Not surprisingly it contains $Z_{\rm Ein}$ because all solutions to Einstein gravity are also solutions to NMG. The second factor in (\[ZcNMG\]) comes from the massive graviton. Here we already took the limit to critical NMG, thus, as a consequence of the degeneration of the equations of motion, this term coincides with the spin-two contribution of Einstein gravity.

Straightforward application of formula (\[zdelta\]) yields $$\label{?ZNMG?}\hskip 1cm
 Z_{\rm cNMG} =Z_{\rm Ein}\cdot\prod_{m\geq0}^\infty|1-q^{m+2}|^{-2-2m} 
                =Z_{\rm Ein}\cdot\prod_{m\geq2}^\infty\prod_{n\geq0}^\infty\frac{1}{|1-q^{m+n}|^2} \,,$$ which, for $q=\bar{q}$,[^4] perfectly agrees with the result in [@Gaberdiel:2010xv]. On the one hand this is a confirmation of the conjecture of [@Denef:2009kn]. On the other hand, since (\[?ZNMG?\]) is equal to the result that was used to support the LCFT conjecture, it tells us, that, to obtain (\[?ZNMG?\]), we already summed over the log quasi-normal frequencies. They are equal to the frequencies of the Einstein quasi-normal modes [@Sachs:2008gt; @Sachs:2008yi] and correspond to the poles of the retarded correlators in the dual CFT [@Birmingham:2001pj]. In the log case these are double poles [@Sachs:2008yi], so we must sum over them twice. Taking a look at (\[ZcNMG\]) we see that we actually did count the spin-two frequencies twice simply because the (square root of the) determinant $\det(-\nabla^2-2)_2^{TT}$ appears twice.

