1 Correction to: Lett Math Phys https://doi.org/10.1007/s11005-017-1027-y
We provide a correction of some formulas. In Proposition 3.1, there is a factor of \(\frac{1}{2}\) missing from the spectral zeta function of the diamond fractal. The correct expression is
The \(\frac{1}{2}\) term appears at the following equation found in the proof
due to the double multiplicity in the spectral decimation polynomial.
Moreover, the following formulas we used from [1] have a minor typo in that they miss the minus signs. The proper formulation is given by
and for \(w=0\)
This updates our results by replacing all calculations of \(\zeta _{\mathcal {L}}'(0)\) with \(-\zeta _{\mathcal {L}}'(0)\). Specifically,
-
(1)
In Proposition 3.1, we have that \(\det \mathcal {L}=2^{\frac{5}{9}}\) and in the comment below the proof we have that \(\log {\det \mathcal {L}}=\frac{5}{9}c\) and
$$\begin{aligned} \log {\det \mathcal {L}_n}=\frac{1}{5}(-2\cdot 4^n +6n+11) \log {\det \mathcal {L}} \end{aligned}$$ -
(2)
In Proposition 4.1, we have that \(\det \mathcal {L}=\frac{2N^{\frac{1}{N-1}}}{(N+2)^{\frac{N-2}{N-1}}}\) and in Corollary 4.1 that
$$\begin{aligned} \log {\det \Delta _n}=c|V_n|+n\log {(N+2)}+ \log {\det \mathcal {L}} \end{aligned}$$ -
(3)
In Proposition 5.2, we have that \(\det \mathcal {L}_{\mu }=\frac{1}{pq}\) and in Corollary 5.1 that
$$\begin{aligned} \log {\det \Delta _n}=|V_n|\left( \log {2}+\frac{\log {(pq)}}{2}\right) +n\log {\frac{(1-q^2)(1-p^2)}{(pq)^2}}+\log {\det \mathcal {L}} \end{aligned}$$
Reference
Derfel, G., Grabner, P., Vogl, F.: The zeta function of the Laplacian on certain fractals. Trans. Am. Math. Soc. 360(2), 881–897 (2008)
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Chen, J.P., Teplyaev, A. & Tsougkas, K. Correction to: Regularized Laplacian determinants of self-similar fractals. Lett Math Phys 108, 1581–1582 (2018). https://doi.org/10.1007/s11005-018-1081-0
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DOI: https://doi.org/10.1007/s11005-018-1081-0