\[sec2\] would not change if we would take the background to be the BTZ black hole, instead of AdS; mainly because the BTZ is obtained by identifications of AdS [@Banados:1992gq]. Moreover, the thermal AdS background used above is also the background of the Euclidean BTZ black hole, with the identification $\tau\to-1/\tau$ [@Giombi:2008vd]. Therefore, we shall blithely permit ourselves to go from one background to the other, as its suits our analysis. In this section our background will be the BTZ black hole.

Quasi-normal modes for excitations around the BTZ black hole were calculated in [@Birmingham:2001pj]. In the context of critical gravity this was first done for log-TMG in [@Sachs:2008gt; @Sachs:2008yi] and a tricritical theory was discussed recently in [@Kim:2012rz].

In the following we will calculate the one-loop partition function of two critical gravity theories in three dimensions, NMG and PET gravity, using the conjecture of [@Denef:2009kn]. We confirm that at the (tri-)critical point the results agree with earlier calculations using heat kernel techniques. Based on that confidence we calculate different truncated one-loop partition functions by subsequently summing over different quasi-normal mode spectra, allowing log-excitations as well as not allowing them.

We will start by recalling the partition function of Einstein 