# Tauberian theorem converse (Wiener-Ikehara)

Jacob Korevaar provides a nice converse to to Wiener-Ikehara tauberian theorem on p. 125 of his Tauberian Theory book: For the non-decreasing, locally of bounded variation function $$s$$, if we have $$\dfrac{s(v)}{v} \to A$$ then the analytic function $$f(z)=\int_1^{\infty}v^{-z}ds(v)$$ on $$x>1$$ has pole type behavior at $$z=1$$ meaning that in some region it behaves like $$\dfrac{A}{z-1}$$. My question is what can we say about the situation that $$0<\liminf\dfrac{s(v)}{v}<\limsup\dfrac{s(v)}{v}<\infty$$ instead? I know that it is possible to get singularity at other part of the line $$x=1$$ in this case, but can we say we have to have pole type behavior?