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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?

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No, one would expect it would come near $C \sqrt{x} \log x$ for an appropriate constant. –  Qiaochu Yuan Jul 19 '11 at 19:58
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On the Riemann Hypothesis, Schoenfeld showed the constant $C$ can be taken as $1/(8\pi)$, and (iirc) this can not be improved on. –  Stopple Jul 19 '11 at 20:33
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I'm confused by all your comments above. The big-O notation means that the error term is bounded by $\sqrt{x}\log(x)$, not that it's asymptotically proportional to it. See en.wikipedia.org/wiki/Big_O_notation. –  André Henriques Jul 19 '11 at 20:52
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@Paul, I just looked up the Schoenfeld result: On RH, $|\pi(x)-li(x)|$ is less than $x^{1/2}\log(x)/(8\pi)$ for $2,657\le x$, no mention that this is optimal. –  Stopple Jul 19 '11 at 21:31
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Once I verified that Schoenfeld's result implies $|\pi(x)-li(x)|<x^{1/2}\log x$ for all $x>2$. –  GH from MO Jul 19 '11 at 22:16
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2 Answers

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.

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It reminds me of the quote "$\log \log x$ approaches infinity but has never been observed to do so." –  Victor Miller Jul 19 '11 at 21:02
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I thought the quote was "$\log\log\log x$ goes to infinity with great dignity." –  François G. Dorais Jul 19 '11 at 21:41
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Montgomery's conjecture is discussed in his paper "The zeta function and prime numbers," Proceedings of the Queen's Number Theory Conference, 1979, Queen's Univ., Kingston, Ont., 1980, 1-31. He calculates, assuming RH and the zeros of the zeta function are linearly independent, the limiting distribution of the quantity $e^{-y/2}(\psi(e^y)-e^y)$. Once one knows this distribution it is natural to conjecture how large $|e^{-y/2}(\psi(e^y)-e^y)|$ can be (infinity often) on probabilistic grounds. –  Mark Lewko Jul 20 '11 at 0:09
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@GH, I temporarily put a copy up at: lewko.files.wordpress.com/2011/07/… the relevant section starts on page 14. –  Mark Lewko Jul 20 '11 at 2:12
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@Mark: Thanks for the paper! @GH: Gonek has a conjecture for the summatory function of the Moebius function which is analogous to Montgomery's conjecture for the error term in PNT. The details are in Nathan Ng's paper "The distribution of the summatory function of the Moebius function" which he has posted on his web-page. –  Micah Milinovich Jul 20 '11 at 14:43
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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.

alt text

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.

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