5
$\begingroup$

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?

Improve this question
$\endgroup$
7
  • 1
    $\begingroup$ You need $\zeta_\Delta$ to define the regularized determinant. The latter can be used to express the Selberg zeta function as you see [projecteuclid.org/euclid.cmp/1104159171]. $\endgroup$
    – user1688
    Commented May 8, 2018 at 7:39
  • $\begingroup$ @Corbennick That's an incredibly helpful paper! I can't tell what they mean by $\Delta_n$ (as in corollary 1) - is that simply the Laplacian? If not, can that corollary be extended to be the determinant of the laplacian? $\endgroup$
    – Nico A
    Commented May 8, 2018 at 14:49
  • $\begingroup$ That's the spinor Laplacian. For n=0 it's the usual, I guess. $\endgroup$
    – user1688
    Commented May 9, 2018 at 18:50
  • $\begingroup$ @Corbennick If you expand that into an answer, I’ll check it! Thank you! $\endgroup$
    – Nico A
    Commented May 9, 2018 at 20:50
  • $\begingroup$ @TreFox: What "intimate connections" do the Minakshisundaram-Pleijel and Selberg $\zeta$ functions have with Artin L-functions? The former two are deeply Riemannian in nature, while the latter are deeply arithmetic (and have significantly more nice properties). $\endgroup$
    – Alex M.
    Commented May 13, 2018 at 16:05

0

Reset to default

You must log in to answer this question.