Historical References For $\pi(n)$

Gauss, 1791

Gauss’ 1791 ‘Some Asymptotic Laws Of Number Theory’ can be found in volume 10 of his collected works. In it he presents his approximation for $\pi(n)$.

$$ \frac{a}{la} $$

Today, this would be written as $n/ \ln(n)$.

Gauss’ 1971 Some Asymptotic Laws Of Number Theory.

Source: http://resolver.sub.uni-goettingen.de/purl?PPN236018647

Legendre, 1797

Legendre in his first edition of ‘Essai Sur La Theorie Des Nombres’ presented his approximation.

$$ \frac{a}{A \text{log}(a)+B} $$

The logarithm is the natural $\ln(a)$. In his 1808 second edition he quantifies the constants.

$$ \frac{x}{\text{log}(x) - 1.08366} $$

Legendre’s 1797 Essai Sur La Theorie Des Nombres.

Source: https://gallica.bnf.fr/ark:/12148/btv1b8626880r/f55. image

Gauss, 1849

Gauss wrote a letter to astronomer Encke dated Decemer 24th 1849, in which he first presents an integral form of a prime counting function. He states this is based on work he started in 1792 or 1793.

Gauss uses the following expression.

$$ \int \frac{dn}{\text{log}n} $$

Today this would be written as the logarithmic integral function.

$$ \int_0^n \frac{1}{\ln(x)} dx $$

First page of Gauss’ 1849 letter to Encke.

Source: https://gauss.adw-goe.de/handle/gauss/199