Note on the absence of remainders in the Wiener-Ikehara theorem

On the absence of remainders in the Wiener-Ikehara and Ingham-Karamata theorems: a constructive approach

We construct explicit counterexamples that show that it is impossible to get any remainder, other than the classical ones, in the Wiener-Ikehara theorem and the Ingham-Karamata theorem under just an additional analytic continuation hypothesis to a half-plane (or even to the whole complex plane).

Generalization of the Wiener-Ikehara theorem

We study the Wiener-Ikehara theorem under the so-called log-linearly slowly decreasing condition. Moreover, we clarify the connection between two different hypotheses on the Laplace transform occurring in exact forms of the Wiener-Ikehara theorem, that is, in "if and only if" versions of this theorem.

Koga版本的Wiener-Ikehara定理的推广

Generalizations of Koga's version of the Wiener-Ikehara theorem

该研究推广改进了Koga的Wiener-Ikehara定理，建立了仅需拉普拉斯变换实部边界假设的新形式，并将其用于快速证明Blackwell更新定理。

On Ikehara type Tauberian theorems with $O(x^γ)$ remainders

Motivated by analytic number theory, we explore remainder versions of Ikehara's Tauberian theorem yielding power law remainder terms. More precisely, for $f:[1,\infty)\rightarrow{\mathbb R}$ non-negative and non-decreasing we prove $f(x)-x=O(x^γ)$ with $γ<1$ under certain assumptions on $f$. We state a conjecture concerning the weakest natural assumptions and show that we cannot hope for more.

An operator theoretic approach to the Prime number theorem

In this short note, we establish an operator theoretic version of the Wiener-Ikehara tauberian theorem, and point out how this leads to a new proof of the Prime number theorem that should be accessible to anyone with a basic knowledge of operator theory.