Advances in Computational Mathematics

, Volume 29, Issue 1, pp 55–70

The prime-counting function and its analytic approximations

π(x) and its approximations
Article

Abstract

The paper describes a systematic computational study of the prime counting function π(x) and three of its analytic approximations: the logarithmic integral \({\text{li}}{\left( x \right)}: = {\int_0^x {\frac{{dt}} {{\log \,t}}} }\), \({\text{li}}{\left( x \right)} - \frac{1} {2}{\text{li}}{\left( {{\sqrt x }} \right)}\), and \(R{\left( x \right)}: = {\sum\nolimits_{k = 1}^\infty {{\mu {\left( k \right)}{\text{li}}{\left( {x^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} } \right)}} \mathord{\left/ {\vphantom {{\mu {\left( k \right)}{\text{li}}{\left( {x^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} } \right)}} k}} \right. \kern-\nulldelimiterspace} k} }\), where μ is the Möbius function. The results show that π(x)<li(x) for 2≤x≤1014, and also seem to support several conjectures on the maximal and average errors of the three approximations, most importantly \({\left| {\pi {\left( x \right)} - {\text{li}}{\left( x \right)}} \right|} < x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}\) and \( - \frac{2} {5}x^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} < {\int_2^x {{\left( {\pi {\left( u \right)} - {\text{li}}{\left( u \right)}} \right)}du < 0} }\) for all x>2. The paper concludes with a short discussion of prospects for further computational progress.

Keywords

Prime-counting function Logarithmic integral Riemann’s approximation 

Mathematics Subject Classifications (2000)

11A41 41A60 11Y35 65G99 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bays, C., Hudson, R.: A new bound for the smallest x with π(x)>li(x). Math. Comp. 69, 1285–1296 (2000)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Berndt, B.C.: Ramanujan’s Notebooks, Part IV. pp. 126–131. Springer, New York (1994)MATHGoogle Scholar
  3. 3.
    Brent, R.P.: Irregularities in the distribution of primes and twin primes. Math. Comp. 29, 43–56 (1975)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    de la Vallée Poussin, C.J.: Sur la fonction ζ(s) de Riemann et le nombre des nombres premiers inférieurs à une limite donnée. Mem. Cour. Acad. Roy. Belg. 59, 1 (1899)Google Scholar
  5. 5.
    Ford, K.: Vinogradov’s integral and bounds for the Riemann zeta function. Proc. London Math. Soc. 85, 565–633 (2002)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Gauss, C.F.: Werke, Vol. II. Königliche Gesellschaft der Wissenschaften zu Göttingen, pp. 444–447 (1863)Google Scholar
  7. 7.
    Gram, J.P.: Undersøgelser angående Mængden af Primtal under en given Grænse. Kong. Dansk. Videnskab. Selsk. Skr. (VI) 2, 183–308 (1884)Google Scholar
  8. 8.
    Ingham, A.E.: The distribution of prime numbers, pp. 105–106. Cambridge University Press, New York (1932)Google Scholar
  9. 9.
    Korobov, N.M.: Estimates of trigonometric sums and their applications. Usp. Mat. Nauk. 13, 185–192 (1958) (in Russian)MathSciNetMATHGoogle Scholar
  10. 10.
    Legendre, A.M.: Essai sur la théorie des nombres, 2ème dition, p.394. Courcier, Paris (1808)Google Scholar
  11. 11.
    Lehman, R.S.: On the difference π(x)−li(x). Acta Arith. 11, 397–410 (1966)MathSciNetMATHGoogle Scholar
  12. 12.
    Littlewood, J.E.: Sur la distribution des nombres premiers. C. R. Acad. Sci. Paris 158, 1869–1872 (1914)MATHGoogle Scholar
  13. 13.
    Riemann, B.: Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsber. Preuss. Akad. Wiss. 1859 671–680 (1859)Google Scholar
  14. 14.
    Rosser, J.B., Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Illinois J. Math. 6, 64–94 (1962)MathSciNetMATHGoogle Scholar
  15. 15.
    Skewes, S.: On the difference π(x)−li(x). II. Proc. London Math. Soc. (3) 5, 48–70 (1955)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    te Riele, H.J.J.: On the sign of the difference π(x)−li(x). Math. Comp. 48, 323–328 (1987)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Vinogradov, I.M.: A new estimate for ζ(1+it) [in Russian]. Izv. Akad. Nauk SSSR, Ser. Mat. 22, 161–164 (1958)MathSciNetMATHGoogle Scholar
  18. 18.
    von Koch, H.: Sur la distribution des nombres premiers. Acta Math. 24, 159–182 (1901)CrossRefMathSciNetGoogle Scholar
  19. 19.
    von Mangoldt, H.: Zu Riemann’s Abhandlung ‘Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse’. J. Reine Angew. Math. 114, 255–305 (1895)Google Scholar
  20. 20.
    Walfisz, A.: Weylsche Exponentialsummen in der neueren Zahlentheorie, pp. 175–188. VEB Deutscher Verlag, Berlin (1963)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Faculty of Electrical EngineeringUniversity of LjubljanaLjubljanaSlovenia

Personalised recommendations