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Claimed Proof of Riemann Hypothesis
An anonymous reader writes "Xian-Jin Li claims to have proven the Riemann hypothesis in this preprint on the arXiv." We've mentioned recent advances in the search for a proof but if true, I'm told this is important stuff. Me, I use math to write dirty words on my calculator.
Me, I use math to write dirty words on my calculator.
Such as 80085?
Summation 2
37047734
By using Fourier analysis on number fields, we prove in this paper E. Bombieri's refinement of A. Weil's positivity condition, which implies the Riemann hypothesis for the Riemann zeta function in the spirit of A. Connes' approach to the Riemann hypothesis. Weather permitting of course. (Just looking on the positivity side)
Man, where's Charles Eppes when you need something explained to you in layman's terms?
Do daemons dream of electric sleep()?
Was reading wikipedia because I have no idea why this is important, but need to know enough to impress my friends (and by that I mean, alienate).
But I noticed this is such a big deal, theres a cool million waiting for the person that proves it. John Nash in "beautiful Mind" tries to prove this one too. Sorry gladiator... not today!
So yeah, Check it out, notice the offer at the end, after all the completely unintelligible mathematicrap:
Riemann hypothesis
The Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous and important unsolved problems in mathematics. It has been an open question for almost 150 years, despite attracting concentrated efforts from many outstanding mathematicians. Unlike some other celebrated problems, it is more attractive to professionals in the field than to amateurs.
The Riemann hypothesis (RH) is a conjecture about the distribution of the zeros of the Riemann zeta-function (s). The Riemann zeta-function is defined for all complex numbers s 1. It has zeros at the negative even integers (i.e. at s = 2, s = 4, s = 6, ...). These are called the trivial zeros. The Riemann hypothesis is concerned with the non-trivial zeros, and states that:
The real part of any non-trivial zero of the Riemann zeta function is ½.
Thus the non-trivial zeros should lie on the so-called critical line, ½ + it, where t is a real number and i is the imaginary unit. The Riemann zeta-function along the critical line is sometimes studied in terms of the Z-function, whose real zeros correspond to the zeros of the zeta-function on the critical line.
The Riemann hypothesis is one of the most important open problems of contemporary mathematics, mainly because a large number of deep and important other results have been proven under the condition that it holds. Most mathematicians believe the Riemann hypothesis to be true.[1] A $1,000,000 prize has been offered by the Clay Mathematics Institute for the first correct proof.[2]
There's nothing Intelligent about Intelligent Design.
Reimann? Like the Noodles right?
Sounds about par for the course for academic hiring, and it sounds like he's still pretty traumatized from it. I hope this works out for him and he can go around flipping off all the hiring committees who turned him down.
What I'm listening to now on Pandora...
According to the http://en.wikipedia.org/wiki/Riemann_hypothesis wikipedia article, this means $1,000,000 if the proof turns out to be valid. Unfortunately, I didn't understand anything else in that article.
arXiv has become the repository for junk that couldn't pass peer review. Wake me up when we see a published journal article.
http://secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/RHproofs.htm
there are "proofs" of the Riemann hypothesis on the arXiv every few weeks. Don't believe it 'til it's vetted.
Part of the reason these problems are so tough because to solve them, you have to understand what the problem is first. I studied the Riemann hypothesis in college for a good week and I'm still not sure where you might begin solving it. Like the Navier-Stokes equations (another big problem with a big prize) solving it will probably require the invention of some new mathematics. Its not simply a matter of dividing by 3 and carrying the 2. I don't know about you but I haven't the slightest idea about how to go about inventing new math. That's the realm of Newton and Einstein, and few others.
New math is the only way to go about solving some of these problems.
Even those who arrange and design shrubberies are under considerable economic stress at this period in history.
http://xkcd.com/113/
Not really, the kind of person who would solve a problem of this nature is probably going to be the Andrew Wiles reclusive genius type - a lot like the Russian gent whose name escapes me who solved the Poincare Conjecture. Thus he's not necessarily going to be too keen to teach/lecture/supervise and so would possibly not be too attractive to prospective employers.
I doubt too many Maths faculties in the world have people working full-time on the Riemann Hypotheses.
Of course I echo your sentiments that his proof is almost certainly flawed though.
Solving the energy crisis is easy.
Use less energy.
Kthxbye.
First Fermat, now this. Is nothing sacred?!
Money. Not much else these days.
mcgrew's razor: Never attribute to stupidity that which can be explained by greedy self-interest
The Riemann zeta function is \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^{s}} [written for LaTeX], or "the sum of 1/(n^s) as n goes from 0 to infinity (increasing by 1 repeatedly)" [in more human-readable form].
