Where does the error term of the Prime Number Theorem touch the predicted asymptotic behavior Since the Riemann hypothesis is equivalent to $\pi(x) = \text{Li}(x) + O(\sqrt x \log x)$,
One would expect that a plot of $|\pi(x) - \text{Li}(x)|$ and $\sqrt x \log x$ would show $|\pi(x) - \text{Li}(x)|$ coming near $\sqrt x \log x$. For values of $x$ up to even $10^8$ this does not happen. Does anyone know when the predicted asymptotic behavior shows up in a plot?
 A: The Riemann Hypothesis is also equivalent to $|\pi(x) - Li(x)| = O(x^{1/2 + \epsilon})$, so let's look at that instead. In other words, $\log$ of the error should be about $(1/2) \log x$.
The sequence of points plotted below is $( \log x,\ \log |\pi(x) - Li(x)|)$ for $x=10^k$, with $1 \leq k \leq 23$. The straight line has slope $1/2$, with constant term chosen by a least squares fit (specifically, the line is $x/2 -1.24878$). Interpreted in this way, you can definitely see the promised asymptotic behavior.
    (source)
(Data set courtesy of Wikipedia)
Note: My $\log$'s are base $10$, since my data set was binned by powers of $10$ already. Of course, that doesn't effect the slope.
A: The $\log x$ factor is a result of Koch (1901) and still the best known consequence of the Riemann Hypothesis, but it is probably very far from the truth. Littlewood (1914) proved that this factor is $\Omega(\log\log\log(x)/\log x)$, while Montgomery conjectures that the truth is around $(\log\log\log x)^2/\log x$. The last information was communicated to me by Pintz around a year ago.
