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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.
We're already being searched at airports, now mathematicians can't carry a protractor or a compass without being looked as being suspicious. When will terrorists learn that attacking math problems never solves anything. Wait, maybe it does...
I have discovered a truly remarkable proof which this post is too small to contain.
Can someone explain exactly what this is and what it means in very small words?
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?
Never confuse volume with power.
seeing as an NSA supercomputer could only refute the hypothesis, and out of the billions of numbers it's already tried there have been exactly zero refutations, i'd put my money on the mathematicians.
specifically, i'd place my bets on the smelliest and most Russian of them.
Cretin - a powerful and flexible CD reencoder
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).
For a second there I thought that said 'Rainman Hypothesis.' Somethine to do with counting cards maybe?
Wrong. Primes do not always plot along one of the axes. Zeroes to the function are always (well, that's the hypothesis) of the form 1/2 + bi. This means they lie on a line parallel to the imaginary axis.
Fault loves the past, worry loves the future, but content enjoys the present.
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 . . .
Well that doesn't make sense... EVERY point lies on a line parallel to an axis. In fact, for each axis there exists a line parallel to that axis that passes through your point.
And they say _I_ suck at math.
;)
-- Minds are like parachutes... they work best when open.
EVERY point lies on a line parallel to an axis
They don't all fall on the *same* line, though.
-- Give me ambiguity or give me something else!
http://www.bearnol.pwp.blueyonder.co.uk/Math/riema nn.htm
Yes, but they don't all lie on the same line.
Patrick Doyle
I mod down every jackass who puts his moderation policy in his sig. Oh, wait a sec....
Man, you must be old, because people have been offering rewards for proofs for hundreds of years.
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.
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.
The mathematician stands a better chance of proving the hypothesis, but the NSA supercomputer stands a better chance of refuting the hyposthesis.
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.
Nope, no sig
Explain sight to the blind.
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.
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.
HallmarkOrnaments.Com
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!
Comment removed based on user account deletion
Yeah, I'm sure they didn't give it out either!
Wake up kids...
And don't bother reading the "privacy policy" anymore... sites just do things anyways.
Get your Unix fortune now!
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.
...that these proofs will not be solved using conventional methods, but they will eventually be solved using SMALL PROGRAMS with SIMPLE RULES. These rules can be run on a simple computer using my program, Mathematica. Easy!
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!!!!!
"I'm an old-fashioned type of guy. I worship the Sun and Moon as gods. And fear them."
Please ignore me. I haven't had my coffee yet. I don't know what I'm saying.
I am totally wrong.
"that God -- with whom he waged a very personal war -- would not let Hardy die with such glory."
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?
Get your Unix fortune now!
He wrote a function called the "zeta function."
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).
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.
... on the Riemann Hypothesis:
Riemann Hypothesis
"I have never let my schooling interfere with my education." - Mark Twain
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.
Or maybe they are targetting all the rich quants who were just laid off On Wall Street.
You'd think they would target people who are good at math.
TRUSTe doing something about a privacy violation?
Score that (+1, Funny)
Once upon a time (1/t), pretty little Polly Nomial was strolling across a field of vectors when she came to the edge of a singularly large matrix.
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."
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
I agree, and have come to the conclusion by a similar method. Whenever I register on a website, I use an e-mail address of "domainname@mydomain.com" Not only does this tell me who is spamming me, but also who has sold my addresses to whom (or who has allowed it to be pilfered.) The nytimes address had to be pointed to the bitbucket about a year ago.
-- Minds are like parachutes... they work best when open.
Well reading thought the article, they seem to miss? a few things.
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....
thank God the internet isn't a human right.
I'm only on chapter 4 of Wolfram's opus 'A New Kind of Science' but reading about the Riemann Hypothesis just screams out connections with Wolfram's work. ANKOS is littered with these odd little diagrams of cellular automata, many of which exhibit prime number relationships.
You can't say "... it turns out to be phenomenally difficult to prove ..."!