You have a slight typo. Should be: "... as n goes from 1 to infinity ..."
Bantam Dominique roosters crow a four-note song. Once you've heard it as "Happy BIRTHday" you can't NOT hear it that way
Charles Eppes: Imagine you have an infinite number of plot holes, and you want to test how they compare to imaginary numbers. The Riemann Hypothesis states that I can use the zeros in this formula to predict how bullets will bounce off of concrete to a degree of statistical accuracy that it will actually give me the social security number of the guilty shooter.
your mother?
If you tried, you'd miss by 1/2.
Weaselmancer
rediculous.
This guys advisor, according to the Math Genealogy Project, is Louis deBranges. DeBranges also claimed to have proven this a few years back, but his proof was not accepted (for reasons unknown to me). The $1M might still be safe.
it has huge effects on prime number distribution
Prime numbers are distributed in pretty much the same way as they were before the proof.
The proof is mathematics for the sake of mathematics. The Riemann Hypothesis has been accepted as true true for over a hundred years, so practical applications that derive from it already exist.
The Continuum Hypothesis is known to be neither provable nor disprovable in the standard axiomatic set theory ZF, enriched with the axiom of choice (ZFC). So I wouldn't really count on someone settling that one either way any time soon. Of course one could come up with a new set of axioms for the set theory and *then* prove or disprove CH but you would be hardpressed to find anyone showing interest in that result. After all, I could just add CH or not(CH) to ZFC and trivially prove or disprove it. So anything in that line first needs to even define what a sensible problem is.
For those who have no clue what I said above:
Continuum hypothesis: There is no set strictly larger than the set of natural numbers and at the same time strictly smaller than the set of real numbers. The size of a set in relation to other is defined in terms of mapping. Positive integers are the same number as even numbers because you can define a bijection between the two. Reals are strictly more than naturals.
ZF: Set theory made axiomatic. Few axioms (like empty set exists, supersets are larger than original sets etc) that you need to believe and most of the set theory believed to follow.
Axiom of Choice: Given a set of sets, one can make a set containing one element from each set. Looks obviously true but in some equivalent but different sounding formulations looks obviously false. Known to be independent to ZF.
Y Independent to axioms X: Believing that Y is true does not yield contradiction together with X unless X itself yield contradictions. Same holds for believing that Y is false.
PS: Apologies for not including links. I am feeling lazy. Wikipedia has nice articles about all of the above. Articles on ZF, CH or Axiom of Choice are the place to start for a fun reading.
Section two of the wiki article (http://en.wikipedia.org/wiki/Riemann_hypothesis) is the great importance here. If indeed there is a proof of Riemann's Hypothesis, then there is a similar proof of the Generalized Riemann Hypothesis, which is in turn a big step in finding the exact distribution of prime numbers.
Finding the distribution of prime numbers has epic consequences, like breaking most encryption, for starters.
I can't believe they are brazenly going forward with research into this subject without knowing if it could possibly lead to the creation of a black hole that will swallow the earth.
Tired?
"hellhole - nice."
No, it's elohlleh, pronounced "elO'-heh-luh", which in the Primitive Quendian proto-language used by the early Elves after their awakening by Eru Ilúvatar, roughly translates to "a dreary, oppressive, or unpleasant place".
Totally different.
Dude, you owe me a monitor.
Note to self: Do not drink coke while reading /.
Do daemons dream of electric sleep()?
I just finally found a simple explanation of complex numbers, and just heard of this Riemann Hypothesis, so I may be way off, but let me try to put what (I think) I've figured out so far in layman's terms for the rest of the lost souls:
Basically, 10 trillian calculations have been done involving certain complex numbers, which all show a clear pattern: if you get an answer of 0, the real part of the number given to the function always seems to be 0.5. As yet, no one has proven this, and so, presumably, no one truly understands why that's the case yet. Also, presumably, when we do understand it, we'll have forward (either in a a step or a leap) in our ability to use complex numbers (and the multi-dimensional calculations they represent.
Or a 'variable' as it is also known.
If you let g0(x)=1/x, then the integral at the bottom of page 38 blows up to Inf.
I don't see a way to fix that, Theorem 8.6 is pretty important to this proof, and probably false. Those bits represent Li's major contribution to the problem, the rest of it is restating previous results.
"The ability to delude yourself may be an important survival tool" - Jane Wagner -
Ah well, not quite right. But let g0(x)=x works, because there's no integrability condition. Thm 8.6 then falls apart because h0 is no longer in L^2(C), or V(h) is not an operator, take your pick.
"The ability to delude yourself may be an important survival tool" - Jane Wagner -
Extremely highly-regarded mathematician Terence Tao has said, in response to a comment left on his blog, that the proof is probably incorrect.