Learn more about the formula for the asymptotic growth of prime numbers?

Questions
1

In the function gallery, there is a formula for the asymptotic growth of prime numbers.

Where can I find more about this? Specifically, how all those terms were found?

2

Prime Formulas -- from Wolfram MathWorld / Formula No. 15

An asymptotic formula for p_n is given by (Cipolla 1902). This asymptotic expansion is the inverse of the logarithmic integral Li(x) obtained by series reversion.

3

To give a slightly longer answer, this goes back to the Prime Number Theorem, which asserts that the number of prime numbers less than n (implemented in the Wolfram language as PrimePi[n]) is approximated by the function LogIntegral[n], in the sense that

Limit[PrimePi[n]/(LogIntegral[n]), n -> Infinity] == 1

The nth prime number Prime[n] is the inverse function of PrimePi, and so we can describe its asymptotic behavior by inverting the LogIntegral function. This gives the series in question.

For a really good introduction to the Prime Number Theorem, I would recommend H.M. Edwards’ book “Riemann’s Zeta Function”.