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

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.