The Zeta Function of the Laplacian on Certain Fractals

We prove that the zeta-function $ζ_Δ$ of the Laplacian $Δ$ 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.

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