Arxiv preprint:

A proof of the Riemann hypothesis

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Interesting--lots of typos to be fixed (e.g. Dedekind does not spell his name "Dedeking") before it's ready for publishing, but it's not just some woo-woo--it's real mathematics (though not in my field so I know the terms and their relevance to the subject at hand, but can't evaluate the correctness of the proof).

Now, we need peer review to decide if this is it or not.

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Well, related to D.I.'s link is this one on the same site:

http://secamlocal.ex.ac.uk/people/st...a/RHproofs.htm

so, there have been lots of serious and not-very-serious "proofs" that failed--just statistically, this casts doubt on the paper, though a peer review still is needed to be sure one way or the other.

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Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

What would it DO? I've read several of the introductions, and I don't really get it. I do have a mathematics dyslexia of sorts, so maybe that's gumming up my works. Is there a plain language, very very general, here's what this means web site? And what does it mean to prove it? I mean you have the dang numbers in front of you already!



CJSF

P.S.
I know that this is important, and I know others understand it.. My post is more about MY frustrations with mathematics, and how it's limited me since school days.

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Wikipedia: Riemann hypothesis

Mathworld: Riemann Hypothesis

Oh. Say no more!

OMG, if it's solved, there'll be sock puppets everywhere!

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0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ...

Big news indeed, if true: RH has generally been regarded as the most famous conjecture in mathematics for more than a century. It would have so many consequences there is even a famous book listing some, because it would in some sense give the complete story about the fabulously irregular distribution of prime numbers.

Now as you would expect, amateur cranks do claim from time to time to have proven RH. However, the author in this case has impressive credentials. Xian-Jin Li earned a Ph.D. in mathematics (1993) from a distinguished department (Purdue University) with a thesis titled The Riemann Hypothesis For Polynomials Orthogonal On The Unit Circle, and his advisor was Louis de Branges, who achieved a landmark result in 1984 when he proved Bieberbach's conjecture (now called the Bieberbach-de Branges theorem). He is on the faculty of the Math Department at Brigham
Young University and has authored various papers on analytic number theory (the field in which one applies analytic tools like Fourier analysis to number theory), which have been published in respectable journals and have been cited by others.

Ordinarily such a stellar background would be very encouraging, but I'm a bit worried, because de Branges himself claimed to have proven RH back in 2004, a claim which has not been accepted by experts on RH! (To be fair to de Branges, his claimed proof of Bieberbach's conjecture was intitially met with sceptism, but in that case the sceptism quickly vanished once experts studied his paper.) Hopefully this simply means that de Branges was onto something after all, and that his former student has now been able to provide the first acceptable proof!

Quite a few famous mathematicians have briefly claimed to have proven RH, in fact this is a bit of running joke. I seem to recall that the legendary English mathematic G. H. Hardy, who was terrified of drowning at sea, once took ship for a conference on the continent. Before leaving he sent a postcard to a friend claiming to have proven RH. When months later his friend incquired after the proof, Hardy blandly replied that he made the whole thing up, on the grounds that God wouldn't sink his ship because then everyone would assume he, Hardy, had indeed proven RH, and according to Hardy, God disliked the idea of anyone thinking that so much he was unable to sink Hardy's ship :wink:

More happily, many many famous mathematicians have outlined programs for proving RH, hitherto none successful. The abstract of the eprint in question mentions one such program, due to Alain Connes and building upon previous work of Enrico Bombieri and Andre Weil, three of the most famous mathematicians of recent times. (Bombieri and Connes have both won Fields Medals, and Weil has often been called the most influential mathematician since Gauss. I can't resist adding that both Bombieri and Connes have written about Penrose tilings, subject of my own diss., which suggests that I possess "good taste" :wink

EDIT: this is beginning to look a lot better!

I now see that two of the finest mathematicians around, Bombieri and Jeffrey Lagarias, have generalized a criterion which was originally proposed by Li. I haven't yet tried to really explain this, but at a glance, Li's paper appears to use many of the ingredients which many of the best mathematicians have expected might play a role in a proof of RH. Indications of a community effort involving the best mathematicians working in this area is generally speaking just what I would expect for the first successful proof of RH. So I see many encouraging signs that Li really might have suceeded in proving RH.

