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More on Riemann Hypothesis
Posted by
michael
on Tue Jul 02, 2002 09:35 AM
from the math-geeks-rejoice dept.
from the math-geeks-rejoice dept.
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.
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Not more attacks! (Score:4, Funny)
You can't stop these attacks (Score:5, Funny)
Even if you are able to get into a cell it can be extremely difficult to stay in and keep your sanity. Many people who do get in just sort of drift off from society and are all but lost. Those few that make it often end up working alone, late at night in the back of dimly lit coffee houses.
There is simply no way to stop someone who is willing to make such sacrifices.
Parent
Some more math humor . . . (Score:2, Funny)
Now Polly was convergent and her mother had made it an absolute condition that she must never enter such an array without her brackets on. Polly, however, who had changed her variables that morning and was feeling particularly badly behaved, ignored this condition on the grounds that it was insufficient, and made her way in amongst the complex elements.
Rows and columns enveloped her on all sides. Tangents approached her surface. She became tensor and tensor. Suddenly two branches of a hyperbola touched her at a single point. She oscillated violently, lost all sense of direction, and went completely divergent. As she reached a turning point she tripped over a square root that was protruding from the erf, and she plunged headlong down a steep gradient. When she was differentiated once more, she found herself, apparently alone, in a non-Euclidean space.
She was being watched, however. That smooth operator, Curly Pi, was lurking inner product. As he numerically analyzed her, his eyes devoured her curvilinear coordinates, and a singular expression crossed his face. Was she still convergent, he wondered. He decided to integrate improperly at once.
Hearing a common fraction behind her, Polly rotated and saw Curly approaching her with his power series expanding. She could see by his degenerate conic that he was up to no good.
"What a symmetric little polynomial you are," he said. "I can see that your angles have lots of secs."
"Oh sir," she protested, "keep away from me. I haven't got my brackets on."
"Calm yourself, my dear", said our suave operator. "Your fears are purely imaginary."
"I, i," she thought. "Perhaps he's homogeneous."
"What order are you?" the brute demanded.
"Seventeen," replied Polly.
"I suppose you've never been operated on?"
"Of course not," Polly cried indignantly. "I'm absolutely convergent."
"Come, come," said Curly. "Let's go off to a decimal place, and I'll take you to the limit!"
"Never!" gasped Polly.
"Abscissa!" he swore, using the vilest oath he knew. His patience was gone. Coshing her over the head with a log until she was powerless, Curly removed her discontinuities. He stared at her significant places and began smoothing her points of inflection. Poor Polly. She felt his hand tending to her asymptotic limit. Her convergence would soon be gone forever.
There was no mercy, for Curly was a heavyside operator. Curly's radius squared itself. Polly's loci quivered. He integrated by parts. He integrated by partial fractions. After he cofactored, he performed Runge-Kutta on her. The complex beast even went all the way around and did a contour integration. Curly went on operating until he satisfied her hypothesis, then he exponentiated and became completely orthogonal.
When Polly got home that night her mother noticed that she was no longer piecewise continuous, but had been truncated in several places. As the months went by, Polly's denominator increased monotonically. Finally she went to l'Hospital and generated a small but pathological function which left little surds all over the place and drove Polly to deviation.
The moral of the story is, "If you want to keep your expressions convergent, never allow them a single degree of freedom."
Eureka! (Score:4, Funny)
Re:Eureka! (Score:4, Funny)
Parent
Forget bigger numbers, how about smaller words? (Score:4, Interesting)
My understanding of the article is that:
A) You can't predict prime numbers.
B) That guy predicted prime numbers.
C) Alot of money goes to whoever proves how the hell he predicted prime numbers.
Ca)If we know how he predicted them we can crack old codes and make new ones?
Re:Forget bigger numbers, how about smaller words? (Score:2, Interesting)
with these sort of mathematical questions, a proof really means understanding the question. The question really is just so many words ("where are the zeroes on the critical line?") but the reason a proof is important is that it tells you why the question was worth asking in the first place.
Phrasing the question in english just doesnt get the point across.... thats what makes mathematics so incredible. take the time to read up on it, even if you dont feel so confident with the maths, it is well worth it.
Other things anyone who really knows can do.... (Score:3, Funny)
Explain sound to the deaf.
Explain intuitive leaps of any kind.
Not every concept maps to a clean explanation in a few simple words. That's why we have the different words. True, most concepts can be mapped somewhat to common language, but come on...give the guy a fucking break. We're talking about advanced mathematics.
Get off YOUR high horse, bubby.
Re:Could you get a bit more arrogant please? (Score:5, Informative)
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
Parent
Re:Forget bigger numbers, how about smaller words? (Score:4, Informative)
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.)
Parent
Re:Forget bigger numbers, how about smaller words? (Score:2)
(1) It's an "on the average" relationship, not an exact relationship - that is, if it says there are 100 primes in a long interval it may be off by a few percent, but if it says there's 1 prime in a short interval it may be off by +/- 1. So it's no good for the bisection search.
