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Mathematics of Computation

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An analytic method for bounding $ \psi(x)$

Author: Jan Büthe
Journal: Math. Comp. 87 (2018), 1991-2009
MSC (2010): Primary 11N05; Secondary 11M26
Published electronically: October 26, 2017
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Abstract: In this paper we present an analytic algorithm which calculates almost sharp bounds for the normalized remainder term $ (t-\psi (t))/\sqrt t$ for $ t\leq x$ in expected run time $ O(x^{1/2+\varepsilon })$ for every $ \varepsilon >0$. The method has been implemented and used to calculate such bounds for $ t\leq 10^{19}$. In particular, these imply that $ li(x)-\pi (x)$ is positive for $ 2\leq x\leq 10^{19}$.

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Additional Information

Jan Büthe
Affiliation: Hausdorff Center for Mathematics, Endenicher Allee 62, 53115 Bonn, Germany

Received by editor(s): November 6, 2015
Received by editor(s) in revised form: August 21, 2016, and January 29, 2017
Published electronically: October 26, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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