The zeta function of the Laplacian on certain fractals

Derfel, Gregory; Grabner, Peter; Vogl, Fritz

doi:10.1090/S0002-9947-07-04240-7

The zeta function of the Laplacian on certain fractals
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by Gregory Derfel, Peter J. Grabner and Fritz Vogl PDF

Trans. Amer. Math. Soc. 360 (2008), 881-897 Request permission

Abstract:

We prove that the zeta function $\zeta _\Delta$ of the Laplacian $\Delta$ on 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, thereby answering a question posed by J. Kigami and M. Lapidus for this class of fractals.

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