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

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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,

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  1. Pafnuty Lvovich Chebyshev, Sur la fonction qui détermine la totalité des nombres premiers,Oeuvres l, 27–48 (1851).

  2. Pafnuty Lvovich Chebyshev, Mémoire sur les nombres premiers,Oeuvres I, 49–70 (1854).

  3. Charles-Jean de la Vallee-Poussin,Ann. Soc. Sci. Bruxelles 20, 183–256, 281–397 (1896).

  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).

  5. Jaques Hadamard,Oeuvres I, 189–210 (1896).

  6. Helge von Koch,Math. Annalen 55 (1902), 441–464.

  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).

  8. Edmund Landau,Vorlesungen über Zahlentheorie, S. Hirzel, Leipzig 1927.

  9. Adrien-Marie Legendre,Théorie des nombres. 2nd edition, 1798, No. 394–401.

  10. Bernhard Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen GröBe,Werke, 136–144.

  11. J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers,Illinois J. Math. 6 (1962), 64–94.

  12. Lowell Schoenfeld, Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II.Math. Comp. 30 (1976), 337–360.

  13. Atle Selberg, An elementary proof of the prime number theorem for arithmetic progressions,Ann. Math. (2)50, 305–313 (1949).

  14. James Joseph Sylvester, On arithmetical series,Collected Works III, 573–587 (1892).

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Correspondence to Friedrich L. Bauer.

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Bauer, F.L. Why legendre made a wrong guess about π-(x), and how laguerre’s continued fraction for the logarithmic integral improved it. The Mathematical Intelligencer 25, 7–11 (2003). https://doi.org/10.1007/BF02984842

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  • Continue Fraction
  • Riemann Hypothesis
  • Weak Approximation
  • Prime Number Theorem
  • Chebyshev Function