Timeline for How good is "almost all" when it comes to the Riemann Hypothesis?
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12 events
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| Jun 5, 2018 at 18:46 | comment | added | Sylvain JULIEN | @Joël : regardless of its potential correctness, would it be possible to derive a concrete arithmetic application of the strategy exposed in arxiv.org/abs/1805.07741 ? | |
| Mar 26, 2014 at 18:06 | comment | added | Joël | I'm sure that we could get some good information we don't already have on the distribution of prime on (relatively, but not too) short intervals $[x,x+y]$, because the contribution of the zeros of the line to the explicit formulae for $\pi(x)$ and $\pi(x+y)$ could be proved to cancel each other to some extent. | |
| Mar 26, 2014 at 18:04 | comment | added | Joël | To complete Greg's comment (which could be an answer) and Stopple's answer, it is not surprising that the $\kappa=1$ conjectures has no consequences because it is compatible with $N(T)-N_0(T) \sim N(T)/\log \log \log \log T$ for example, meaning that even if "almost all" zeros are on the critical line, still almost as many of them are not. Now if we had a good proof that $\kappa=1$, with a good upper bound for $N(T)-N_0(T)$, we could certainly get some arithmetic application. For example, in the extreme case that $N(T)-N_0(T)=O(1)$ (finitely many zeros off the critical line),... | |
| Mar 26, 2014 at 16:23 | history | edited | RHarris | CC BY-SA 3.0 |
Removed the second question about the correctness of the preprint.
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| Mar 26, 2014 at 16:06 | history | edited | Stopple |
retagged
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| Mar 26, 2014 at 16:06 | answer | added | Stopple | timeline score: 27 | |
| Mar 26, 2014 at 13:48 | comment | added | Joël | About the appropriateness of (2), there is a consistent policy on MO not to allow general discussions on the correctness of preprints, by fear of endless controversy. Discussions about particular points are Okay, though, like "I don't understand the proof of Lemma 3.14 in such preprint". I share your curiosity about (2) (obviously, when a striking result is announced but no complete proof given, there is a large room for skepticism), but I think you should just leave question (1), which is very interesting. | |
| Mar 26, 2014 at 6:07 | comment | added | Greg Martin | One answer to (1): the Riemann hypothesis implies that $\big| \#\{$primes${}\le x\} - \int_2^x \frac{dt}{\log t} \big| < 2\sqrt x\log x$ for all $x$. This is known to be false if there's even a single violation of the Riemann hypothesis - then there would be arbitrarily large values of $x$ for which $\big| \#\{$primes${}\le x\} - \int_2^x \frac{dt}{\log t} \big| > x^{1/2+\delta}$ for some fixed $\delta>0$. | |
| Mar 26, 2014 at 3:18 | history | edited | RHarris | CC BY-SA 3.0 |
added 127 characters in body
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| Mar 26, 2014 at 2:57 | review | Close votes | |||
| Mar 26, 2014 at 18:22 | |||||
| Mar 26, 2014 at 2:52 | review | First posts | |||
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| Mar 26, 2014 at 2:37 | history | asked | RHarris | CC BY-SA 3.0 |