(2) The Zeta function [wolfram.com] is not that easy to compute anyhow.
Re:Forget bigger numbers, how about smaller words? (Score:3, Informative)
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.
Re:Here it is in small words (Score:5, Informative)
Parent
Re:Here it is in small words (Score:2)
And they say _I_ suck at math.
;)
Re:Here it is in small words (Score:2, Redundant)
They don't all fall on the *same* line, though.
Re:Here it is in small words (Score:2)
Re:Here it is in small words (Score:5, Informative)
The function had already been discussed by Euler.
For some reason, primes always plot along one of the axes. No one can figure out why.
Actually, that's easy. Primes (at least over the integers) are real numbers, and the zeta function maps real numbers greater than one to real numbers, which is evident from the definition as a Dirichlet series.
Quite a few proofs in analytical number theory rely on the fact that in certain areas on the right side of the line {z : Re z = 1/2} contain no zeroes of the zeta function. So far, mathematicians have tried to carefully choose these areas in order to get good results (so that you can still use them efficiently, but you can also prove that no zeroes lie in it). If we knew that no such zeroes exist at all (the Riemann Hypothesis), we could avoid all these rather technical details and theorems would improve considerably as well (for example, the error term in the Prime Number Theorem).
Parent
Rainman Hypothesis? (Score:2)
Proofs delicate? (Score:2, Interesting)
Wha-wha? I was under the impression that proofs are rock-solid demonstrations of a particular fact given a set of well-defined mathematical laws . . .
Re:Proofs delicate? (Score:4, Insightful)
Think about it in terms of spacecraft. A couple of vehicles were perfect and landed on Mars. One had a small defect, it wasn't complete (meters and miles were mixed up). It was lost.
Parent
Re:Proofs delicate? (Score:2)
But it's easy to prove... (Score:3, Funny)
Full Circle (Score:2, Interesting)
Someone here is clueless, but whom? (Score:2, Informative)
There is no simple way to tell if a number is prime, and that is the basis for most modern encryption schemes. Solving the hypothesis could lead to new encryption schemes and possibly provide tools that would make existing schemes, which depend on the properties of prime numbers, more vulnerable.
AFAIK, modern PK encryption depends on either the RSAP (RSA problem), related to the IFP (integer factorization problem), or the DHP (Diffie-Hellman problem), related to the DLP (discrete logarithm problem). (Then there are elliptic curves, but those aren't used much except in some proprietary systems; they haven't been studied as much and therefore aren't considered as trustworthy.)
I fail to see how breakthroughs in prime distribution theory would affect either the IFP or the DLP, or lead to new cryptosystems. So, am I clueless, or has the NYT done it again?
It's the Times... (Score:5, Informative)
...and your description of the mathematical basis for modern encryption is essentially correct. One could argue that there is a relationship between finding out if a number is prime and determining its prime factors. But such a relationship has so far eluded mathematicians.
The statement "there is no simple way to tell if a number is prime" is true only for a limited definition of "simple way." The impossibly complex ways of the past have been replaced with complex-but-definitely-possible techniques, which would definitely make encryption vulnerable if encryption depended on the properties of prime numbers. Which it doesn't. (As you correctly point out.)
Another quote from the article (paraphrased by the submitter), is also erroneous unless something has changed:
I believe the Clay Mathematics Institute award specifically excludes refutation. (Presumably because someone could refute the hypothesis simply by stumbling onto a single counter-example: a zeta-function zero which does not lie near the complex axis. This would be the mathematical equivalent to hitting the lottery and might do little to advance mathematics.)
Parent
Re:It's the Times... (Score:2)
I would argue that it would do much to enhance mathematics, if for no other reason than by freeing up the time of all of the good mathematical minds who are trying to figure out the proof.
When I finsh my Linux Xbox port (Score:2, Funny)
A proof that is worth millions to MAN kind (Score:2, Funny)
Keep in mind this proof looks much better if you can actually use the square root symbol
The problem:
Prove that women are all evil.
(With written proof, men don't have to worry about women arguing this fact anymore
The proof:
Given that:
Proceede with the proof:
See what an undergrad in Mathematics, an undergrad in C.S., and a Master's in C.S. gets you
Seriously, I wish someone could prove that P=NP. I hated graduate Algorithms! This would have eliminated a portion of my least favorite topic in that course (NP and NP-completeness). If this world is not truely hell, someone will prove that and share it to help prevent the suffering of innocent C.S. graduate students.
Re:A proof that is worth millions to MAN kind (Score:2)
Lea
Re:A proof that is worth millions to MAN kind (Score:3, Informative)
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?
ZetaGrid (Score:5, Informative)
Apparently there's a distributed computing project called ZetaGrid [hipilib.de] 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 [wolfram.com]
Riemann Zeta Function [wolfram.com]
Also, there's some rather technical details on the subject, from Stephen Wolfram's (A New Kind of Science) pet site.
I am confident... (Score:3, Funny)
Either that, or you can solve them by buying REAL ESTATE with NO MONEY DOWN! or by placing SMALL ADS in NEWSPAPERS with your own 900 NUMBER!!!!!
