# Correction to: Regularized Laplacian determinants of self-similar fractals

• Joe P. Chen
• Alexander Teplyaev
• Konstantinos Tsougkas
Correction
• 79 Downloads

## 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
\begin{aligned} \zeta _{\mathcal {L}}(s)=\frac{4^s(4^s-1)}{6}\left( \frac{4}{4^s-4}+\frac{2}{4^s-1}\right) \zeta _{\Phi ,0}(s). \end{aligned}
The $$\frac{1}{2}$$ term appears at the following equation found in the proof
\begin{aligned} \zeta _{\Phi ,-1}(s)= & {} \sum \limits _{\begin{array}{c} \Phi (-\mu )=-1 \\ \mu>0 \end{array}} \mu ^{-s}=\frac{1}{2}\sum \limits _{\begin{array}{c} \Phi (-4\mu )=-2 \\ \mu>0 \end{array}} \mu ^{-s}=\frac{1}{2} 4^s \sum \limits _{\begin{array}{c} \Phi (-4\mu )=-2 \\ \mu >0 \end{array}} (4\mu )^{-s}\\= & {} \frac{1}{2} 4^s \zeta _{\Phi ,-2}(s) \end{aligned}
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
\begin{aligned} \zeta _{\Phi ,w}(0)=0 \quad \text { and } \quad \zeta '_{\Phi ,w}(0)=-\frac{\log {a_d}}{d-1}-\log {(-w)} \end{aligned}
and for $$w=0$$
\begin{aligned} \zeta _{\Phi ,0}(0)=-1 \quad \text { and } \quad \zeta '_{\Phi ,0}(0)=-\frac{\log {a_d}}{d-1}. \end{aligned}
This updates our results by replacing all calculations of $$\zeta _{\mathcal {L}}'(0)$$ with $$-\zeta _{\mathcal {L}}'(0)$$. Specifically,
1. (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. (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. (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

1. 1.
Derfel, G., Grabner, P., Vogl, F.: The zeta function of the Laplacian on certain fractals. Trans. Am. Math. Soc. 360(2), 881–897 (2008)

## Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

## Authors and Affiliations

• Joe P. Chen
• 1
• Alexander Teplyaev
• 2
• Konstantinos Tsougkas
• 3
1. 1.Department of MathematicsColgate UniverstyHamiltonUSA
2. 2.Department of MathematicsUniversity of ConnecticutStorrsUSA
3. 3.Department of MathematicsUppsala UniversityUppsalaSweden