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 n
th 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".