## 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 $\psi (x)$.

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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