The Zeta Function of the Laplacian on Certain Fractals

Abstract

We prove that the zeta-function ${\zeta }_{\Delta }$ of the Laplacian $\Delta$ on a self-similar fractals with spectral decimation admits a meromorphic continuation to the whole complex plane. We characterise the poles, compute their residues, and give expressions for some special values of the zeta-function. Furthermore, we discuss the presence of oscillations in the eigenvalue counting function.

Added an unconditional proof for the presence of non-real poles of the zeta-function for the class of fractals under consideration