Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11N05, 11M26

Retrieve articles in all journals with MSC (2010): 11N05, 11M26


Additional Information

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

DOI: https://doi.org/10.1090/mcom/3060
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

American Mathematical Society