If so, this would be a Really Huge achievement!!!

Almost any mathematician would probably agree that RH has always been expected to have much more importance in mathematics than FLT (which was finally proven by Andrew Wiles toward the close of the last century, after being open for an even longer time than RH)--- not to take anything from that achievement, which was a landmark by any standard.

Further reading:

The website mentioned by ToSeek is an excellent source of information; for more detail try

Harold M. Edwards,
Riemann's Zeta Function,
Academic Press, New York, London, 1974.

Donald Zagier,
"The first 50 million prime numbers",
Mathematical Intelligencer 0 (1977), 7-19.

Terry Tao (another Fields medalist),
Structure and randomness in the prime numbers

This has been slashdotted, but more sober/knowledgeable comments are already appearing in the math blogs, e.g. see these comments at the blog of Peter Woit (Math, Columbia). I am seeking comment from Tao and Lagarias.

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Fermat's Last Theorem, despite occupying so many mathematicians' time (and dominating some careers at the expense of anything that would have actually advanced their names) for 300 years or so before finally being proved, doesn't have any applications that I know of, even applications like "it can be applied to prove another important result".

However, the quest for the proof of the theorem resulted in the development of an awful lot of mathematics that DID turn out to be useful. The "ideal" (a mathematical object, not a concept of perfection) was invented by Kummer in an attempt to fix up a flawed "proof" of FLT. These "ideals" turned out to be integral (which is a pun, but not obviously so if you don't study the subject) to algebraic geometry, commutive algebra, algebraic number theory, and for that matter, algebra. Would that mathematics have been discovered without FLT pointing the way? Possibly, but it's not obvious that it would. Applications ultimately derived from this work include cryptography, advanced techniques in solving polynomial equations (I know there are applications for that, can't think of any at the moment), computational geometry and its application to, for example, robotics and current attempts at machine vision (not to mention video games), and for that matter, manifold theory which is important in, for example, the formulation of General Relativity, which went into the design of the GPS system.

Now, for the Riemann Hypothesis, attempts at proof no doubt have lead to important advancements in mathematics. As for the usefulness of the conjecture itself--I have come across algorithms, whose proof of correctness assumed the Riemann Hypothesis (deep in the recesses of my memory--can't remember how RH was used). A proof of the RH would help users of such algorithms sleep better. On the other hand, I'm sure the algorithm writers figured, either it's true and the algorithm works, or the algorithm fails and we find a counterexample to RH (either that or, far more likely, a bug in the software). In other words, those algorithms are really heuristics that happen to work in practice unless or until RH is proven.

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Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

It seems that Terry Tao has identified a serious error:
Silly me, having missed that! :blush:

And Alain Connes wrote:
(These comments were posted at their respective blogs.)

Li has posted a third version attempting to address these criticisms, but from the sound of things, I now consider it unlikely the gaps will be easily repaired.

[EDIT 5 July 2008: the day after Connes and Tao posted their comments, Li withdrew his eprint. Because he did the right thing--- which cannot have been easy--- I'd like to go out of my way to applaud the courage of anyone bold enough to attack RH. Even if the effort ends in defeat, as happened on this occasion, the attempt is noble, and there is no great shame in joining the ranks of leading mathematicians who have briefly believed they had proven RH.]

The website of Murray Watkins (already cited twice above) has some really nice information on why the RH is so important in mathematics. An awful lot of wonderful mathematics would immediately follow if it is true (almost everyone thinks it is true).

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after reading most of 9 posts so far- my eyes started to glaze over and i went into a near catatonic state during some of the longer posts.. and after clicking on a few links here..
i have absolutely no idea what this is all about.
since i've been overdosing on Stargate DVD's lately- i'm up to the middle of season 8- i will use a metaphor from that show to explain my cluelessness..
i feel like O'Neill when Carter is explaining how something involving some new alien technology works...

