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Let $P_n$ be independent variables which are 1 with probability $1/\log n$ and $0$ with probability $1-1/\log n$ and let $$\Pi(x) = \sum_{n\leq x} P_n.$$

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

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

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Edit: H. L. Montgomery has given an unpublished probabilistic argument that suggests

$$\limsup_{x\rightarrow \infty} \frac{|\psi(x)-x|}{\sqrt{x} (\log\log\log x)^2} = \frac{1}{2\pi}.$$

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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Let $P_n$ be independent variables which are 1 with probability $1/\log n$ and $0$ with probability $1-1/\log n$ and let $$\Pi(x) = \sum_{n\leq x} P_n.$$

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

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

where

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

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