New explicit formulae for the prime counting function

Mladen Vassilev-Missana
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 19, 2013, Number 1, Pages 44–49
Full paper (PDF, 159 Kb)

Details

Authors and affiliations

Mladen Vassilev-Missana
5 V. Hugo Str., 1124 Sofia, Bulgaria

Abstract

In the paper new explicit formulae for the prime counting function π are proposed and proved. They depend on arbitrary positive arithmetic function which satisfies certain condition. As a particular case a formula for π depending on Euler’s function φ is obtained. To the author’s best knowledge such kind of formulae are proposed for the first time in the mathematical literature.

Keywords

  • Prime number
  • Composite number
  • Arithmetic function

AMS Classification

  • 11A25
  • 11A41

References

  1. Atanassov, K. A new formula for the n-th prime number. Comptes Rendus de l’Academie bulgare des Sciences, Tome 54, 2001, No. 7, 5-6.
  2. Atanassov, K. On a new formula for the n-th prime number. Notes on Number Theory and Discrete Mathematics, Vol. 10, 2004, No. 1, 24.
  3. Atanassov, K., A formula for the n-th prime number, Comptes Rendus de l’Academie bulgare des Sciences, Tome 66, No. 4, 2013, 503-506.
  4. Atanassov, K., M. Vassilev-Missana. On explicit formulae for prime and twin prime numbers. Italian Journal of Pure and Applied Mathematics, No. 20, 2006, 103-120.
  5. Ribenboim, P. The New Book of Prime Number Records (3rd Edition), Springer-Verlag, New York, 1996.
  6. Hardy, G. H., E. M. Wright An Introduction to the Theory of Numbers (5th Edition), Oxford, England, Clarendon Press, 1979.
  7. Vassilev-Missana, M. Three Formulae for n-th Prime and Six for n-th Term of Twin Primes. Notes on Number Theory and Discrete Mathematics, Vol. 7, 2001, No. 1, 15–20.
  8. Sándor, J., B. Crstici. Handbook of Number Theory II, Kluwer, London, 2004.
  9. Sierpiński,W. Elementary Number Theory (2nd Edition), North Holland, Amsterdam, 1988.

Related papers

Cite this paper

Vassilev-Missana, M. (2013). New explicit formulae for the prime counting function. Notes on Number Theory and Discrete Mathematics, 19(1), 44-49.

Comments are closed.