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Mathematics of Computation

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Estimating $\pi (x)$ and related functions under partial RH assumptions


Author: Jan Büthe
Journal: Math. Comp. 85 (2016), 2483-2498
MSC (2010): Primary 11N05; Secondary 11M26
DOI: https://doi.org/10.1090/mcom/3060
Published electronically: December 1, 2015
MathSciNet review: 3511289
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Abstract:

We give a direct interpretation of the validity of the Riemann hypothesis for all zeros with $\Im (\rho )\in (0,T]$ in terms of the prime-counting function $\pi (x)$ by proving that Schoenfeld’s explicit estimates for $\pi (x)$ and the Chebyshov functions hold as long as $4.92\sqrt {x/\log (x)} \leq T$.

We also improve some of the existing bounds of Chebyshov type for the function $\psi (x)$.


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Additional Information

Jan Büthe
Affiliation: Mathematisches Institut, Endenicher Allee 60, 53115 Bonn, Germany
MR Author ID: 1017601
Email: jbuethe@math.uni-bonn.de

Received by editor(s): November 13, 2014
Received by editor(s) in revised form: March 11, 2015, and March 15, 2015
Published electronically: December 1, 2015
Article copyright: © Copyright 2015 American Mathematical Society