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I think I would go for a Euler-Maclaurin expansion. of the second term of
$$ -\sum_{k=a}^N \log(e^{k\kappa}-1) = -\kappa\frac{(N+a)(N-a+1)}{2}-\sum_{k=a}^N\log(1-e^{-k\kappa}). $$
I think I would go for a Euler-Maclaurin expansion.
I think I would go for a Euler-Maclaurin expansion of the second term of
