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Mathematics of Computation

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The impact of $ \zeta(s)$ complex zeros on $ \pi(x)$ for $ x<10^{10^{13}}$

Authors: Douglas A. Stoll and Patrick Demichel
Journal: Math. Comp. 80 (2011), 2381-2394
MSC (2010): Primary 11A41, 11M06, 11M26, 11N05
Published electronically: April 1, 2011
MathSciNet review: 2813366
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Abstract: An analysis of the local variations of the prime counting function $ \pi(x)$ due to the impact of the non-trivial, complex zeros $ \varrho_k$ of $ \zeta(s)$ is provided for $ x<10^{10^{13}}$ using up to 200 billion $ \zeta(s)$ complex zeros. A new bound for $ \vert\mathrm{li}(x)-\pi(x)\vert<x^{1/2}(\mathrm{log} \mathrm{log} \mathrm{log} x+e+1)/e \mathrm{log} x$ is proposed consistent with the error growth rate in Littlewood's proof that $ \mathrm{li}(x)-\pi(x)$ changes sign infinitely often. This bound is also consistent with all presently known cases where $ \pi(x)>\mathrm{li}(x)$ including many new examples listed. This implies that Littlewood's constant $ \mathrm{K}=1/e$, the lower bound for Skewes' number is $ 3.17\times 10^{114}$ and the positive constant $ c$ in the Riemann Hypothesis equivalent $ \vert\mathrm{li}(x)-\pi(x)\vert<c \mathrm{log}(x)x^{1/2}$ is less than $ 3\times 10^{-27}$.

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Additional Information

Douglas A. Stoll
Affiliation: Boeing Research and Technology, Seattle, Washington

Patrick Demichel
Affiliation: Hewlett-Packard, Les Ulis, France

Keywords: Skewes’ number, zeta zero distribution, Riemann Hypothesis
Received by editor(s): November 4, 2009
Received by editor(s) in revised form: August 6, 2010
Published electronically: April 1, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.