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Old 2018-12-13, 15:18   #1
ssybesma
 
"Steve Sybesma"
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Default distributed computing project to prove/disprove Riemann Hypothesis

All I've read so far on the attempts seem to be from lone individuals working on this. Is there any way that a program can be written to search for examples that exist in the critical strip which are not on the critical line? It seems like a computer application that's working in a distributed framework can input numbers a lot faster than one individual can. I've looked around a bit but not seen this idea mentioned, yet I cannot possibly be the first one to think of this.

The program if created could be used to map out the results of each input and try to predict where a likely candidate would be.

Some inputs could be pre-programmed into it such as commonly known irrational/transcendental constants out to millions of decimal points in case a proof of something existing in the critical strip apart from the critical line can be found that is directly or indirectly related to one of them.

Last fiddled with by ssybesma on 2018-12-13 at 15:30 Reason: elaboration
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Old 2018-12-13, 15:44   #2
ssybesma
 
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Thinking on this some more, if a number is found in the critical strip not part of the critical line, I would think it would have to have some kind of significance if it's very rare, meaning that it's possibly related in some way to a known constant. I'm speaking completely out of ignorance, but just trying to reason this in my head. Examples, inverse of pi, or inverse of square root of 2. Some creativity in thinking up logical examples to look at would be something the program could do.

Last fiddled with by ssybesma on 2018-12-13 at 15:53
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Old 2018-12-13, 16:01   #3
ssybesma
 
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Default ZetaGrid project

I happened to find this project which ended in 2005.

https://web.archive.org/web/20131005....zetagrid.net/

I wonder if it can be restarted and if any of the concepts I mentioned were in this.

13 years on we have much more powerful computers.
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Old 2018-12-13, 16:04   #4
petrw1
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You might start here:

https://www.youtube.com/watch?v=d6c6uIyieoo

Around 14:00 mark he talks about how much computational work has already been done to try to find a counterexample.
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Old 2018-12-13, 16:22   #5
ssybesma
 
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Quote:
Originally Posted by petrw1 View Post
You might start here:

https://www.youtube.com/watch?v=d6c6uIyieoo

Around 14:00 mark he talks about how much computational work has already been done to try to find a counterexample.
I saw that yesterday and posted a question there. Since then I had more ideas. That in fact is what got me so interested. That Russian guy makes it so darn interesting. Sebastian Wedeniwski's ZetaGrid project seemed to have been using a brute force method. I was thinking of an effort using more defined inputs and mapping the results (like dots on a graph) to see if a pattern can be detected. I feel if a zero is found in that area of the critical strip not on the 1/2 line that it must be related to some significant number.

Last fiddled with by ssybesma on 2018-12-13 at 16:28
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Old 2018-12-13, 17:51   #6
ssybesma
 
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Quote:
Originally Posted by ssybesma View Post
Thinking on this some more, if a number is found in the critical strip not part of the critical line, I would think it would have to have some kind of significance if it's very rare, meaning that it's possibly related in some way to a known constant. I'm speaking completely out of ignorance, but just trying to reason this in my head. Examples, inverse of pi, or inverse of square root of 2. Some creativity in thinking up logical examples to look at would be something the program could do.
Forgot inverse of Golden Ratio and inverse of 'e'. All those can be plugged into the app in some way.

Speaking of Golden Ratio, this guy plotted the Golden ratio (starting at 6:30)...

https://www.youtube.com/watch?v=sj8Sg8qnjOg

...in a way that reminded me a bit of the Riemann zeta function plotted on the opening of this video:

https://www.youtube.com/watch?v=sD0NjbwqlYw

Last fiddled with by ssybesma on 2018-12-13 at 18:00
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Old 2018-12-13, 21:34   #7
ssybesma
 
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Quote:
Originally Posted by ssybesma View Post
Forgot inverse of Golden Ratio and inverse of 'e'. All those can be plugged into the app in some way.

Speaking of Golden Ratio, this guy plotted the Golden ratio (starting at 6:30)...

https://www.youtube.com/watch?v=sj8Sg8qnjOg

...in a way that reminded me a bit of the Riemann zeta function plotted on the opening of this video:

https://www.youtube.com/watch?v=sD0NjbwqlYw
Forgot Euler's constant...plus any others that you can think of.
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Old 2018-12-13, 22:05   #8
chalsall
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Quote:
Originally Posted by ssybesma View Post
Forgot Euler's constant...plus any others that you can think of.
To put on the table, it's relatively easy to instance a central co-ordinating server for a distributed computing effort.

The hard part is writing the software which runs on the clients' kit.

And, then, finding people motivated to run the software.

Best of luck with that.
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Old 2018-12-18, 03:21   #9
ssybesma
 
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Since my posts I gained an appreciation of just how difficult a problem it is to find a zero that is not on the critical line after tens of trillions of zeroes on the critical line were found already. I sincerely hope that Michael Atiyah is onto the answer. I read about the guy who discovered the proof for Fermat's Last Theorem, and even when he thought he had the answer, a couple months later a minor error was found and he had to repair his proof. Hope with RH that it's really getting closer. Would be so great.
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Old 2018-12-18, 04:07   #10
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It's worth pointing out that checking that no zeros have been missed is a nontrivial task, and it's not obvious at a glance that the task is solvable at all (there might be a pair of zeros very close together that you haven't noticed, for example).
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Old 2018-12-18, 13:26   #11
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Quote:
Originally Posted by CRGreathouse View Post
It's worth pointing out that checking that no zeros have been missed is a nontrivial task, and it's not obvious at a glance that the task is solvable at all (there might be a pair of zeros very close together that you haven't noticed, for example).
Pairs of zeros very close together are called Lehmer's Phenomenon in honor of Derrick Lehmer, who found the first such pair. Another (potential) headache is a zero of order (multiplicity) greater than 1. No such zero has been discovered.

In Riemann's zeta function, at the end of Chapter 7 (The Riemann-Siegel Formula), H. M. Edwards says,

Quote:
Moreover, the discoveries of Lehmer's phenomenon (Section 8.3) and of the fact that Z(t) is unbounded (Section 9.2) completely vitiate any argument based on the Riemann-Siegel formula and suggest that, unless some basic cause is operating which has eluded mathematicians for 110 years, occasional roots \rho off the line are altogether possible. In short, although Riemann's insight was stupendous it was not supernatural, and what seemed "probable" to him in 1859 might seem less so today.
Curiously, the same year as the Academic Press copyright on Edwards's book (1974), Deligne proved the third of the Weil conjectures, AKA the "Riemann hypothesis conjecture." Thus, an analog of RH in another context was proven just about the time the Academic Press edition with Edwards's statement came out.

Last fiddled with by Dr Sardonicus on 2018-12-18 at 13:28
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