Entropy Calculation from Regularized Hamiltonian

Question

I refer to Srednicki's 'Entropy and Area' paper here.I want to derive the expression for entropy (20), given by 
$$S_{l}(n,N) = \xi_{l}(n)[-\log\xi_{l}(n) + 1], \xi_{l}(n) = \frac{n(n+1)(2n+1)^{2}}{64l^{2}(l+1)^{2}} + O(l^{-6}) $$ from the given quantum field theoretic Hamiltonian in equation (18).

I have managed to compute the entropy of $N$ coupled harmonic oscillators (Hamiltonian given by equation (7)) and I understand that I have to perform the calculation for this Hamiltonian analogously. I need to know how to go about this:

1. What does it mean to perform the calculation perturbatively?

2. Does $l >> N$ in equation (18) mean that we can ignore the middle term in the Hamiltonian (18)?

3. What are the steps I need to go through to calculate $S_{l}$, in an analogous fashion to the quantum mechanical entropy for $N$ coupled harmonic oscillators?

  [1]: https://arxiv.org/abs/hep-th/9303048