Why legendre made a wrong guess about <Emphasis Type="Bold">π</Emphasis>-(<Emphasis Type="Italic">x</Emphasis>), and how laguerre’s continued fraction for the logarithmic integral improved it

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,