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!



What is $e^{- \zeta_{\Delta} '(0)}$ for a $\Delta$ the Laplacian of a manifold?
If I am not mistaken, this cannot be true in general. For example, one may scale the metric for $\mathbb{S}^{1}$ proportionally and the determinant of the Laplacian remains the same independent of the length of the circle. For dimension 2, the determinant of the Laplacian obtains its maximum when $M$ has [constant sectional curvature](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.486.558&rep=rep1&type=pdf). If we rescale the metric proportionally at all places, similar phenomenon happens. One may work with a sphere or a torus as $\mathbb{R}^{2}/L$ with $L$ being a lattice to check explicitly how the $\zeta$-function behaves under dilation.


There is (allegedly) [a related conjecture](https://arxiv.org/pdf/1401.6989.pdf) by Nicolas Bergeron, Mehmet Haluk Sengun, Akshay Venkatesh, which claims analytic torsion grows exponentially with respect to the covering of 3-manifolds. I am not entirely sure whether the statement is true (I heard people found counter-examples and the conjecture has to be modified). I think the motivation of $\tau(S)$ may be from elsewhere (like Weyl's paper).


