Some Bits Crossing My Mind This Month

A miscellany today...:

1)  FIRST, in case you've been living under a rock... on the planet Zorka... in Galaxy 134-18B this last week and don't know, TOMORROW (Monday) Michael Atiyah is giving a 45-min. talk entitled simply, "The Riemann Hypothesis" at the Heidelberg Laureate Forum, claiming "a simple proof using a radically new approach." I have no idea how serious of a "proof" this is (other than Atiyah being a serious mathematician, but still hard to take this at face-value, given how many radically new, simple approaches have already been tried). Either way, math cyberspace should be abuzz tomorrow with commentary following the presentation:

I believe the talk will be live-streamed and recorded at the HLF YouTube channel here:

https://www.youtube.com/user/LaureateForum

Also, a couple of the Aperiodical bloggers will be in attendance and reporting on the meeting (I assume they'll check in after the dust settles, but maybe they'll do some live-blogging or tweeting  as well? -- surely there will be some live-tweeting from #HLF18).

==> ADDENDUM 9pm. 9/23... the proof, by contradiction, has now been posted here (h/t to @sigfpe on Twitter):

https://www.dropbox.com/s/pydoj0a8hguebc6/2018-The_Riemann_Hypothesis.pdf?dl=0

ADDENDUM II 9/24:  one of the live-tweeted threads from Atiyah's talk (now over) is here:
https://twitter.com/mpoessel/status/1044131977950109696

...needless to say, a lot of skepticism being expressed across the Web by those who understand the math/logic; no doubt there will be a lot more commentary today, and even if negative, much food-for-thought may still emerge from this.

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2)  Speaking (loosely) of proofs... logician George Boolos would’ve been 78 this month… had he not died at the relatively young age of 55 in 1996. For any relative newbies, one of my favorite mathy pages on the Web is his famous, delightful single page explaining Gödel’s second incompleteness theorem “in words of one syllable” (worth reading for fun at least once every year!):

https://www2.kenyon.edu/Depts/Math/Milnikel/boolos-godel.pdf

It can probably even make a nice introduction to Gödel for younger folks.

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3)  Curses, curses, curses to Jordan Ellenberg who has gotten me regularly reading Martin Shkreli’s prison-composed blog, ever since Jordan cited it in a tweet.  I first wrote about it back here:

https://math-frolic.blogspot.com/2018/08/calling-all-number-theorists.html

And in his 9/6/18 entry, after seeking help with some math he was working on, Shkreli ended by writing:

“Thank you to all the professors, postdocs and other math professionals who have reached out to help me. It has been great to communicate with you all. Bear with me as I order my thoughts and respond in the limited way I can.”

I don’t know if this is bluster, bluff, or actuality, but if it is for real, I’d sure be curious to hear about what substantive math any “math professionals” have taken up with Martin, if you’d care to share? Ought to be some sort of interesting backstory there.

[...Also, Martin regularly recommends Bio-Pharm stocks to buy or avoid (or short), and even though I don't dabble in bio-pharm stocks myself I'd be curious if anyone else has found his judgments useful/profitable.]

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4)  For those interested in cognition, science writer John Horgan has a new Web-accessible volume out on mind-body problems. I’ve enjoyed John’s writing in the past, but also tire a bit of this topic that seems forever shrouded in sound and fury, without much ever resolved. As John says, the book “offers my subjective takes on my subjects’ subjective takes on subjectivity.” So I wasn’t expecting too much from his latest, but in fact enjoyed it immensely, partly because of the portraits it paints of specific diverse, fascinating thinkers; their foibles and makeup, in addition to their academic or cerebral selves, while delving into their thoughts on mind/body issues. You can read the whole volume here: 

https://mindbodyproblems.com

…or you can download it from the Web for a small price.

