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?


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