Advertisement

The Mathematical Intelligencer

, Volume 25, Issue 3, pp 7–11 | Cite as

Why legendre made a wrong guess about π-(x), and how laguerre’s continued fraction for the logarithmic integral improved it

  • Friedrich L. Bauer
Article
  • 70 Downloads

Abstract

Carl Friedrich Gauβ, in 1792, when he was 15, found by numerical evidence that π(x), the number of primes p such that p ≤ x, goes roughly with x/in x (letter to Encke, 1849). This was, as can be seen from Table 1, a very weak approximation with an error of about 10%. In 1798 and again in 1808,

Keywords

Continue Fraction Riemann Hypothesis Weak Approximation Prime Number Theorem Chebyshev Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Pafnuty Lvovich Chebyshev, Sur la fonction qui détermine la totalité des nombres premiers,Oeuvres l, 27–48 (1851).Google Scholar
  2. Pafnuty Lvovich Chebyshev, Mémoire sur les nombres premiers,Oeuvres I, 49–70 (1854).Google Scholar
  3. Charles-Jean de la Vallee-Poussin,Ann. Soc. Sci. Bruxelles 20, 183–256, 281–397 (1896).Google Scholar
  4. Paul Erdős, On a new method in elementary number theory which leads to an elementary proof of the prime number theorem,Proc. Natl. Acad. Sci. U.S.A. 35, 374–384 (1949).CrossRefGoogle Scholar
  5. Jaques Hadamard,Oeuvres I, 189–210 (1896).Google Scholar
  6. Helge von Koch,Math. Annalen 55 (1902), 441–464.CrossRefMATHGoogle Scholar
  7. Edmond Nicolas Laguerre, Sur la réduction en fractions continues d’une fonction que satisfait à une équation différentielle linéaire du premier ordre dont les coefficients sont rationelles,J. Math. Pures et Appl. (4)1 (1885).Google Scholar
  8. Edmund Landau,Vorlesungen über Zahlentheorie, S. Hirzel, Leipzig 1927.MATHGoogle Scholar
  9. Adrien-Marie Legendre,Théorie des nombres. 2nd edition, 1798, No. 394–401.Google Scholar
  10. Bernhard Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen GröBe,Werke, 136–144.Google Scholar
  11. J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers,Illinois J. Math. 6 (1962), 64–94.MATHMathSciNetGoogle Scholar
  12. Lowell Schoenfeld, Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II.Math. Comp. 30 (1976), 337–360.MATHMathSciNetGoogle Scholar
  13. Atle Selberg, An elementary proof of the prime number theorem for arithmetic progressions,Ann. Math. (2)50, 305–313 (1949).Google Scholar
  14. James Joseph Sylvester, On arithmetical series,Collected Works III, 573–587 (1892).Google Scholar

Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  • Friedrich L. Bauer
    • 1
  1. 1.KottgeiseringGermany

Personalised recommendations