There are probably many Douglas Hofstadter fans out there, so as one sample chapter, I recommend Chapter Two which is with Dr. Hofstadter (p.s… one small side-note that I learned here, and didn’t even realize before, is that David Chalmers did his PhD. under Hofstadter):

https://mindbodyproblems.com/chapter-two/

Speaking of books, Scott Alexander (just a bit behind the times) offers a long review of Nassim Taleb’s “The Black Swan” here (followed by 250+ comments):

http://slatestarcodex.com/2018/09/19/book-review-the-black-swan/

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5)  For those with the chops to follow it, Steve Strogatz recently passed along this history of the Langlands Program:

http://www.ams.org/journals/bull/2018-55-04/S0273-0979-2018-01609-1/S0273-0979-2018-01609-1.pdf

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6)  And I'll close out with this gem that surfaced on my Twitter feed yesterday:
https://twitter.com/jayvanbavel/status/1042838461173116928

If Gluten-free Flour Is Your Thing....

H/T to Nalini Joshi yesterday 2 days ago, for tweeting the above bag of Anthony’s gluten-free, almond flour. You get the feeling that Anthony, at heart, is a frustrated mathematician-wannabe… the lower left corner of the bag reads as follows:

“For a free bag on us, and a personal high five from Anthony, prove that the real part of every nontrivial zero of the function below is equal to 1/2”

:)))

[no mention that with the $1 million in prize money you'd receive for proving the Riemann hypothesis you could purchase several dozen bags of Anthony’s fine flour AND have money left over for coffee (even expresso).]

Further, in the comments to Nalini’s tweet someone mentions that a new attempt to prove the RH was recently submitted to arXiv:

https://arxiv.org/abs/1804.04700

https://arxiv.org/pdf/1804.04700.pdf

So seriously let us know what if any status that submission has.

Finally, drawing again from the David Wells' book that I mentioned yesterday ("The Penguin Book of Curious and Interesting Mathematics") a couple of old quotes from David Hilbert:

Upon being asked "What technological achievement would be the most important?" Hilbert replied, "To catch a fly on the moon. Because the auxiliary technical problems which would have to be solved for such a result to be achieved imply the solution of almost all the material difficulties of mankind." 

Then asked what mathematical problem was the most important, he responded, "The problem of the zeros of the zeta function, not only in mathematics, but absolutely most important!"

Riemann In the News...

Lots of interesting mathy stuff out there this week, but hey, you can’t go wrong with the crown jewel of number theory, so I’ll direct you to two pieces on the Riemann Hypothesis, if you’ve not seen them:

First, a brief interview with Barry Mazur and William Stein, authors of “Prime Numbers and the Riemann Hypothesis” (one of my favorite 2016 books):

http://www.cambridgeblog.org/2017/04/a-brief-interview-with-barry-mazur-and-william-stein/

…and then the incomparable Natalie Wolchover summarizing the latest intriguing approach from physicists to Riemann’s 150+ year-old, million-dollar conundrum:

https://www.quantamagazine.org/20170404-quantum-physicists-attack-the-riemann-hypothesis/

The actual (physics) work was published last year but is just now being widely disseminated on popular media:

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.130201

There are a great many other introductions to RH on the internet, including some video ones such as these:

From Numberphile:

https://www.youtube.com/watch?v=d6c6uIyieoo

…and from 3 Blue1Brown:

https://www.youtube.com/watch?v=sD0NjbwqlYw

An Incidental Remark...

Sunday reflection:

"In [his 1859 paper], Riemann made an incidental remark -- a guess, a hypothesis. What he tossed out to the assembled mathematicians that day has proven to be almost cruelly compelling to countless scholars in the ensuing years...

"...it is that incidental remark -- the Riemann Hypothesis -- that is the truly astonishing legacy of his 1859 paper. Because Riemann was able to see beyond the pattern of the primes to discern traces of something mysterious and mathematically elegant at work -- subtle variations in the distribution of those prime numbers. Brilliant for its clarity, astounding for its potential consequences, the Hypothesis took on enormous importance in mathematics. Indeed, the successful solution to this puzzle would herald a revolution in prime number theory. Proving or disproving it became the greatest challenge of the age...

