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Topic: Mathematician claims proof for Riemann hypothesis (Read 7473 times) |
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Barukh
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Re: Mathematician claims proof for Riemann hypothe
« Reply #1 on: Jun 10th, 2004, 12:07am » |
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Very intriguing!
Is there anybody at this forum with the necessary background to verify the proof? 
I don't understand why "..His latest efforts have neither been peer reviewed nor accepted for publication..."?
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Sir Col
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Re: Mathematician claims proof for Riemann hypothe
« Reply #2 on: Jun 10th, 2004, 12:53am » |
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on Jun 10th, 2004, 12:07am, Barukh wrote:| Is there anybody at this forum with the necessary background to verify the proof? |
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I'm almost there, but could someone tell me what the zeta and gamma functions are? 
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ThudnBlunder
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Re: Mathematician claims proof for Riemann hypothe
« Reply #3 on: Jun 10th, 2004, 2:48am » |
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According to another mathematician, it is not the first time that he has claimed to have proved it.
Check out this page which shows that in the last 20 years or so he has wangled himself more than $400,000 in funding in order to prove, disprove, or fail to prove RH. Nice work if you can get it. The following self-fulfilling claim, taken from one of his abstracts, might be the way for us laggards to go:
"The solution of the Riemann Hypothesis is indeed the Holy Grail of analysis, and the principal investigator is by most accounts the most capable person in the world to attempt a solution."
on Jun 10th, 2004, 12:07am, Barukh wrote:
I don't understand why "..His latest efforts have neither been peer reviewed nor accepted for publication..."? |
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(And I don't understand why - if he is gainfully employed full-time by Purdue University in order to teach and do research with pen and paper - a financial award is necessary. Indeed, how has this 400 grand aided him in his heretofore fruitless search?)
"While mathematicians ordinarily announce their work at formal conferences or in scientific journals, the spirited competition to prove the hypothesis -- which carries a $1 million prize for whomever accomplishes it first -- has encouraged de Branges to announce his work as soon as it was completed."
''I invite other mathematicians to examine my efforts,'' said de Branges, who is the Edward C. Elliott Distinguished Professor of Mathematics in Purdue's School of Science. ''While I will eventually submit my proof for formal publication, due to the circumstances I felt it necessary to post the work on the Internet immediately.''
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Barukh
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Re: Mathematician claims proof for Riemann hypothe
« Reply #4 on: Jun 10th, 2004, 4:41am » |
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You may want also look here (Introduction will suffice) for a critique of earlier versions of his proof.
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Sir Col
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Re: Mathematician claims proof for Riemann hypothe
« Reply #5 on: Jun 12th, 2004, 2:58am » |
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I just noticed an interesting reference to de Branges' latest paper here:
http://mathworld.wolfram.com/
Riemann Hypothesis "Proof" Much Ado About Nothing
A June 8 Purdue University news release reports a proof of the Riemann Hypothesis by L. de Branges. However, both the 23-page preprint cited in the release (which is actually from 2003) and a longer preprint from 2004 on de Branges's home page seem to lack an actual proof. Furthermore, a counterexample to de Branges's approach due to Conrey and Li has been known since 1998. The media coverage therefore appears to be much ado about nothing.
[copied in case the item is removed]
What was interesting, and unknown to me also, was the iminent victory over the twin prime conjecture (also found on the same page above):
Twin Prime Proof Proffered
A May 26 preprint by Vanderbilt University mathematician R. F. Arenstorf appears to come close to settling the longstanding question of the infinitude of twin primes. While a hole has recently been found in the proof, mathematicians remain hopeful that the proof can be fixed up as needed.
[Article: http://mathworld.wolfram.com/news/2004-06-09/twinprimes/]
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Michael
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Re: Mathematician claims proof for Riemann hypothe
« Reply #6 on: Aug 29th, 2004, 7:47am » |
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What? - "I'm almost there, but could someone tell me what the zeta and gamma functions are?"
If you are not familiar with zeta and gamma functions you are clearly not almost there.
Michael Dagg
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Speaker
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Re: Mathematician claims proof for Riemann hypothe
« Reply #8 on: Sep 15th, 2004, 11:36pm » |
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Ha ha. Towr, I clicked on Joke, it was a joke. 
Somedays, I'll click on anything.
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Ted
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Any one on this thread good at answering questions/problems in abstract algebra?
Thanks,
Ted
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Michael Dagg
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What is the algebra question?
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Ken Wiley
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Re: Mathematician claims proof for Riemann hypothe
« Reply #11 on: Jan 29th, 2005, 4:27pm » |
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Just curious, but are you the same Michael Dagg that is known for having written all the ODE solvers?
