For example, we can express P(x) in terms of P(s), the PN density at scale x=s (which amounts to interpreting the constant “c” in the original post)

P(x, P(s)) = P(s) / ( 1 + P(s) log(x/s))

Note that P(x, P(y, P(z))) = P(z) / ( 1 + P(z) log (z/x) ) which follows from the RG assumptions stated in the paper.

Further, if we heuristically argue that P(s) is small for x near s, we can use perturbation theory to derive the next order of approximation to the solution to P’ = -P^2/ ( x – P); namely, if we denote this next functional approximation by PP,

PP(x, P(s)) = P(s) / ( 1 + P(s) log(x/s) * ( 1 + P(s) * loglog(x/s) ) )

is the solution to P’ = -P^2/x – P^3/x^x.

[1] A. Petermann, “The so-called renormalization group method applied to the specific prime number logarithmic decrease”, Euro. Phys. J. C, 17, p367 (2000).DOI 10.1007/s100520000469. First page. Preprint.

]]>Here’s something else from Frank Morgan: The Prime Number theorem says that the probability P(x) that a large integer x is prime is about 1/log x. At about age 16 Gauss apparently conjectured this estimate after studying tables of primes…….

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