Correction to: Regularized Laplacian determinants of self-similar fractals

  • Joe P. Chen
  • Alexander Teplyaev
  • Konstantinos Tsougkas
Correction
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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)MathSciNetCrossRefMATHGoogle Scholar

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

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