Ken
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Michael Dagg
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Re: Mathematician claims proof for Riemann hypothe
« Reply #12 on: Feb 2nd, 2005, 9:31pm » |
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I have wrote many but certainly not all.
Regards,
MD
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Ken Wiley
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Re: Mathematician claims proof for Riemann hypothe
« Reply #13 on: Feb 6th, 2005, 11:09am » |
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Interesting. That Trench ODE package that you wrote can be found where? What language did you write it in? I see it on the internet in various places such as netlib but I can't find the right link any more.
My experience with ODE solvers has been hot but cold mostly cold when it comes to trying to make an adaptive one that works good. Most of the good solvers have so many input parameters that it becomes confusing as to how to use them as well as what some of the parameters really mean, even when there are good descriptions.
I worked on an adaptive solver for a while in C what used Euler multistep method combined with a Runge-Kutta 5 but it did not perform well at all even when I tricked out the step size and the number of max iterations > 100 for the adaptive bailout.
It would mostly go bannas in all but the simple ODEs. My choice in picking the Euler method was to reduce the computation time of say a Runge-Kutta 4 and then if the error tols looked good it would use the Runge-Kutta 4 value as the solution.
Out of the box, what would be the best general purpose ODE solver? - I have heard of DVERK and EPSODE but never used them.
Ken
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Michael Dagg
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Re: Mathematician claims proof for Riemann hypothe
« Reply #14 on: Feb 10th, 2005, 3:12pm » |
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The Trench ODE software is packaged with the students solutions guide to Trench's text (Brooks/Cole) - wrote in Delphi. I don't manage any links to it - that is for Brooks/Cole to deal with.
Be sure that Euler's method is not multistep. You might be thinking of Heun's method which is but is a PC method that uses Euler as the predictor and the trapezoidal rule as the corrector;it will also go bananas on you with just a mildly nonlinear DE and will likely jump to another trajectory in a
neighborhood of a singular point.
It is clear why your adaptive solver does not work very well. I assume that you mean you are using Euler with a RK4 or RK5 but not both. Using Euler with a RK4 or RK5 in an adaptive scheme is like using a sledgehammer(Euler) and an axe(RK) in a log cutting study if you are not just trying to break the skin. The regions of stability don't line up very well. Another way to say it is that their measure spaces will quickly become disjoint.
Using an ODE solver is a completely different art form than writing your own. Most all non-trivial solvers have a fair set of calling parameters and if you are not familiar with
the descriptions be sure to study the method so you will be able to accurately interpret the results. Most all professionally written solvers are accompanied by driver routines that will give you some clues as to the calling parameters and how to
use them - read up on the material by Lawrence Shampine.
Finally, if some extra complexity is permissible for non-stiff ODEs then, out of the box, use a BDF method or some other implicit scheme. DVERK is a RK56 and is OK for non-stiff problems. EPSODE is a driver for the EPISODE package by Hindmarsh & Byrne. There is a BDF method in EPISODE. There is a java device at the bottom of this page that lets you fiddle with some BDF methods http://www.cse.uiuc.edu/eot/modules/ode/bdf/
Regards,
MD
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Sir Col
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Re: Mathematician claims proof for Riemann hypothe
« Reply #15 on: Feb 10th, 2005, 4:34pm » |
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I find the language of odes convoluted by the complex and formal stanzaic structure. Actually I used to think that they were all Greek.
Don't take this the wrong way, Michael, as I'm sure what you've written is very good, but in terms of more recent odes you cannot beat the works of Keats and Coleridge.
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Michael Dagg
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Re: Mathematician claims proof for Riemann hypothe
« Reply #16 on: Feb 11th, 2005, 10:29am » |
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It is worth mentioning that Edwin Moise made a similar statement to me about odes back in the late 1980's.
It was mystified when I read it but I do agree.
Regards,
MD
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Ken Wiley
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Re: Mathematician claims proof for Riemann hypothe
« Reply #17 on: Mar 5th, 2005, 9:56am » |
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Your response is very much appreciated - sorry for my delay in checking this board!
I am not surprised that the solver problem is based on my own poor selection and I can now actually say I understand why it did not work: basically, I am using a very less accurate method in conjunction with a more accurate method. This makes
complete sense to me now and I certainly see that most all adaptive solvers use methods that are 1 or so orders a part.
But what gets me about what you said about possiblity of the Heun method jumping to another trajectory really puzzles me because [the] initial value problem, if there is a solution, it is _unique_. Therefore, if it is unique how could the solver jump to
another trajectory?
Also, about some of the methods you mentioned, don't they require computation of some partial derivatives?
