… Let S be a compact Riemann surface, which we will regard as the quotient of the upper
half-plane H by a discontinuous group r of hyperbolic transformations. We will assume that H
is endowed with the metric y-2((dx)2 + (dy)2), and we will denote the area of S by A. Let O =
Xo < xi < X2 < ?* be the eigenvalues corresponding to the problem Af + V = 0 on S, where A is
the Laplace operator on S, derived from the metric induced on S by that of H. … If E> 0, and
we observe, using the power series for e-'t, that fo (1 - e't) ts dt/t can be continued to the left of …

We give two equivalent analytic continuations of the Minakshisundaram–Pleijel zeta function
ζU/K (z) for a Riemannian symmetric space of the compact type of rank oneU/K. First we
prove that ζU/Kcan be written asζ U/K (z)= e iπ (z− N/2) VU/Kζ G/K (z)+ F (z), whereN=
dimU/K, VU/Kis the volume ofU/K, ζG/K (z) is the local zeta function forG/K (the noncompact
symmetric space dual toU/K), andF (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 …