# Questions tagged [tauberian-theorems]

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### Karamata's Abelian/Tauberian Theorem in the complex plane

The following result is well known (a particular case of Karamata's Tauberian Theorem for Power Series in Corollary 1.7.3 of Regular variation by Bingham, Goldie & Teugels): Fix $c, \rho>0$. If ...
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### Tauberian lower bound for a series

Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of positive number such that $\sum_n a_n < +\infty$ (i.e. $a_n \in \ell^1$) but $\sum_n r^n a_n = +\infty$ for every $r > 1$. Given $\sigma \in (0,1)$, ...
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### Convergence speed of the tail of distribution using Tauberian remainder theorem

This question may be related to this one. Now I try to make some statistical estimator using Laplace transform, but I face the following serious problem. Let $f$ be some one-sided probability ...
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### Modern treatment of Delange's Tauberian Theorem

Tauberian theorems abound in the literature. One of the most general, powerful, and versatile is due to Delange, and appears as Theorem I of the paper: H. Delange - Généralisation du théorème de ...
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### Reference request: Extensions of Wiener's Tauberian Theorem

Wiener's Tauberian Theorem says that linear combinations of translations of a function $f$ are dense in $L^1(\mathbb{R})$ if and only if the zero set of the Fourier transform of $f$ is empty. This is ...
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### Karamata's proof of Hardy-Littlewood Tauberian theorem

I understand Karamata's proof of the Hardy-Littlewood Tauberian theorem as in http://individual.utoronto.ca/jordanbell/notes/karamata.pdf, but what on earth is the motivation behind Lemma 4 - i.e, ...
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### A question concerning Tauberian theory

Let $D(s) = \sum_{n=1}^\infty a_n n^{-s}$ be a Dirichlet series with $a_n ≥ 0$ and abscissa of convergence $\sigma_a = 1$. Further, we assume that $D(s)$ is holomorphic in each point $\Re(s) = 1$ ...
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### A Generalized Wiener-Ikehara Theorem with multiple poles on the line

One version of the Wiener-Ikehara Theorem says that if $$f(s) = \sum \frac{a(n)}{n^s}$$ is a Dirichlet series with nonnegative coefficients that converges absolutely for $\text{Re}(s) > 1$ and ...
This is a fairly vague question. Suppose we have a sequence of positive numbers $(c_n)_n$ and we want to find an asymptotic formula for $S(x) = \sum_{n \leq X} c_n$. In favorable circumstances, ...