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The paper is noted as being withdrawn due to a mistake on page 29. Sounds like a false alarm.

The Riemann Hypothesis is one of the Milenium Problems. Here is a link to a piece by Bombieri written for the Clay Institute describing the RH and why it is important in mathematics.

http://www.claymath.org/millennium/R...is/riemann.pdf

novaderrik: I know how you feel! Well at least about the "over dosing" of stargate! lol... great show, I watch it when ever it's on sci fi.... anyways, don't you think they would have checked their work more carefully before subminting it?

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I'm doing my best to understand this thing. I'm on line 11 of that link so far.

Can someone please give me an example of a zero for the Riemann zeta function? Is a zero a complex variable s, with a real part greater than 1, that satisfies the equation Zeta(s) = 0. Or am I missing something?

clop

edit: I'm beginning to think that a "zero" in this context means a minimum? i.e. the lowest points in a contour map of the function?

A "zero" of the Zeta function is any complex number s (not variable) that satisfies Zeta(s) = 0. Since the zeta function is "obviously" zero in certain locations of no interest, those cases are colloquially called the "trivial" zeroes, and no one is interested in those, they have real part equal to or less than zero. It is only possible zeroes with a real part greater than zero that are of interest. The Riemann hypothesis is that all the non-trivial zeros have real part equal to precisely 1 - it has plenty of zeroes there, in a fashion related to the prime numbers. Since it can't possibly have any zeroes with real part (strictly) between 0 and 1, it is zeroes with real part greater than one (which the Riemann hypothesis asserts not to exist) that would be the really interesting ones.

Incidentally, you and CFerro shouldn't be worried about being totally unable to understand a mathematics paper. Although I have a good degree in mathematics from a leading British university, and indeed having studied in the very best bit of that university (I had some of the same tutors that Andrew Wiles had when he was a student), with a mere first degree I understand far too little of what goes on in cutting edge mathematics even to be able to follow the gist of most published papers. It is generally acknowledged that doctorate students in mathematics need to do a lot more taught study before they are ready to be able to engage in mathematics at that level. Even though I was borderline for a 1st class degree (in the days when that meant something), I could see that the top students were on another planet from me. I made a good tactical decision to take a masters degree in Economics, another subject where an undergraduate degree is insufficient to prepare you to do research.

Thank you Ivan, very helpful. For what it's worth, I have an MSc in Control Systems from UMIST in the UK. I even took the optional advanced mathematics module but I am still lost. In control theory, poles were conditions that systems tried to move away from, and zeroes were stable conditions that systems moved closer to. A normal die in contact with a surface would have 6 zeroes (lying on one of the flat faces), 12 + 8 poles (teetering on an edge or on a point) and lots of unstable conditions in between. Is a pole in the context of the zeta function a complex number s where the series is non-convergent?

clop

Yes. See here.

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Actually the Riemann Hypothesis is that all non-trivial zeros of the zeta function have real part equal to 1/2. See the Bombieri description in the link in the earlier post.

Not only do doctorate students in mathematics need a lot of study to tackle this problem, it is widely recognized as the most difficult problem in mathematics. It has been an open problem since Hilbert gave his famous address at the beginning of the 20th century, and remains unsolved. It is the only one of the milennium problems that was also a Hilbert problem.

"If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven? "-- David Hilbert

The answer is yes, but with a caveat. The zeta function is an analytic function with a simple pole at s =1. Everywhere else it is defined and is analytic. Being analytic means that it can be locally described by a convergent power series. But that power series can vary with location, and in the case of the zeta function it does. As you and others have noted the series sum (1/n^s) diverges unless re(s)>1. But it is possible to start with that series for and do what is called "analytic continuation" to define the zeta function for all complex s, except for the simple pole at s=1. By doing that one obtains other expressions for the zeta function. For a more complete description you might take a look at the book [I]Riemann's Zeta Function[I] by Edwards.

