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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Estimating $\pi (x)$ and related functions under partial RH assumptions
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by Jan Büthe PDF
Math. Comp. 85 (2016), 2483-2498 Request permission

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 $\psi (x)$.

References
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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
  • © Copyright 2015 American Mathematical Society
  • Journal: Math. Comp. 85 (2016), 2483-2498
  • MSC (2010): Primary 11N05; Secondary 11M26
  • DOI: https://doi.org/10.1090/mcom/3060
  • MathSciNet review: 3511289