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More on Riemann Hypothesis
Anonymous Coward writes "The NYTimes has a little story on a recent conference at New York University's Courant Institute where mathematicians gathered to discuss potential attacks on the Riemann hypothesis. The Clay Mathematics Institute had announced an award of a million dollars for a proof (or refutation) of the Riemann hypothesis during the millenial celebrations. That million dollars won't be worth much if it takes as long as that Last Theorem by Fermat to solve. There were some interesting observations such as the statistical distribution of the zeros looked just like calculations on the energy levels of large atoms." We did a related story on hard math problems two years ago.
He wrote a function called the "zeta function." Any number, when fed into this equation, will yield a result somewhere on a plane. For some reason, primes always plot along one of the axes. No one can figure out why.
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A good example of a computer proving a hypothesis, with a great deal of human help, of course, is the map coloring problem. The current best-case proof that the minimum number of colors required to color any map is four utilizes a brute-force approach where the solution space is broken down into a finite (but large) number of possibilities that the computer can then attack individually.
Apparently there's a distributed computing project called ZetaGrid which has calculated the first 50 billion zeros out ... if you're bored of SETI@Home, this might be a nice change of pace.
Riemann Hypothesis
Riemann Zeta Function
Also, there's some rather technical details on the subject, from Stephen Wolfram's (A New Kind of Science) pet site.
We can neither love nor pity nor forgive. If you make a slip in handling us you die!
First off, not being able to prove or disprove something doesn't mean it's not true or untrue, just that one can't prove it either way. Incompleteness specficially means that there are true statements in the system that can't be proven or derived in the system. It doesn't mean that "not everything has to necessarily be true or untrue."
... the equivalent of "this statement is unprovable" (if you prove it's true, you've contradicted yourself. if you can't prove it's true, then it's true, but you're not able to prove it so it's incomplete). The overwhelming majority of mathematics is complete and consistant, and there's no reason to expect it not to be and give up prematurely.
Secondly, iirc, Gödel showed that sufficiently complex systems have to either be inconsistant or incomplete using a very specific paradox
Finally, who's being "hasty"? What exactly are you suggesting? That they give up the search for a proof because there's a tiny chance that it may be unprovable? Why doesn't the entire field of theoretical math just stop right now, then?
The following sentence is true. The preceding sentence was false.
A) You can't predict prime numbers.
B) That guy predicted prime numbers
Riemann discovered a function that reasonably well matches the number of primes found within long intervals of numbers. It can't find primes per se, it just predicts how many you'll find between 'm' and 'n'. And it's no help for factoring a product of two primes, so it won't crack codes.
Of course, winning the prize (by taking Riemann's work a few steps further) might happen to suggest a method for factoring a product of primes, but it's more likely it will be of interest only to those few mathematicians that can remember what Riemann's hypothesis was in the first place. (I used to know but no longer remember, and that d!@#d article didn't give an equation or otherwise say anything really useful.)
With current technology [...] it is impossible for a computer to prove it.
No. It has been theoretically possible for computers to solve mathematical proofs ever since the first Turing-esque computers (the only missing element being "infinite" storage capacity) were built. And if a proof of Riemann requires more than a terabyte of statements and reasoning, then it's also beyond the capabilities of human mathematicians.
I know that somebody is researching some theorem-proving capable AI, but it seems that they didn't succeed yet in proving whether it can exist or not
They can exist, and people are working on them.
Almost...
A) You can't predict prime numbers
B) That guy came up with a formula that SEEMS to predict prime numbers
C) A lot of money goes to whoever proves that the formula REALLY DOES predict prime numbers
Just because a formula SEEMS to work for every number you try doesn't mean that it REALLy DOES work for all numbers. The classic example
All numbers are less than 43 billion.
I call this "The Rinker Hypothesis"
Is it true? It seems to be..I tried 1, 2 & 3 and it worked. I tried every number up to one million and it worked. In fact, I tested the hypothesis for every number up to one billion and it was true for all those numbers. This example is rather trivial and silly, but it demonstrates the point: simply because a mathematical hypothesis (aka a conjecture) is true for every number you try doesn't mean that it's true for ALL NUMBERS.
Riemann's hypothesis seems true for every number they try, but they haven't proved that it's true for ALL NUMBERS.
If you can't explain something in ordinary words to a layman, then you really don't understand it.
:)
... on the horizontal axis.
:)
I'm about halfway through writing up my PhD thesis on some applications of homological algebra to knot theory and low-dimensional geometric topology (provisional title liber rerum dementiae, but it'll probably end up being called something more mathematically appropriate).
In principle, yes, I could explain the details of my research to a suitably motivated layman. But I suspect it would take rather a long time.
