For a compact Riemannian manifold M of dimension N with eigenvalues of the Laplace–Beltrami operator, the zeta function is given for sufficiently large by . 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 are normalized eigenfunctions. This can be analytically continued to a meromorphic function of s for all complex s, and is holomorphic for. The only possible poles are simple poles at the points for N odd, and at the points for N even. If N is odd then vanishes at. 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 indicates that the quotient of both the sides tend to 1 when T tends to. The function can be recovered from 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 n2 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 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 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 is scalar curvature, the trace of the Ricci curvature, on M. 2) Weyl Asymptotic Formula Let M be a compact Riemannian manifold, with eigenvalues with each distinct eigenvalue repeated with its multiplicity. Define N to be the number of eigenvalues less than or equal to, and let denote the volume of the unit disk in. Then as. Additionally, as, This is also called Weyl's law, refined from the Minakshisundaram–Pleijel asymptotic expansion.