You intuition from control theory is applicable to the theory of complex variables up to a point. In control theory the poles and zeros relate to rational functions (ratios of polynomials) that arise from treating a system of ordinary differential equations using the Laplace transform. In the time domain you deal with a matrix differential equation of the form dX/dt = Ax + Bu. The poles that come up are related to eigenvalues of the A matrix, and stability requires that they have negative real part. The complex variable theory involved is simplified by the fact that the functions that arise are rational. The work on the zeta function uses a bit more of the general theory of analytic functions of a complex variable, and such functions may be more complicated than rational functions, and their singularities may be more complicated as well. In the case of the zeta functions the singularities are not complicated, but the process for expressing the function is more involved than it is for rational functions.

Basically what happens is this function has a power series and a radius of convergence about any point that you pick in the complex plane. But the radius of convergence only extends from the central point to the point 1 on the real line. So the specific power series that describes zeta varies with the central point that you pick. This leads one to search for expressions other than the power series to define zeta and to work with it. For a simpler function to think about with respect to this phenomena consider 1/x. It has a simple pole at x = 0, and it has a power series representation centered at any point other than 0. But the power series varies with the point about which you expand it, and has a finite radius of convergence. The zeta function is more complicated in that a simple global formula (like 1/x) is not readily available either.

Here is Terrence Tao's original rebuke of the paper:

Source.

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So, progressing by doing the "mathematics that would follow if it were proven" as if it were, would lead to serious issues when applied to practical matters? Why not try some of those new mathematics as if it were true and see what benefits we might get? I mean if we're reasonable sure it is true, why not move forward?

CJSF

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I was looking up past the city lights
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See the constellation ride across the sky
No cigar, no lady on his arm
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by They Might Be Giants

See one of my previous posts in this thread, the one on Fermat's Last Theorem--there are two answers given there:

1. Already done.

2. The journey toward a proof leads to serious applications.

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I hate those page 29 errors!

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I assume you are referring to the Riemann hypothesis (RH)? If so, before reading on, if you haven't already done so, please read some of the items I already cited for some information about what kinds of marvelous mathematics would follow from RH if it is true.

If I understand you correctly, you ask: if all the evidence strongly suggests RH is true, and many marvelous benefits of RH are known, why not assume its true and forget about trying to rigorously prove it?

There are many possible answers. I'll just offer a few brief thoughts:

• Surely the most characteristic concept in all of mathematics is the notion of proof. Your suggestion is incompatible with the fundamental nature of the mathematical enterprise.
• History shows that the greatest advances in mathematics most often occur when someone succeeds in settling a long-open problem. Success often involves inventing powerful new mathematical machinery which has many other applications in mathematics, and which almost always raises new questions; often, a new subfield within mathematics results from huge advances of this kind. For example, K-theory arose in the context of operator theory (with a triumphant application in proving the Atiyah-Singer index theorem), but also has important and powerful applications in algebra, topology, and dynamical systems.
• The most beautiful work in mathematics often is associated with such novel machinery, for example K-theory arose in the context of operator theory but also has important and powerful applications in algebra, topology, and dynamical systems.
• Even if a long-open conjecture for which much evidence has long been known is eventually settled in the affirmative, the insights required to finally prove that the conjecture is true are generally extremely deep, beautiful and powerful. But if it is actually wrong, the ideas required to explain why would almost surely represent deep and extremely novel insights. If RH is wrong, proving this would probably require revolutionary insight comparable with Goedel's incompleteness theorem.

Hmm... add this book to my recommended reading:

Benjamin H. Yandell,
The Honors Class.
A. K. Peters, 2002.

This book concerns the Hilbert problems, whose resolution dominated twentieth century mathematics. This is a very readable book, and mostly pretty good, which is no doubt due to the fact that Yandell has a fine undergraduate math education--- the very best popular math books, IMO, are by leading mathematicians.

(The weakest chapter in Yandell's book is unfortunately on my current favorite topic, the Schubert calculus, a classic topic in enumerative geometry. The late nineteenth century work on this is generally acknowledged to be the first appearance of what we now know as cohomology.)