You see, and this really isn't meant to sound arrogant, supercilious, or dismissive, university-level mathematics is pretty damned difficult, and the details of most cutting-edge research really doesn't make sense until you've spent several years learning the background (the mindset, the language, the fundamental concepts).
My current area of research is essentially the applications of homological algebra to knot theory and low-dimensional geometric topology. To explain this to a non-mathematician, I'd first have to teach them a lot of background stuff (group theory, a bit of stuff about rings and modules, point set topology, basic algebraic topology (the fundamental group, (co)homology theory), some geometric topology (basic course in knot theory, some stuff about 3-manifolds), a bit of category theory, and some homological algebra (broad overview of the (co)homology theory of groups and algebras)).
It's taken me nearly nine years (3-year BA, 1-year MSc specialising in topology and knot theory, plus nearly five years doing a (part-time) PhD) to get to this point myself. If I were a bit cleverer (or didn't have a `proper' job as well) I might have been able to shave a couple of years off that.
My friend Steve has a physics degree. I managed, in ten minutes one evening, with much handwaving, to give him some idea of what my thesis is all about. It helped that he knew what a group was already though. But for me to explain it fully to him would probably necessitate him doing at least one mathematics degree first. And that's not really something I'd wish on one of my friends
Now this really isn't meant in an arrogant way, and I hope you won't read it like that, but Euclid was right: There is no royal road to geometry.
I can have a go at explaining the Riemann hypothesis, though. To fully understand what it's about and why it's so damned difficult you'll need to do an advanced course in complex analysis (which isn't my field either).
A complex number is a sort of two-dimensional number, which you can regard as a point in a plane (the `complex plane' or `argand diagram'). You add them together coordinate-wise, and you multiply them together in a weird manner which involves something which behaves like a `square root of -1' (engineers also like to think of it as a sort of 90-degree phase-shift operator, I'm told).
There's a particular function (`Riemann's zeta function') defined on the complex plane (it takes one complex number as input and returns one complex number). For some complex numbers (`the zeros of the function'), the value of this function is zero.
The `trivial' zeros occur at the points -2, -4, -6,
The `non-trivial' zeros (that is, all the other points for which zeta is zero) all seem to occur on the line parallel to the vertical axis that intersects the horizontal axis at +0.5. Indeed, nobody's ever found one which doesn't.
The Riemann Hypothesis is that *all* the non-trivial zeros lie on this line. It's known to be true for the first (large number which temporarily escapes me), but it turns out to be phenomenally difficult to prove that it's true in every case.
Now that's the basic idea, but it doesn't (and I can't - it's not my field) explain *why* it's so difficult that some of the greatest minds (Hardy, Littlewood, Ramanujan, etc) of the past 150 years have failed to prove it, and why the Clay institute are willing to pay a million dollars to someone who can.
- nicholas (we don't just sit around doing big sums, you know
>You do that in Graduate School? Up here in Canada that is second year undergrad material.
;)
If your school is CIPS (Canadian Information Processing Society) accredited... which just about every University CS program is... I would be somewhat suspicious of this claim.
You may be confusing "Analysis of Algorithms" with "Complexity Theory" which are different (though of course, related) things. Yes, most programmes give an introduction to P vs. NP in second year, but I would be surprised if you are doing serious complexity theory simply because a 2nd year CS undergrad just doesn't have the mathematical tools to do this yet (not to mention that with the CIPS cirriculum requirements.. there isn't anywhere to *put* courses to aquire said background).
That being said: Prove me wrong. What school are you at, and are they hiring?
Li(x) is defined as the integral from 0 to x of (1/t) dt. And apparently the number of primes below x (usually denoted pi(x)) is pretty well approximated by Li(x).
Not quite: note the obvious initial logarithmic divergence. Informally, you can just change the integrand from what you had to dt/(log t), but you really ought to work around the singularity at 1. Some people change the bounds of integration to start at 2 to avoid this. It simply shifts the function by a small constant (about 1.05)
I'm a bit surprised no one here has mentioned Pierre Deligne's 1974 proof of the Weil conjectures, in particular the analogue of RH for smooth projective varieties over finite fields (for which he was awarded a Fields medal in 1978). This is perhaps the strongest "evidence" for the original hypothesis (unless you find the brute force calculation convincing), and it has other interesting consequences, for example the resolution of Ramanujan's tau conjecture (ref: Hartshorne's Algebraic Geometry).
There is a nice discussion of potential avenues of attack on the Riemann Hypothesis at the end of chapter 5 in Patterson's text on the Zeta Function (Cambridge Studies 14), including some vague ideas on why a purely analytic strategy is not likely to be successful.
"Your notation sucks!" -- Serge Lang (1927-2005)