The relation between π(x) and certain arithmetic functions

Magdalena Corciovei-Bănescu
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 17, 2011, Number 3, Pages 1—9
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Authors and affiliations

Magdalena Corciovei-Bănescu
Faculty of Mathematics and Informatics, Constanța
Bd. Mamaia 124, 8700 Constanța, Romania
National College “George Cosbuc”, Bucharest, Romania
Str. Olari 29-31 Sect.2 Bucharest

Abstract

We prove an improvement Rosser–Schoenfeld inequalities, more precisely:
\frac{ x^{k+1}}{(k+1) \log x +0,1 k^2 +0,1 k -0,99}< A_{k}(x)< \frac{ x^{k+1}}{(k+1) \log x -0,1 k^2 -0,2 k -1,11},
where Ak(x) = Σp ≤ x pk and k ≥ 0.

Keywords

  • Arithmetic functions
  • Inequalities

AMS Classification

  • 11N05
  • 11N64
  • 11Y60

References

  1. Apostol, T. Introduction to Analytic Number Theory. Springer-Verlag, 1976.
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  3. Dusart, P. Sharper bounds for φ, θ, π, pn. Rapport de recherche 1998 – 2006, Universite de Limoges.
  4. Massias, J.-P., G. Robin. Bornes effectives pour certaines fonctions concernant les nombres premiers. Journal de Theorie des Nombres de Bordeaux, Tome 8, 1996, No. 1, 215-242.
  5. Rosser, J., L. Schoenfeld. Approximate formulas for some functions of prime numbers. Illinois, Volume 6, 1962, Issue 1, 64–94.
  6. Rosser, J., L. Schoenfeld. Sharper bounds for the Chebyshev function θ and φ. Mathematics of computation, Vol. 29, January 1975, No. 129, 243–269.
  7. Schoenfeld, L. Sharper bounds for the Chebyshev function θ and φ. Mathematics of computation, Vol. 30, January 1976, 337–360.

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Cite this paper

APA

Corciovei-Bănescu, M. (2011). The relation between π(x) and certain arithmetic functions. Notes on Number Theory and Discrete Mathematics, 17(3), 1-9.

Chicago

Corciovei-Bănescu, Magdalena. “The Relation between π(x) and Certain Arithmetic Functions.” Notes on Number Theory and Discrete Mathematics 17, no. 3 (2011): 1-9.

MLA

Corciovei-Bănescu, Magdalena. “The Relation between π(x) and Certain Arithmetic Functions.” Notes on Number Theory and Discrete Mathematics 17.3 (2011): 1-9. Print.

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