A generalized effective Newman's Tauberian theorem is established via an effective Riemann-Lebesgue lemma for p-variation functions and applied to abstract analytic semigroups to yield explicit-error prime number theorem and Mertens function results.
Diamond, Chebyshev estimates for Beurling generalized prime numbers,Proceedings of the AMS39(1973), 503–508
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Newman's Tauberian theorem, the Riemann-Lebesgue Lemma, and abstract analytic number theory
A generalized effective Newman's Tauberian theorem is established via an effective Riemann-Lebesgue lemma for p-variation functions and applied to abstract analytic semigroups to yield explicit-error prime number theorem and Mertens function results.