The Ramanujan Journal

, Volume 45, Issue 1, pp 227–251 | Cite as

Explicit estimates of some functions over primes

  • Pierre Dusart


New results have been found about the Riemann hypothesis. In particular, we noticed an extension of zero-free region and a more accurate location of zeros in the critical strip. The Riemann hypothesis implies results about the distribution of prime numbers. We get better effective estimates of common number theoretical functions which are closely linked to \(\zeta \) zeros like \(\psi (x),\vartheta (x),\pi (x)\), or the \(k\mathrm{{th}}\) prime number \(p_k\).


Number theory Arithmetic functions Chebyshev’s functions Estimates of prime numbers 

Mathematics Subject Classification

Primary 11N56 Secondary 11A25 11N05 



The author wishes to thank the anonymous reviewers for their helpful comments.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.XLIM - UMR CNRS 7252, Faculté des Sciences et TechniquesUniversité de LimogesLimogesFrance
  2. 2.Département de MathématiquesLimogesFrance

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