This is a theorem of Flajolet and Odlyzko (I think): Let $f(z)$ be a function analytic in a domain $D = \{z : |z| \leq s_1, |\text{Arg}(z-s)| > \frac{\pi}{2} - \eta \},$ where $s, s_1 > s,$ and $\eta$ are three positive real numbers. Assume that, with $ \sigma(u) = u^\alpha \log^\beta u$ and $\alpha \notin \{0, -1, -2, \dots \}$, we have $ f(z) \sim \sigma \left( \frac{1}{1-z/s} \right) \qquad \text{ as } z \rightarrow s \text{ in } D.$ Then, the Taylor coefficients of $f(z)$ satisfy $[z^n]f(z) \sim s^{-n} \frac{\sigma(n)}{n \Gamma(\alpha)}.$
My question is whether, with the same premise, except that $\alpha = 0, \beta \in \{1, 2, 3, \dots\}$, we have $[z^n]f(z) \sim \beta s^{-n} n^{-1} \log^{\beta-1} n.$