Timeline for A question concerning Tauberian theory

8 events

when toggle format	what		by	license	comment
Dec 4, 2017 at 20:43	vote	accept	Johann Franke
Nov 20, 2017 at 16:51	answer	added	js21		timeline score: 5
Nov 20, 2017 at 14:31	comment	added	Johann Franke		This is interesting. In fact I know that the residues of all possible poles are bounded by a constant $K$ only dependent on the $\epsilon_0$ above (I also know that all poles of $D(s)$ have at most order $k$) such that $\Gamma(s)D(s)$ is kind of fast decreasing. Also I assume that there should be a pole free region which approaches the line $\Re(s) = 1$ when $t \to \infty$, but I haven't shown this yet. Does the inverse Mellin transform suffice to show the question?
Nov 20, 2017 at 14:16	comment	added	Johann Franke		Thank you very much so far. I am familiar with Delange's theorem but unfortunately the leading term does not suffice in my case.
Nov 19, 2017 at 23:22	comment	added	reuns		The $o(x)$ term depends on the density of poles and their residues. The first question is if $\Gamma(s) D(s)$ is fast decreasing so that $\sum_{n=1}^\infty a_n e^{-nx} = \sum_\beta x^{-\beta} P_\beta(\log x) +o(x^{-1+\epsilon})$ as $x \to 0$
Nov 19, 2017 at 15:58	comment	added	Peter Humphries		Delange's generalisation of Ikehara's Tauberian theorem gives you the leading order term, at the very least.
Nov 19, 2017 at 15:24	history	edited	Johann Franke	CC BY-SA 3.0	deleted 31 characters in body
Nov 19, 2017 at 15:11	history	asked	Johann Franke	CC BY-SA 3.0