 Spectral zeta function

For a nonnegative elliptic operator $L$ on a closed manifold, sum its positive eigenvalues with multiplicity. The defining series converges in a right half-plane. If $n_0=\dim \ker L$ and $K(t)=\mathrm{Tr}(e^{-tL})$ is the heat trace, thenSubtracting finitely many terms of the short-time heat kernel expansion gives a meromorphic continuation. Zero eigenvalues must be excluded: assigning them a nonexistent inverse power does not define the spectral zeta function.