Historical References For π(n)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