Good intro... (Score:5, Funny)
That has to be the funniest things I've read, today.
Is it me or does it seem that all "hard" mathematicians are either at war with God or trying to "refute"/"prove"/divide/discover/humiliate him/her/it/Taco?
Here's some background info... (Score:2, Redundant)
Riemann Hypothesis [wolfram.com]
harmony (Score:3, Interesting)
Of course primes have a generally log distribution, because every prime you find provides a factor later on down the line so the primes become more sparse.
Then there's the atoms thing, sfaik shells/energy levels are basically harmonic and a harmonic is more-or-less the opposite of a prime.
since harmonics and the increasing sparseness of primes could be taken as identical you're going to get the same distribution patterns out.
here goes
primes v harmonics
2 is prime and a harmonic root
3 is prime and a discord (root)
4 is non prime, and the second octave of the first root
5 is prime and a discord (root)
6 is non prime, and cord of the first and second roots
7 is prime and a discord (root)
8 is non prime, and third octive of the first root
9 is non prime, and first octave of the second root
etc....
Re:Who stands a better chance? (Score:2, Insightful)
specifically, i'd place my bets on the smelliest and most Russian of them.
Re:Who stands a better chance? (Score:4, Interesting)
The mathematician stands a better chance of proving the hypothesis, but the NSA supercomputer stands a better chance of refuting the hyposthesis.
With current technology, it's extremely unlikely that the mathematician would refute the hypothesis or the computer might prove it (although it is possible).
Finally, props goes out to the Courant Institute of Mathematical Sciences. The best, my favorite, and my current graduate school (@ nyu).
Parent
Re:Who stands a better chance? (Score:2, Insightful)
With current technology it is extremely unlikely, although possible, that a mathematician would refute the hypothesis, but it is impossible for a computer to prove it.
While a mathematician can't try some random still untested number and hope to get the "right" one (all of the "small" numbers have been tried), he could always try and build some "special" class of numbers that could refuse the hypothesis and test those, or he could find some logical contradiction etc.
On the other side, with current technology computers can only try more and more cases, so that if the hypothesis is false they eventually find an example, but they just can't try every number (not in a finite time :) ), nor actually prove a theorem by logical means.
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, so it will be quite some time before those could be available, if ever.
Re:Who stands a better chance? (Score:3, Informative)
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 [google.com], and people are working on them [google.com].
Re:Who stands a better chance? (Score:2, Informative)
I am an idiot. Soy estupido. (Score:2, Troll)
I am totally wrong.
But no one would care about that (Score:3)
If a computer disproves it by finding a prime that happens to map wrong on the zeta theorem, mathemeticians will still want to know why this one didn't work, when all the others have.
BTW You have also determined a relative probability -- "better chance" -- of something that may be undefined. If the theorem is in fact true, then a computer's chance of disproving it is exactly equal to a mathemetician's chances: zero.
Re:Why so much money? (Score:2)
Re:Why so much money? (Score:2, Funny)
You'd think they would target people who are good at math.
Re:Log in blues? (Score:4, Interesting)
I keep a spreadsheet of user info I give when I register with various sites. All the data is serialized in a way which I can instantly identify which source the data came from. Its pretty effective for tracking down spam.
I've never had any spam directly from NYT, but they have clearly sold the information to 3rd parties, and those 3rd parties have probably sold it again and again.
I used two methods to verify it came from NYT - the e-mail address was a numbered account made on my own mail server, and my last named had a serial number appended. Both elements are present in the same I've traced to them.
The spam has come from:
- company trying to sell me HGH
- company trying to sell me diploma's
- company trying to sell me financial services
So just because they don't spam you doesn't mean that haven't enabled someone else to do it.
Parent
Re:Log in blues? (Score:2)
Re:Log in blues? (Score:2)
TRUSTe doing something about a privacy violation?
Score that (+1, Funny)
Re:Log in blues? (Score:2)
Wake up kids...
And don't bother reading the "privacy policy" anymore... sites just do things anyways.
Re:Let's not be too hasty (Score:4, Informative)
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?
Parent
Re:If we're so smart... (Score:2)
Wrong. Computers are very useful when a theorem needs to be checked for a number of cases that is finite but very large, in the millions or billions, and each check is tedious to do by hand but very easy to do programatically. The four-color map theorem was proved by a computer this way in 1976, but a lot of mathematicians had a hard time accepting it because it would take so long to check and verify thousands of graphs manually. Which really isn't necessary if you admit the software they used into the proof, but there was some resistance to the possible consequences of allowing computers and computer programs into a mathematical proof. And math would get pretty boring if all anyone did was smash all the interesting problems with brute-force trial and error. And the proof generated by checking all the thousands of possible graphs seems almost like cheating- it doesn't really give you any insight into why maps only need four colors, does it? It doesn't explain the reason in the way that you'd imagine a nice one-page proof would. It proves that all maps only need four colors, and that's it. But it's still a valid mathematical proof.