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Posted 3 months ago
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 In the function gallery, there is a formula for the asymptotic growth of prime numbers. http://functions.wolfram.com/NumberTheoryFunctions/Prime/06/01/0006/Where can I find more about this? Specifically, how all those terms were found?
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Posted 3 months ago
 http://mathworld.wolfram.com/PrimeFormulas.html / 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.
 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] == 1The 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".