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Let $\zeta(M,s)$ be the Minakshisundaram-Pleijel zeta function, which encodes the eigenvalues of the Laplace-Beltrami operator. Where can I find a proof or reference of the following identity? If $M$ is a surface: $$\zeta'(\Delta, 0) = \frac{1}{12}\int_M K dA$$

Where $K $ is the Gaussian Curvature.

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    $\begingroup$ Isn't this just follow by differentiating the Mellin transform and using heat kernel asymptotics? $\endgroup$
    – Bombyx mori
    Commented Apr 19, 2018 at 17:14
  • $\begingroup$ @Bombyxmori Could you expand on that a little and post it as an answer? $\endgroup$
    – Nico A
    Commented Apr 19, 2018 at 17:18
  • $\begingroup$ I thought about it, the approach does give the right number, but it is not rigorous and I think a better approach is needed. I am not sure if this is address in their original paper. $\endgroup$
    – Bombyx mori
    Commented Apr 20, 2018 at 21:58

2 Answers 2

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Look for "A Panoramic View of Riemannian Geometry" by Marcel Berger: you will find what you are looking for in sub-subchapter 9.7.2 "Great Hopes" (pages 421-422). Taking a look at 1.8.5 "Second Way: the Heat Equation" (page 100) where things are done for surfaces with boundary embedded in Euclidean spaces will improve your understanding. Berger's text does not give all the details, but is full of further references that will lead you further.

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If you read French, it seems Marcel Berger adresses this question in his review article in Development of Mathematics 1950-2000, Birkhaüser.

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