Newman's Tauberian theorem, the Riemann-Lebesgue Lemma, and abstract analytic number theory

TL;DR

This paper develops an effective version of Newman's Tauberian theorem and the Riemann-Lebesgue Lemma, applying them to analytic number theory problems like the prime number theorem.

Contribution

It introduces a generalized, effective form of Bekehermes' improvement of Newman's Tauberian theorem and applies it to derive explicit results in number theory.

Findings

Effective version of Bekehermes' improvement of Newman's Tauberian theorem

Explicit error term for Mertens' function

Proof of a version of the prime number theorem

Abstract

We give a generalized and effective version of Bekehermes' improvement of Newman's Tauberian theorem. To do so we prove an effective version of the Riemann-Lebesgue Lemma for functions of bounded $p$-variation. We apply our Tauberian theorem to abstract analytic semigroups and prove a version of the prime number theorem as well as an estimate for Mertens' function with explicit error term.