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New bounds for $ \psi(x)$


Authors: Laura Faber and Habiba Kadiri
Journal: Math. Comp. 84 (2015), 1339-1357
MSC (2010): Primary 11M06, 11M26
DOI: https://doi.org/10.1090/S0025-5718-2014-02886-X
Published electronically: October 21, 2014
Corrigendum: Math. Comp. 87 (2018), 1451-1455.
MathSciNet review: 3315511
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Abstract: In this article we provide new explicit Chebyshev bounds for the prime counting function $ \psi (x)$. The proof relies on two new arguments: smoothing the prime counting function which allows one to generalize the previous approaches, and a new explicit zero density estimate for the zeros of the Riemann zeta function.


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Additional Information

Laura Faber
Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta, T1K 3M4 Canada
Email: laura.faber2@uleth.ca

Habiba Kadiri
Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta, T1K 3M4 Canada
Email: habiba.kadiri@uleth.ca

DOI: https://doi.org/10.1090/S0025-5718-2014-02886-X
Keywords: Prime number theorem, $\psi(x)$, explicit formula, zeros of Riemann zeta function
Received by editor(s): July 18, 2013
Received by editor(s) in revised form: September 9, 2013
Published electronically: October 21, 2014
Additional Notes: The first author was funded by a Chinook Research Award.
The second author was funded by ULRF Fund 13222.
The authors’ calculations were done on the University of Lethbridge Number Theory Group Eudoxus machine, supported by an NSERC RTI grant.
Article copyright: © Copyright 2014 American Mathematical Society

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