The Zeta Function of the Laplacian on Certain Fractals

We prove that the zeta-function ${\zeta }_{\mathrm{\Delta }}$ of the Laplacian $\mathrm{\Delta }$ 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.