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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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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,

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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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Keywords

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