All Questions

Let $(a_n)_{n \ge 1}$ be a sequence of complex numbers, and suppose that the sequence has a well-defined "average", in the sense that $$ \lim_{N \to \infty} \frac{1}{N}\sum_{i = 1}^N a_i = R$...

I encountered this statement about Dirichlet series but couldn't find a similar result in Korevaar's "Tauberian Theory". Is this statement valid? Statement: Let $f(s) = \sum_{n=1}^{\infty} \...

I'm wondering about a possible extension of Ikehara's theorem for Dirichlet series with coefficients that are not necessarily positive. Consider the following: Let $D(s) = \sum_{n \geq 1} \frac{a(n)}{...

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 ...

I am trying to prove or disprove $$\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} ,$$ where $\sum c_{k}<\infty, \sum c_{k}^{2}<\infty\text{ and }\frac{\...