I'm trying to apply the Tauberian theorems for Laplace transforms:
\begin{align} f(t)&\simeq t^{\rho-1}L(t)\\ &\Updownarrow\\ f(s)&\simeq \Gamma(\rho)s^{-\rho}L(1/s) \end{align}
where $0<\rho<\infty$ and $\Gamma(\rho)$ is the Gamma function.
This theorem should be applied to the waiting time PDF with the asymptotic behavior:
\begin{align} \psi(t)\sim\frac{1}{t\ln^\beta(t)} \end{align} where $\beta>1$.
That is I want to calculate $\psi(s)$. But when I try to apply the theorem, I get $\rho=0$, which is not allowed. I don't know if I can make a trick to avoid this?