For a connected, finite graph $G$, let $\lambda_1, \ldots, \lambda_n$ denote the nonzero eigenvalues of the graph Laplacian. We define $\zeta_G = \Sigma_{i = 1}^n \lambda_i^s$.

Then Kirkoffs Matrix-Tree theorem can be reformulated as saying that $e^{ - \zeta_G'(0)} / |G| = \tau(G)$, where $\tau$ is the number of trees of the graph $G$. (See here for more: http://www.math.ucsd.edu/~fan/lattice.pdf )

If $M$ is a Riemannian manifold, and $\Delta$ is it's Laplacian, we can define a similar zeta function: $\zeta_{\Delta}$. ( https://en.wikipedia.org/wiki/Minakshisundaram%E2%80%93Pleijel_zeta_function )

Question: Is there a reasonable meaning that can be assigned to $e^{- \zeta_{\Delta}'(0)} / Vol(M)$?

I think (but am not sure) that this quantity doesn't make much sense as stated. The reason is because:

$\zeta_{\Delta}(s) \sim (4 \pi s)^{-n/2} \Sigma_{m = 0}^{\infty} a_m s^m$ (see application one on the wikipedia page) , which has no derivative at zero.

Thank you!

analytic torsionof the Riemannian manifold. It is, of course, a regularized value of the determinant of the Laplace operator. What you look for may be the Ray-Singer conjecture (Adv. Math., 1971) and its proofs by Cheeger (Analytic torsion and Reidemeister torsion, 1979) and Muller (Analytic torsion and R-torsion of Riemannian manifolds, Adv. Math., 1978). $\endgroup$