Boundary condition dependence of spectral zeta functions.

In this work, we provide the analytic continuation of the spectral zeta function associated with the one-dimensional regular Sturm-Liouville problem and the two-dimensional Laplacian on the annulus. In the one-dimensional setting, we consider general separated and coupled boundary conditions, and on the annulus we restrict our work to Dirichlet-Robin boundary conditions. In both cases, we use our results to calculate the coefficients of the asymptotic expansion of the associated heat kernel. In the one-dimensional case, we additionally use the analytically continued spectral zeta function to compute the determinant of the Sturm-Liouville operator.