Timeline for The zeta function and classical mechanics
Current License: CC BY-SA 3.0
38 events
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| Mar 26, 2019 at 2:30 | review | Close votes | |||
| Mar 26, 2019 at 19:01 | |||||
| Nov 4, 2014 at 0:07 | comment | added | martin | @AndréLeClair Thanks, yes, I certainly will - hopefully it will clear up a few things for me. | |
| Nov 4, 2014 at 0:05 | comment | added | André LeClair | I recommend you have a look at our most recent paper, from a few weeks ago. Still not complete, but gaps are closing. It helps to explain why there is a unique solution for the equation for the n-th zero that we proposed. | |
| Nov 3, 2014 at 23:25 | comment | added | martin | @AndréLeClair thank you for responding and clarifying - I must admit, I was not absolutely clear on what was being implied in the comments, but without a full answer to the question, there is only so much that one can say. I really appreciate your clarification here. | |
| Nov 3, 2014 at 20:24 | comment | added | André LeClair | We could not claim a proof of the RH, but not for the reasons discussed here. GH from MO, I think is referring to the fact that N(T), the number of zeros on the entire critical strip, is known, so if one assumes the RH, one can obtain our formula. But that is not how we obtained our formula. We did not assume the RH nor N(T). The Lambert expression is only an approximate solution to an exact equation for the n-th zero. However we could not prove that there was a solution to this equation for every n. | |
| Oct 26, 2014 at 21:12 | history | protected | Todd Trimble | ||
| Oct 26, 2014 at 20:41 | comment | added | GH from MO | @joro: The alleged formula merely says, assuming standard notation, that the imaginary part of the $n$-th zero on the critical line is asymptotically $2\pi n/\log n$. This is true if and only if asymptotically 100% of all nontrivial zeros are on the critical line (when ordered by imaginary part). In particular, the alleged formula is true under RH. These are all simple facts. | |
| Oct 26, 2014 at 16:52 | comment | added | joro | @GHfromMO does RH imply something about the validity of the alleged formula with Lambert W? If you have an oracle that computes the $n$-th zero (possibly on the critical line) and compare it with with the alleged formula, does this imply something about the validity of the formula or RH? (for some reason can't join the chat). | |
| Oct 26, 2014 at 13:20 | comment | added | martin | @GHfromMO apologies for earlier - I was travelling at the time and went completely out of signal area. I see your point though that $W(x)$ is asymptotically $\log(x)$. | |
| Oct 26, 2014 at 10:53 | comment | added | GH from MO | I continued in chat. | |
| Oct 26, 2014 at 10:48 | comment | added | martin | @GHfromMO yes! So would "assuming the RH" be more accurate? | |
| Oct 26, 2014 at 10:47 | comment | added | GH from MO | In general I am skeptic about proofs by physicists. They have different rules, different motivations, different styles. | |
| Oct 26, 2014 at 10:47 | comment | added | martin | Let us continue this discussion in chat. | |
| Oct 26, 2014 at 10:46 | comment | added | martin | @GHfromMO sorry, I misunderstood - now deleted. Your point is interesting - I didn't realise this. | |
| Oct 26, 2014 at 10:44 | history | edited | martin | CC BY-SA 3.0 |
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| Oct 26, 2014 at 10:41 | comment | added | GH from MO | We have no asymptotic for the zeros on the critical line, because we have no idea how many zeros are on the critical line. We know more than 40%, but the ratio might fluctuate between say 67% and 93% as far as I know. Please delete the remark "as GH from MO pointed out below, this only applies to the zeros on the critical line", because I never said this. | |
| Oct 26, 2014 at 10:38 | comment | added | martin | @GHfromMO ah, ok, so it would be a weaker asymptotic | |
| Oct 26, 2014 at 10:35 | comment | added | GH from MO | Yes. If there are more than 0% nontrivial zeros off the critial line, then the equation fails for the zeros on the critical line. The reason is simple: the equation is true for all the zeros in the critical strip, because we know asymptotically the number of such zeros up to a given height. | |
| Oct 26, 2014 at 10:32 | comment | added | martin | @GHfromMO Are you saying that if there are zeros off the critical line, the formula is not valid asymptotically for the ones that are on it? | |
| Oct 26, 2014 at 10:29 | history | edited | martin | CC BY-SA 3.0 |
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| Oct 26, 2014 at 10:28 | comment | added | GH from MO | @martin: Yes, and that is how I understood it. The claimed equation for the zeros on the critical line would imply that only 0% of the nontrivial zeros are off the critical line. We certainly don't have a proof of that. | |
| Oct 26, 2014 at 10:26 | comment | added | martin | @GHfromMO sorry, yes - should have stated this only applies to zeros on the critical line - will update. | |
| Oct 26, 2014 at 10:24 | comment | added | GH from MO | The first equation would imply that asymptotically 100% of the nontrivial Riemann zeros are on the critical line. This is a famous open problem, so I doubt the authors have a valid unconditional proof. | |
| Oct 26, 2014 at 10:12 | history | edited | martin | CC BY-SA 3.0 |
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| Oct 26, 2014 at 10:05 | history | edited | martin | CC BY-SA 3.0 |
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| Oct 26, 2014 at 9:44 | history | edited | martin | CC BY-SA 3.0 |
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| Oct 26, 2014 at 9:28 | history | edited | martin | CC BY-SA 3.0 |
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| Oct 26, 2014 at 9:21 | comment | added | martin | @მამუკაჯიბლაძე Thanks for your response. Please see update. | |
| Oct 26, 2014 at 9:20 | history | edited | martin | CC BY-SA 3.0 |
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| Oct 26, 2014 at 6:42 | comment | added | მამუკა ჯიბლაძე | Do the graphs of $v$ and $2\pi/\kappa$ intersect at the zeros or at their approximations? And do you know anything about accuracy of the approximation? From what I was able to understand the error term grows like logarithm of $y$, or is this wrong? | |
| Oct 26, 2014 at 5:18 | history | edited | martin | CC BY-SA 3.0 |
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| Oct 26, 2014 at 0:04 | comment | added | martin | eg $y=1$ gives *first equation*$=14.5213\dots\sim\Im\rho_1,$ etc. | |
| Oct 25, 2014 at 23:57 | comment | added | martin | $y$ corresponds to the $y$th zeta zero. | |
| Oct 25, 2014 at 23:18 | comment | added | Felipe Voloch | What does the first equation mean? Specifically, what is y? | |
| Oct 25, 2014 at 23:13 | history | edited | martin | CC BY-SA 3.0 |
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| Oct 25, 2014 at 20:51 | history | edited | martin | CC BY-SA 3.0 |
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| Oct 25, 2014 at 20:43 | history | edited | martin | CC BY-SA 3.0 |
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| Oct 25, 2014 at 20:35 | history | asked | martin | CC BY-SA 3.0 |