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A still sharper region where $ \pi(x)-{\mathrm{li}}(x)$ is positive


Authors: Yannick Saouter, Timothy Trudgian and Patrick Demichel
Journal: Math. Comp. 84 (2015), 2433-2446
MSC (2010): Primary 11-04, 11A15, 11M26, 11Y11, 11Y35
DOI: https://doi.org/10.1090/S0025-5718-2015-02930-5
Published electronically: February 12, 2015
MathSciNet review: 3356033
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Abstract: We consider the least number $ x$ for which a change of sign of $ \pi (x)-\mathrm {li}(x)$ occurs. First, we consider modifications of Lehman's method that enable us to obtain better estimates of some error terms. Second, we establish a new smaller upper bound for the first $ x$ for which the difference is positive. Third, we use numerical computations to improve the final result.


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Additional Information

Yannick Saouter
Affiliation: Institut Telecom Brest, Department Informatique, CS 83818, 29238 Brest, Cedex 3 France
Email: Yannick.Saouter@enst-bretagne.fr

Timothy Trudgian
Affiliation: The Australian National University, Mathematical Sciences Institute, Building 27, ACTON, ACT 0200 Australia
Email: timothy.trudgian@anu.edu.au

Patrick Demichel
Affiliation: Hewlett-Packard France, 91947 Les Ulis, Cedex France
Email: patrick.demichel@hp.com

DOI: https://doi.org/10.1090/S0025-5718-2015-02930-5
Received by editor(s): June 11, 2013
Received by editor(s) in revised form: December 4, 2013
Published electronically: February 12, 2015
Additional Notes: The second author was supported in part by ARC Grant DE120100173.
Article copyright: © Copyright 2015 American Mathematical Society