Let us define the zeta function of an elliptic differential operator $H$ with eigenvalues $\lambda_n$ like so: \begin{aligned} \zeta_H(s) := tr( H^{-s} ) \\ := \sum^\infty_{n=1} \frac{1}{(\lambda_n)^s} \end{aligned} Then, for a compact Riemannian manifold $M$ with Laplace-Beltrami operator $\Delta$, the Minakshisundaram–Pleijel zeta function is given by $\zeta_\Delta(s)$.

The Selberg Zeta Function of a Riemannian surface has a lot more setup, although a definition can be found in Definition 4.1 here.

As I study the Minakshisundaram–Pleijel zeta function, I see that it has a lot in common with the Selberg Zeta Function, in terms of analogous objects in algebraic geometry (both have intimate connections to Artin L-functions). My question - is there anything connecting one of these functions to another? I've seen mentioned that the Selberg Zeta Function can be considered a zeta function of a twisted Dirac operator, which would make both special cases of the zeta function defined above, but that's not all that direct of a connection.

Are there any rigorous connections between these two functions?