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The Ramanujan Journal

, Volume 28, Issue 3, pp 435–442 | Cite as

On an inequality of Ramanujan concerning the prime counting function

  • Mehdi Hassani
Article

Abstract

We discuss on the sign of \(\mathcal{R}_{\alpha }(x):=\pi(x)^{2}-\frac{\alpha x}{\log x}\pi(\frac{x}{\alpha })\) for x sufficiently large, and for various values of α>0. The case α=e refers to a result due to Ramanujan asserting that \(\mathcal{R}_{e}(x)<0\). Related by this inequality, we obtain a conditional result that gives the number N>530.2 such that \(\mathcal{R}_{e}(x)<0\) is valid for x>e N . Moreover, we show that under assumption of validity of the Riemann hypothesis, the inequality \(\mathcal{R}_{e}(x)<0\) holds for x>138,766,146,692,471,228. Then, in various cases for α, we find numerical values of x α in which \(\mathcal{R}_{\alpha }(x)\) is strictly positive or negative for xx α .

Keywords

Prime numbers Inequalities 

Mathematics Subject Classification (2000)

11A41 26D20 

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References

  1. 1.
    Berndt, B.C.: Ramanujan’s Notebooks (Part IV). Springer, Berlin (1994) zbMATHGoogle Scholar
  2. 2.
    Dusart, P.: Inégalités explicites pour ψ(X), θ(X), π(X) et les nombres premiers. C. R. Math. Acad. Sci. Soc. R. Can. 21(2), 53–59 (1999) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Barkley Rosser, J., Schoenfeld, L.: Sharper bounds for the Chebyshev functions θ(x) and ψ(x). Math. Comput. 29(129), 243–269 (1975) zbMATHGoogle Scholar
  4. 4.
    Schoenfeld, L.: Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II. Math. Comput. 30(134), 337–360 (1976) MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ZanjanZanjanIran

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