Table of Contents
Journal of Discrete Mathematics
Volume 2013, Article ID 491627, 11 pages
Research Article

Efficient Prime Counting and the Chebyshev Primes

1Institut FEMTO-ST, CNRS, 32 Avenue de l’Observatoire, F-25044 Besançon, France
2Telecom ParisTech, 46 rue Barrault, 75634 Paris Cedex 13, France
3Mathematical Department, King Abdulaziz University, Jeddah, Saudi Arabia

Received 17 October 2012; Revised 28 January 2013; Accepted 29 January 2013

Academic Editor: Pantelimon Stǎnicǎ

Copyright © 2013 Michel Planat and Patrick Solé. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. H. M. Edwards, Riemann's Zeta Function, Academic Press, New York, NY, USA, 1974. View at Zentralblatt MATH · View at MathSciNet
  2. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, The Clarendon Press Oxford University Press, Oxford, UK, 5th edition, 1979. View at MathSciNet
  3. H. Davenport, Multiplicative Number Theory, vol. 74 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 1980. View at Zentralblatt MATH · View at MathSciNet
  4. S. Skewes, “On the difference π(x)li(x) (I),” The Journal of the London Mathematical Society, vol. s1-8, no. 4, pp. 277–283. View at Publisher · View at Google Scholar · View at MathSciNet
  5. G. Robin, “Sur la différence Li (θ(x))-π(x),” Annales de la Faculté des Sciences de Toulouse, Série 5, vol. 6, no. 3-4, pp. 257–268, 1984. View at Google Scholar · View at MathSciNet
  6. J. Sándor, D. S. Mitrinović, and B. Crstici, Handbook of Number Theory. I, Springer, Dordrecht, The Netherlands, 1995. View at MathSciNet
  7. J. B. Rosser and L. Schoenfeld, “Approximate formulas for some functions of prime numbers,” Illinois Journal of Mathematics, vol. 6, pp. 64–94, 1962. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. L. Schoenfeld, “Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II,” Mathematics of Computation, vol. 30, no. 134, pp. 337–360, 1976. View at Google Scholar · View at MathSciNet
  9. W. J. Ellison and M. Mendès-France, Les Nombres Premiers, Hermann, Paris, France, 1975. View at MathSciNet
  10. A. E. Ingham, The Distribution of Prime Numbers, Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK, 1990. View at MathSciNet
  11. T. W. Cusick, C. Ding, and A. Renvall, Stream Ciphers and Number Theory, vol. 55 of North-Holland Mathematical Library, North-Holland, Amsterdam, The Netherlands, 2005. View at MathSciNet
  12. N. J. A. Sloane, “The on-line encyclopedia of integer sequences,” 2011,
  13. E. A. Weinstein, Prime counting function, Mathworld.