zeta function zUrK z. for a Riemannian symmetric space of the compact type of rank one UrK. First we prove that zUrK can be written as zUrK z.seip zyNr2.VUrKzGrK z. qF z., where Nsdim UrK, VUrKis the volume of UrK, zGrK z. is the local zeta function for GrK the noncompact symmetric space dual to UrK., and F z. is an analytic function which is given explicitly as a contour integral cf. Eq. 4.11... To prove the above formula we use a relation first noticed by Vretare.between the scalar degeneracies of the Laplacian on UrK and the Plancherel measure on GrK. The second expression we obtain for zUrK z. is in terms of a series of generalized.Riemann zeta functions zR z, q. cf. Eq. 5.9... The doubly connected case of real projective spaces is also discussed.