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 . The case of a compact region of the plane was treated earlier by . Definition. For a compact Riemannian manifold "M" of dimension "N" with eigenvalues formula_1 of the Laplace–Beltrami operator formula_2, the zeta function is given for formula_3 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 formula_6 are normalized eigenfunctions. This can be analytically continued to a meromorphic function of "s" for all complex "s", and is holomorphic for formula_7. The only possible poles are simple poles at the points formula_8 for "N" odd, and at the points formula_9 for "N" even. If "N" is odd then formula_10 vanishes at formula_11. If "N" is even, the residues at the poles can be explicitly found in terms of the metric, and by the Wiener–Ikehara theorem we find  as a corollary the relation where the symbol formula_13 indicates that the quotient of both the sides tend to 1 when T tends to formula_14. The function formula_15 can be recovered from formula_10 by integrating over the whole manifold "M": Heat kernel. The analytic continuation of the zeta function can be found by expressing it in terms of the heat kernel as the Mellin transform In particular, we have where is the trace of the heat kernel. The poles of the zeta function can be found from the asymptotic behavior of the heat kernel as "t"→0. Example. If the manifold  is a circle of dimension "N"=1, then the eigenvalues of the Laplacian are "n" for integers "n". The zeta function where ζ is the Riemann zeta function. Applications. Apply the method of heat kernel to asymptotic expansion for Riemannian manifold (M,g) we obtain the two following theorems. Both are the resolutions of the inverse problem in which we get the geometric properties or quantities from spectra of the operators. 1) Minakshisundaram–Pleijel Asymptotic Expansion Let (M,g) be an "n"-dimensional Riemannian manifold. Then, as "t"→0+, the trace of the heat kernel has an asymptotic expansion of the form : In dim=2, this means that the integral of scalar curvature tells us the Euler characteristic of M, by the Gauss–Bonnet theorem. In particular, where S(x) is scalar curvature, the trace of the Ricci curvature, on M. 2) Weyl Asymptotic Formula Let M be a compact Riemannian manifold, with eigenvalues formula_25 with each distinct eigenvalue repeated with its multiplicity. Define N(λ) to be the number of eigenvalues less than or equal to formula_26, and let formula_27 denote the volume of the unit disk in formula_28. Then as formula_30. Additionally, as formula_31, This is also called Weyl's law, refined from  the Minakshisundaram–Pleijel asymptotic expansion.