It happens that no-one has managed it yet.
If someone does manage it, perhaps it will seem simple.
Even if it doesn't, will it be possible to prove that it is the simplest possible proof?
Your logic is seriously faulty. The Riemann Hypothesis may never be proved whereas there is a non-zero quantity of >=$1M contracts floating around at all times.
To condense njj's excellent post on the topic, if you can explain a concept to a layman in such a way that they actually understand it, then virtually by definition, the concept was not advanced, as it did not require extensive pre-education.
However, advanced concepts exist. Therefore, there exist concepts which by definition can not be adequately explained to a layman.
In fact, sir, the arrogance is yours. You are arrogant in assuming that it's possible for a layman, i.e., you, to understand everything, and that if you don't understand it, the fault must lie with the explanation.
There are things which are hard. Deal with it. (This arrogance is regrettably quite popular.)
Thanks. Great explanation.
Could you elaborate and tie this in with the number of primes between m and n?
Okie doke, forgive me if I'm missing something here, but is Fermat's Last Theorem the same as the conjecture mentioned in the article? The one that took Dr. Andrew Wiles seven years to solve?
You're missing the point. Fermat's Last Theorem was created in 1630 and solved in 1993 (363 years later). The Riemann hypothesis was composed in 1859, so at that rate it won't be solved until 2222. $1M may not be worth much in 220 years.
Andrew Wiles may have spent seven years of dedicated time on Fermat's Last Theorem, but this doesn't mean that the conjecture itself was solved in seven years. Huge leaps in mathematical theory needed to take place before Wiles could realize his proof. Anyway, Wiles didn't prove FLT so much as he proved the Taniyama-Shimura-Weil conjecture, which was of interest because someone else had already proved that FLT would follow from TSW.
TSW conjectures that "all semistable elliptic curves with rational coefficients are modular". This statement would probably have sounded like gibberish to Fermat, so it really trivializes the problem to claim that the conjecture was solved in seven years. Whole branches of mathematics had to be invented before FLT could be proved.
-a
How to rationalize theft.
Yes! The first part in your response about the axioms is what I meant. Choosing different axioms yeilds different theory (and possibly rubbish); for example the Axiom of Choice is necessary for basically all modern analysis, but you can have a lot of classical analysis without it. It is a requirement for measure theory (a measure cannot be constructed without it). Still the Axiom of Choice allows for some very non-intuitive results: for example you can break the unit sphere (3d) into finitely many (however immeasuralbe) pieces and then proceed to construct two unit spheres out of those (Banach-Tarski Decomposition). The axiom itself is however very intuitive and is part of established mathematics (from 1920s on I think). One can only wonder... Anyway excellent page here. Includes comment by Jerry Bona: The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma? (the three are equivalent). Luckily the mathematics we now have seems to portray nature rather well, so I think we can rest assured.
You still fail to understand the meaning of axioms. Let's forget the name 'axiom' and talk about just assumptions. For example you might implicitly use some basic assumption when calculating 2+2=4 (at least, you apparently are calculating in Z, not in Z_2 for example). You see, every time you try to set up some proposal or theorem you need to assume something. Without assuming some underlaying construct what is there to deduce (based on nothing)? The reason we talk about 'axioms' is because we wish to emphasize the importance of these basic assumptions. You should go to some mathematician you respect and discuss the matter with him, if cannot convey it over here.
The point however is indisputable: all mathematics is logically based on some set of axioms (or assumptions, if you will). These assumptions need to be correct for the mathematics to be correct. The actual process of axiomatization has got nothing to do with this; here you are mixing history with mathematical constructs to prove something. In mathematics, the most important thing is to completely understand what you are doing. It may be your intuition that is guiding you: intuition is necessary but can just as easily lead you to wrong theorems. Only by complete understanding and carefull verifiying should you be confident on your results. This is however very difficult; recently a friend of mine had to 'cancel' several of his published articles, because he was using an established ten year old result that was proven to be wrong. So mathematics (remember the 'empirical' point) is not unerring.