Picking up on what Ivan said about background needed to follow recent papers in high level number theory, it is widely acknowledged that this is among the most difficult of all subjects in which to do research, since so much background is needed (easily two or three times what is needed to get started in many other interesting and important areas). That said, I can recommend a very readable undergraduate level text which gives excellent insight into the kind of ideas used in analytic number theory (where we apply the tools of analysis, e.g. function theory--- undergraduate complex analysis is basically the elementary theory of complex-valued holomorphic functions of one complex variable--- to number theory):

G. J. O. Jameson,
The Prime Number Theorem,
LMS student texts 53,
Cambridge University Press, 2003.

The prime number theorem is the essential background for the genesis of RH, incidently.

In algebraic number theory we apply algebraic tools (e.g. the theory of number fields) to number theory. For this try

H. P.F. Swinnerton-Dyer,
A Brief Guide to Algebraic Number Theory,
LMS student texts 50
Cambridge University Press, 2001.

In both books you will see that zeta functions play a role; the most famous example of a zeta function is the Riemann zeta function, the locations of whose nontrivial zeros is the subject of RH. The notion of zeta functions is an important unifying idea in mathematics.

Casual surfers can browse Wikipedia's article on prime geodesics for an example of a zeta function which arises somewhere other than number theory.

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Thanks, Chris.. I can't promise I can look into the suggested reading any time soon, but I appreciate the recommendations. I'm sure many of the principles behind the image processing and enhancement algorithms I use every day at work have some similar stories.

CJSF

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Two years ago moved from my town
I was looking up past the city lights
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See the constellation ride across the sky
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Just a guy made of dots and lines

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by They Might Be Giants

Quite often applications for new developments in pure mathematics follow those developments my many years, if ever. But the insights gained from those developments can have far reaching effects. Witness the application of Riemannian geometry to special relativity, the calculus of variations to optimal control theory, or the theory of Fourier transforms to signal analysis.
In the case of the Riemann Hypothesis the immediate implications are for number theory, in particular distribution of prime numbers. That might have implications in coding, but I suspect that any applications will not be immediate.

The move forward is in the development of the techniques and insights necessary to prove something like the Riemann Hypothesis, more so than in the theorem itself. Mathematics, particularly research mathematics, is not engineering. I have done both and there is at best a tenous relationship.

It is sometimes difficult to describe the work of a pure mathematician. But if you want to try to understand, it is a good idea to read the words of real experts. In the case of the Riemann Hypothesis you could do no better than to read the piece written by Bombieri (link posted earlier). He is an expert in the true sense of the word, which is precisely why he was chosen by the Clay Institute to write that piece as part of their program for "Milenium Prizes". The Riemann Hypothesis is the only Millenium Prize problem that was also one of the original Hilbert Problems. It is probably the deepest of those problems. Solve it and you can claim a million dollar prize, which Jim Carlson would be delighted to award.

Very much a layman's take on this, based on the "small" part of mathematics I've played with (prime factorizations, I just cracked another C136 from Brent's list):

The RH amongst other things puts strong limits on the expected frequency of prime vs. composite numbers as you go to larger numbers.
This has the result that the expected behavior of many algorithms can be calculated if you assume RH and is basically unpredictable if you don't.
Part of why the RH is considered likely to be true is that those algorithms actually behave as you'd expect if it was.

In a way, actually using the algorithms IS going ahead as if it was true, but the practical impact if it wasn't would be that the time spent would be different than expected or possibly that the algorithm would never halt(ok, then it wouldn't be an algorithm, and might be a counter-proof to RH, but never halting is difficult to prove), not that the results would be wrong.

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This is interesting. Do you actually need the full strength of the Riemann Hypothesis for your application?

It is known that lim (pi(x)ln(x)/x) = 1, liimit taken as x goes to infinity, where pi(x) is the number of primes less than or equal to x. This is the classical "prime number theorem". I take it that this is not strong enough for your needs.

I don't get it. Without knowing whether the reimann hypothesis is valid or not, couldn't you still use it to break internet security by assuming it is true and building your possibly-pseudo-prime factorization scheme off of that assumption?

And if it doesn't work, it may be an indirect disproof.

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