prime numbers - What does the probabilistic model suggest the error term in the PNT should be?

Let $P_n$ be independent variables which are 1 with probability $1/log n$ and $0$ with probability $1-1/log n$ and let

$$Pi(x) = sum_{nleq x} P_n.$$

Then Cram'{e}r showed that, almost surely,

$$limsup_{xrightarrow infty} frac{|Pi(x)-ell i(x)|}{sqrt{2x}sqrt{frac{loglog x}{log x}}} = 1$$

where

$$ell i (x) = int_2^x frac{dt}{log t}.$$

See page 20 here: http://www.dms.umontreal.ca/~andrew/PDF/cramer.pdf

Edit: H. L. Montgomery has given an unpublished probabilistic argument that suggests

$$limsup_{xrightarrow infty} frac{|psi(x)-x|}{sqrt{x} (logloglog x)^2} = frac{1}{2pi}.$$

This is announced in: H.L. Montgomery, "The zeta function and prime numbers," Proceedings of the Queen's Number Theory Conference, 1979, Queen's Univ., Kingston, Ont., 1980, 1-31.

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