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Estimates of $ \psi,\theta$ for large values of $ x$ without the Riemann hypothesis


Author: Pierre Dusart
Journal: Math. Comp. 85 (2016), 875-888
MSC (2010): Primary 11N56; Secondary 11A25, 11N05
Published electronically: July 20, 2015
MathSciNet review: 3434886
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Abstract: The enlargement of known zero-free regions has enabled us to find better effective estimates for classical number-theoretic functions linked to the distribution of prime numbers. In particular we draw the quintessence of the method of Rosser and Schoenfeld on the upper bounds for the usual Chebyshev prime and prime power counting functions to find an upper bound function directly linked to a zero-free region.


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Additional Information

Pierre Dusart
Affiliation: XLIM - UMR CNRS $n^{∘}$7252, Université de Limoges, France
Address at time of publication: Département de Mathématiques, 123 avenue Albert THOMAS, 87060 Limoges Cedex, France
Email: pierre.dusart@unilim.fr

DOI: https://doi.org/10.1090/mcom/3005
Keywords: Number theory, arithmetic functions
Received by editor(s): March 17, 2014
Received by editor(s) in revised form: October 4, 2014
Published electronically: July 20, 2015
Article copyright: © Copyright 2015 American Mathematical Society