Ken
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Michael Dagg
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Re: Mathematician claims proof for Riemann hypothe
« Reply #18 on: Mar 15th, 2005, 11:16am » |
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It is the very nature of numeric approximation that leads to potential trajectory-jumping and has nothing to do with the uniqueness theorem except that when it happens we know something went wrong.
A solver can jump to another trajectory for any number of reasons which may include loss of stability, floating point error, taking too large of a leap, and other things that have to
do with the nature of the ODE and the solver itself.
A non-adaptive solver will jump to another trajectory around a singular point or an asymptote most of the time if the family of trajectories extend to the other sides of the singular point or asymptote and will do so all of the time if the code does not make any attempt to detect singular points or floating point errors between successive increments/decrements of time - for example, by detecting one or more divisions by
zero, numeric over/under-flows, undefined operations, etc. in the time interval. Numeric over/under-flow is treated as a special case since neither implies division by zero or discontinuity but could indicate that there is a division by zero or asymptote nearby or just simply the computation requires floating point result that is out of range.
Adaptive solvers are less prone to trajectory-jumping for numerous reasons but the main reason is that they have some sort of error control implemented such that if the error
constraint is not satisfied after a number of iterations for one reason or the other, the solver terminates. The nature of symmetry, if any, of a family of trajectories about a singular point plays an interesting role in trajectory-jumping.
Run your favorite RK45 on these two ODEs y' = -x/y and y' = x/y and see if you can get it to do some trajectory-jumping in some disc containing (0,0). One of the ODEs is more
prone to trajectory-jumping than the other but notice that there does not appear to be much difference between the two ODEs except for the minus sign.
Finally, yes, methods that use the Jacobian use partial derivatives. Most codes numerically approximate the Jacobian but codes that contain a symbolic differentiation algorithm can compute an analytic Jacobian which reduces the successive numeric approximations to successive function
evaluations instead.
Regards,
MD
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uMRod
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Re: Mathematician claims proof for Riemann hypothe
« Reply #19 on: Mar 17th, 2005, 8:01pm » |
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I have been watching this thread about diff equ's and the question that I have is that how can you use a Runge-Kutta with implicit autonomous diffs such as y'=-x/y or y'=x/y?
Also, doesn't solution = tracjectory in this same sense?
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Michael Dagg
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Re: Mathematician claims proof for Riemann hypothe
« Reply #20 on: Mar 19th, 2005, 10:50am » |
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Let me first mention that neither of the two ODEs are autonomous. This is so because x appears in each.
RK methods can be applied to explicit and implicit ODEs; and, in fact, can be used on a wide variety of equations involving derivatives such as differential algebraic equations and many others.
I am guessing that you probably thought that since the two ODEs are implicit, perhaps only an implicit solver could be used to solve them. Turns out that explicit and implicit solvers can be used to solve explicit and implicit ODEs. The reality is that the choice depends the nature of the ODE. For example, implicit solvers are superior for solving stiff ODEs while for many non-stiff ODEs explicit solvers generally work pretty well.
Yes, solution curve = trajectory.
Regards,
MD
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ThudnBlunder
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Re: Mathematician claims proof for Riemann hypothe
« Reply #21 on: Mar 19th, 2005, 1:04pm » |
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on Mar 17th, 2005, 8:01pm, uMRod wrote:| I have been watching this thread about diff equ's |
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Hmm...is that what this thread is about?
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uMRod
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Re: Mathematician claims proof for Riemann hypothe
« Reply #22 on: Mar 26th, 2005, 8:36am » |
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I see. What about coming up with a power series equation for the solution of dy/dx = xy, y(0)=1? One would think that it is easy.
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Ghost9155
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Re: Mathematician claims proof for Riemann hypothe
« Reply #23 on: Mar 28th, 2005, 6:27am » |
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Let y = sum_j=1_to_inf [aj*x^j] and substitute it for y in the right hand side of the diff equ and then
differentiation it with respect to x and set it as the left side of the diff equ. Combine the terms
in the two sums into one sum by shifting the index of summation. You can then determine the form of
the coefficients for each power of x in the sum by using the initial conditions.
They should involve powers of numbers and factorials.
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Icarus
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Re: Mathematician claims proof for Riemann hypothe
« Reply #24 on: Mar 28th, 2005, 7:36pm » |
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Actually, Ghost9155, its even easier to simply solve this diffE: divide both sides by y, then integrate to get ln(y) = x2/2 + C. Exponentiating and evaluating for y(0)=1 gives
y = ex^2/2 = [sum] x2n/2nn!
But I thing uMRod was only giving this as an example in his on-going discussion with Michael Dagg, rather than offering it as a proplem he was having trouble solving.
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