Timeline for How differently would we model the distribution of primes if prime gap is larger?
Current License: CC BY-SA 4.0
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| Jan 21, 2021 at 19:00 | history | edited | Turbo | CC BY-SA 4.0 |
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| Jan 21, 2021 at 18:44 | answer | added | Will Sawin | timeline score: 2 | |
| Jan 21, 2021 at 18:29 | history | edited | Turbo |
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| Jan 21, 2021 at 7:01 | history | edited | Turbo | CC BY-SA 4.0 |
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| Jan 21, 2021 at 7:01 | comment | added | Turbo | Replaced 'predicts'. | |
| Jan 20, 2021 at 12:08 | comment | added | KConrad | @EmilJeřábek yes, I was careless there. Monach and Montgomery conjectured that $\pi(x) - {\rm Li}(x)$ is $O(\sqrt{x})$, with $|\pi(x) - {\rm Li}(x)|/\sqrt{x}$ being $O((\log\log\log x)^2/\log x)$ and $\Omega((\log\log\log x)^2/\log x)$. | |
| Jan 20, 2021 at 10:48 | comment | added | Emil Jeřábek | @KConrad In fact, RH does imply the error term is $O(\sqrt x\log x)$. (The $O(\sqrt x(\log x)^2)$ bound is for $\psi(x)-x$.) | |
| Jan 20, 2021 at 4:30 | comment | added | Turbo | @KConrad yes I agree $O(\cdot)$ is not $\Omega(\cdot)$ but the problem is not about these issues. I mention 'So if this much larger gap were true..' and so the point is what if the truth is way off. | |
| Jan 20, 2021 at 3:55 | comment | added | KConrad | RH implies $\pi(x) = {\sf Li}(x) + O(\sqrt{x}(\log x)^2)$, but that $(\log x)^2$ is an artifact of what can be proved, not an indication that it is in some way a reflection of the true order of magnitude of the error term. There is no claim that the error term can't be $O(\sqrt{x}\log x)$ or even $O(\sqrt{x})$; it's just that nobody has ever shown something like that. In contrast, the exponent $1/2$ in the $\sqrt{x}$ piece of the error term is known to be optimal, since shrinking that would lead to a known false result (because there are nontrivial zeros with real part $1/2$). | |
| Jan 20, 2021 at 3:49 | comment | added | KConrad | For comparison, Bach showed GRH implies that for composite $m \geq 3$, the least witness for the Miller-Rabin test on $m$ is $O((\log m)^2)$, but numerical data suggest the the least witness is $O(\log m)$. Only being able to prove $O((\log m)^2)$ from GRH does not mean the true order of magnitude can't be smaller than what that bound suggests. | |
| Jan 20, 2021 at 3:49 | comment | added | KConrad | An estimate $O(g(x))$ need not be as large as $g(x)$, but is just upper-bounded by a constant multiple of $g(x)$, e.g., $\sin x = O(x^{55})$ as $x \to \infty$. So when you say RH predicts a gap of $O(\sqrt{p_n}(\log p_n)^2)$, that does not mean it predicts a gap on the order of $\sqrt{p_n}(\log p_n)^2$. Such a $O$-estimate is simply the sharpest people have been able to prove; we are dealing here with a lack of suitable technique to push the upper bound down to what is expected to be the true order of magnitude. (contd.) | |
| Jan 20, 2021 at 3:00 | history | asked | Turbo | CC BY-SA 4.0 |