A quantified Tauberian theorem for Laplace-Stieltjes transform

TL;DR

This paper establishes a quantified Tauberian theorem for Laplace-Stieltjes transforms, extending previous results and applicable to special Dirichlet series, with implications for functions of bounded variation.

Contribution

It introduces a generalized quantified Tauberian theorem involving Laplace-Stieltjes transforms for functions of bounded variation, extending prior work by Batty and Duyckaerts.

Findings

Theorem applies to functions of bounded variation.

Generalizes results of Batty and Duyckaerts.

Applicable to special Dirichlet series.

Abstract

We prove a quantified Tauberian theorem involving Laplace-Stieltjes transform which is motivated by the work of Ingham and Karamata. For this, we consider functions which are locally of bounded variation and, therefore, get a generalisation of some results of Batty and Duyckaerts. We show that our theorem can be applied to special Dirichlet series.