With your deduction about Gödel Incompleteness theorem you are also mixing things; namely mathematical constructs and mathematicians themselves. As with mathematics (and with all kinds of logic), if you choose a set of assumptions which is allready conflicting within itself you can prove anything. This will have nothing to do with nature however. So, the Gödel Incompleteness (GI) result applies because of the following first-order logic: GI applies to all mathematical constructs which include at least the Peano axioms AND Number Theory as a mathematical construct includes Peano axioms => GI applies to number theory. It couldn't get more simple! (hope I got my assumptions right...)
As to Number Theory being the 'Queen of Mathematics'; this is the general opinion. I myself do think that Complex Analysis is the most beautiful part of mathematics (eloquent proofs, non-intuitive results (at first), all accessible to a first or second year student). Anyway I've allways disliked purely discrete things (such as integers). I don't study complex analysis by the way; I've done research on Markov operators (stochastics stuff) and now I'm back to basic applied stuff (cutting and packing; you even get to see actual results!).
Even if it doesn't, will it be possible to prove that it is the simplest possible proof?
yes, or at least you could find the simplest proof in finite time
[as a proof it would have to be constructable in a finite number of steps from 1st order logic and the axioms of set theory. Given N, the number of steps the proof took, there is a finite set s(N-1) of all statements taking less than N steps, these could be exausted in finite time by a computer and examined to see if any where proofs of the theorem.]
Actually, he is quite correct that it continues to be difficult to prove. Even if the proof can be contained in a statement that's Einteinianly simple (E=m_0c^2), the road to reach that proof has still proven to be phenomenally difficult.
I've been trying to publish my proof of Goldbach's conjecture, and it's just 12 pages long. (I'm serious.) I'm discovering a lot of barriers in Academia to getting heard. But as simple as the result sounds, the road to get there took weeks for direction and years for refinement... And I doubt it could be attained by someone who weren't a multidisciplinary scientist-and-artist, becuase the problem-solving required both the logic and the thinking-outside-the-box to deconstruct known methods into untried ones.
Thanks. Great explanation.
Very kind of you to say, thanks.
Could you elaborate and tie this in with the number of primes between m and n?
I'm a little less confident about this, but here goes...
As I understand it (and bear in mind that I've not done any complex analysis for several years, and number theory has never really been my forte) sometime during the 19th century Gauss noticed that the distribution of primes was approximated pretty well by a function he called the `logarithmic integral'.
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).
Now this is where I start to lose track of things.
As I understand it, it was proved at the beginning of the 20th century that the validity of the Riemann hypothesis is equivalent to the assertion that the deviation of Li(x) from the actual value of pi(x) is of the order of sqrt(x)*ln(x).
If I remember correctly, some work of the three great British mathematicians Hardy, Littlewood, and Hardy-Littlewood showed that pi(x) actually oscillates around Li(x) infinitely many times (although it really doesn't do it very quickly - the first value of x for which the graphs cross is very big indeed).
qv: the Clay Institute's page
and Chris Caldwell's page for better explanations.
As to whether there's a relatively simple proof out there, I don't know. My (non-specialist) suspicion is that there isn't, because some really clever people have tried to find one for a century and a half and failed. I'd be interested and impressed to be proved wrong on this, though.
Andrew Wiles' celebrated proof of Fermat's Last Theorem (if you haven't done so, read Simon Singh's book on the subject, and if possible watch the BBC Horizon documentary - the transcript is available) was pretty complicated and, I'm told (algebraic geometry isn't my field either - there's an awful lot of diverse mathematics out there) introduced some genuinely new ideas and methods.
The general feeling is that if there were a simple proof of either Fermat or Riemann (or, for that matter, the Goldbach or Poincare Conjectures) then someone would have found it by now - some really top brains have worked on all of these over the years (including several Fields medallists, FRSes, and the like).
