Timeline for Why is so much work done on numerical verification of the Riemann Hypothesis?
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Jul 16 at 12:57 | history | protected | Carlo Beenakker | ||
| Nov 6, 2021 at 17:07 | answer | added | TravorLZH | timeline score: 12 | |
| S Mar 24, 2019 at 10:47 | history | suggested | Rafael Marazuela | CC BY-SA 4.0 |
I've added a link to the Wikipedia.
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| Mar 24, 2019 at 9:16 | review | Suggested edits | |||
| S Mar 24, 2019 at 10:47 | |||||
| Mar 23, 2019 at 1:05 | comment | added | Nell | @user1729: if I understand correctly, there is even tradeoff between (G)RH computations and the verification you mention in (2): the proof of ternary Goldbach uses a (G)RH check up to a finite height, and had it used a smaller height (presumably still above a certain level), it would have needed a verification up to a higher $n$. | |
| Mar 22, 2019 at 17:24 | vote | accept | Hollis Williams | ||
| Mar 21, 2019 at 19:01 | comment | added | Thomas Sauvaget | Note also that there is interesting work of Voros arxiv.org/pdf/1703.02844.pdf not directly on Riemann zeros but on modified Keiper-Li sequences that are fully explicit and can capture behaviour of zeros off the critical line (but see the discussion p27 onwards "Now the true current challenge..."). | |
| Mar 21, 2019 at 18:14 | comment | added | literature-searcher | Zagier famously gave 300 million zeros as the level of experiment needed to convince him, and when computations hit 200 million, the computationalists were cajoled to extend this so that a bet with Bombieri could be won. maths-people.anu.edu.au/~brent/pub/pub081.html | |
| Mar 21, 2019 at 18:05 | comment | added | J Fabian Meier | Let me add a different angle: Going through millions of zeros is experimental evidence for the hypothesis. Whether it is stronger or weaker than a possible proof that is probably so complicated that only a dozen people in the world will understand it, is up to you. | |
| Mar 21, 2019 at 17:31 | history | edited | YCor | CC BY-SA 4.0 |
edited tags; edited title
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| Mar 21, 2019 at 17:04 | comment | added | reuns | Note there is some computable $C$ such that $\psi(n) =1_{x > 1} \log \lfloor \exp(n-\sum_{|\Im(\rho)| < Cn} \frac{(n+1/2)^\rho}{\rho})+\frac12(1- \log 2\pi-\log(1-(n+1/2)^{-2}))\rfloor $. The RH aims at replacing $Cn$ by $C' n^{1/2} \log^3 n$. From the small non-trivial zeros you know $\psi(n)$ for $n$ small, and from $\psi(n)$ for $n$ small and the knowledge that the RH is true for $\Im(s)$ small, you know the approximate imaginary part of the non-trivial zeros. | |
| Mar 21, 2019 at 16:14 | answer | added | user1728 | timeline score: 50 | |
| Mar 21, 2019 at 16:02 | comment | added | Hollis Williams | Yes, I am familiar with the work Turing did with his computers to see if he could find a counterexample: I suppose the modern work with supercomputers could be viewed as an extension of that research. | |
| Mar 21, 2019 at 15:48 | history | became hot network question | |||
| Mar 21, 2019 at 15:39 | answer | added | Conrad | timeline score: 27 | |
| Mar 21, 2019 at 15:20 | comment | added | Richard Stanley | The Riemann hypothesis may be false. Even if one feels that the chance that RH is false is very small, the expected payoff for finding a counterexample is still large. | |
| Mar 21, 2019 at 14:26 | comment | added | user1729 | Regarding your last paragraph: The proof of the Ternary Goldbach Conjecture proceeded in two steps: 1) prove the result for all numbers bigger than a certain, known number $n$, and 2) use a computer to verify the result for all numbers less than $n$. It is conceivable that a similar method of proof would work for the Riemann hypothesis. | |
| Mar 21, 2019 at 14:24 | comment | added | Nell | Is there really such a huge amount of (human) work on numerical verifications? I am not aware of that many papers. | |
| Mar 21, 2019 at 14:16 | answer | added | Nell | timeline score: 25 | |
| Mar 21, 2019 at 13:52 | history | asked | Hollis Williams | CC BY-SA 4.0 |