Minakshisundaram–Pleijel zeta function

The Minakshisundaram–Pleijel zeta function is a zeta function encoding the eigenvalues of the Laplacian of a compact Riemannian manifold. It was introduced by Subbaramiah Minakshisundaram and Åke Pleijel (1949). The case of a compact region of the plane was treated earlier by Carleman (1935).

For a compact Riemannian manifold M of dimension N with eigenvalues ${\displaystyle \lambda _{1},\lambda _{2},\ldots }$ of the Laplace–Beltrami operator Δ the zeta function is given for ${\displaystyle \operatorname {Re} (s)}$ sufficiently large by

(where if an eigenvalue is zero it is omitted in the sum). The manifold may have a boundary, in which case one has to prescribe suitable boundary conditions, such as Dirichlet or Neumann boundary conditions.

More generally one can define

for P and Q on the manifold, where the f_n are normalized eigenfunctions. This can be analytically continued to a meromorphic function of s for all complex s, and is holomorphic for P≠Q.

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