(There's a guy who posts regularly to sci.math who reckons he's got a simple proof of Fermat which doesn't resort to all that scary stuff about elliptic curves or modular forms. The consensus seems to be that he's a nutter, though - his `proofs' contain obvious flaws which he refuses to acknowledge, claiming instead the existence of an enormous academic conspiracy against his work.)
It's also often the case (and this was true for the Fermat theorem) that proofs of such intractible problems, even those which are subtly flawed, introduce new ideas and methods of attack.
This is why otherwise sensible mathematicians have a go at these problems - even if they don't manage to solve them, the chances are that the attempt will inspire them to find new methods or potentially important partial results. Even had Wiles' original (flawed) proof turned out to be irrepairable, it was a pretty major piece of work which introduced some important new ideas which could well be useful in solving related problems.
My guess (as an interested non-specialist) is that while a proof of RH would be complicated and elegant, it would also involve some new twist or idea. As for who might do it, my money would be on Prof Louis de Branges of Purdue University - he demolished the (similarly intractible) Bieberbach Conjecture in the 1980s and thus seems to know what he's doing. Or it might be someone else entirely, someone who's spent seven years locked in their attic (as Wiles did).
nicholas
Ok, I'll try to give out a dummy proof for Gödel's Incompleteness theorem (the whole thing is apparently about 30 pages, I'll admit I haven't read the whole thing; I've read a partial proof in Russel's and Norwig's 'Artificial Intelligence'). This should clear things out a little bit and give insight to the discussion.
We'll start with the observation that in number theory we have names for all the natural numbers. This is seen as follows: let's say we have the successor function S and a single constant 0; then let S(0) denote 1, S(S(0)) denote 2 etc. By induction we have names for all the natural numbers.
Gödel also included the following function symbols: +, * and Exp and also the usual set of logical connectives and qualifiers in first-order logic. It is now obvious that that the set of sentences we can write in this language can be enumerated (order the symbols in alphabetical order, then do the same with sentences of lenght 1, then with 2 and so on). We can therefore number any sentence a with a unique natural number #a (the Gödel number). Therefore: Number theory contains a name for each of it's own sentences!!! In the same way we can number each possible proof P with a Gödel number G(P) because a proof is a finite sequence of sentences.
Then let us assume that we have an arbitrary set A of true statements about natural numbers. Because A can be named by a given set of integers we propose that it is possible to write the following sentence in our language: a(j,A) =
All i for which i is not the Gödel number of a proof of the sentence whose Gödel number is j, where the proof uses only premises in A.
Furthermore, let r be the sentence r(#r,A) i.e. a sentence that states its own unprovability from A. Can such a sentence exist for all A? Don't ask me, but apparently Gödel would have said that the answer is yes.
The rest is rather simple alltough rather ingenious. We need to prove that r is true. We'll go with reductio ad absurdum: Let's first suppose that r is provable from A (that r actually is false statement! remember that r was stating it's unprovability from A). But this would mean that we have a false statement provable from A. Therefore A cannot consist of only true sentences. This is a contradiction since according to our premises A consists of only true sentences! Therefore r must not be provable from A which is exactly what r claims.
So from the above (assuming that we believe the sentence r can be constructed) we have seen that for any set A of true sentences in number theory we have statements that cannot be proven from A. As a special case we can choose A = axioms of the number theory. Hence number theory containts statements that cannot be proven!
Feel free to complain about the inaccuracies in the above; all I can do is to suggest you get Gödel's proof into your hands. Anyway to my mind (if I do not miss any subtleties) the above goes on to establish that we can never prove all the theorems of mathematics within any given system of axioms (as the above problem appears allready with the natural numbers). This is apparently why Hilbert was pissed about Gödel's proof.
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)
Theoretically in the sense that it's useful for many other theorems. Quite a few things are based on numbers having a unique (down to the order) factorization as a product of primes. That proof is reasonably simple, and we can easily find pseudoprimes (which for most intents and purposes are fine) and that covers what we mostly need. The Riemann hypothesis is more of purely theoretical and cryptological interest though.
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