"It has become clear that the Riemann Hypothesis, whose resolution seems to hang tantalizingly just beyond our grasp holds the key to a variety of scientific and mathematical investigations. The making and breaking of modern codes, which depend on the properties of the prime numbers, have roots in the Hypothesis. In a series of extraordinary developments during the 1970s, it emerged that even the physics of the atomic nucleus is connected in ways not yet fully understood to this strange conundrum. ...Hunting down the solution to the Riemann Hypothesis has become an obsession for many -- the veritable 'great white whale' of mathematical research. Yet despite determined efforts by generations of mathematicians, the Riemann Hypothesis defies resolution."  

-- John Derbyshire, from the dustjacket description of Prime Obsession 



==> p.s... check out my new interview with science/math writer Brian Hayes over at MathTango:
https://mathtango.blogspot.com/2016/07/brian-hayes-award-winning-science-writer.html

Exquisite 12+ Minutes

Just another GREAT offering from those wonderful folks over at Numberphile today, this time on the Riemann Hypothesis, zeta function, and L-functions:




Meanwhile, over at MathTango I've been feelin' the burn of Jason Wilkes' unconventional treatise on math instruction:
http://mathtango.blogspot.com/2016/05/feel-burn.html

Looking Forward and Backward (at books)

Since listing my favorite math books of 2015, I was recently reminded that the new Barry Mazur/William Stein volume on the Riemann Hypothesis is due out at the end of January 2016:
http://amzn.to/1lZuxxb

Too late for Christmas, but what a great start to the new year. David Mumford calls it "a soaring ride." I suspect once out, this short volume will be THE book (out of many available) to introduce folks to possibly the most important unresolved, far-reaching conjecture in all of mathematics. (...Perhaps I already know my favorite book of 2016!)

Meanwhile, I just obtained a couple of fine prior books on paradoxes, and feel safe recommending both well-ahead of finishing them. Roy Cook's 2013 "Paradoxes" is a good, fairly standard treatment of what I believe is one of the most important topics in all of math/philosophy, for bright high-school-level-and-above students.

Stanley Farlow's 2014 "Paradoxes In Mathematics" looks to be an especially wonderful introduction to several of the classics for middle-to-high-school students particularly, in breezy but broad-covering fashion. I was previously unaware of this succinct little volume from Dover, and am delighted to have stumbled upon it. Again a great stocking-stuffer for that distinctively math-inclined youngun on your list.

$1 Million Still Awaits...

Riemann Zeta Function along critical line Re(s) = 1/2

Long-time readers know this blog is as interested as any in news of the Riemann Hypothesis. I won't even dignify it with a link or any names, but there was a story this week of the Riemann Hypothesis being proved by a Nigerian mathematician... uhhh, yeah, sure.

The first problem was that I saw the story a couple days after the fellow had apparently announced the proof -- any genuine proof would've hit various legit math websites I follow within 30 mins. of being announced; maybe 3 minutes! The two places I initially saw the story were... well, let's just say, NOT the brightest bulbs in the world of journalism (although some more legitimate sources embarrassingly picked up the story-blurb later). And finally, call me prejudiced, but my gut reaction at this point, to ANY odd story emanating from Nigeria is, "FA-A-A-AKE!" (don't blame me, Nigeria has allowed it's own credibility to be trashed).

Anyway, plenty of others have voiced their skepticism, although admittedly, I've not yet seen the story specifically unmasked as a hoax or case of crackpottery from the get-go, or alternatively as someone with actual math credentials sincerely making an over-the-top claim that doesn't pan out (if someone by now knows the full details or backstory feel free to elucidate in the comments).

For now at least, seems safe to say that Riemann's 156-year-old mystery still awaits a solution that will send legions of mathematicians into paroxysms of jubilation(!), and $1 million (Clay Millennium Prize) still awaits the person who can do it.

-------------------------------------------------- 

ADDENDUM:  A couple of folks have emailed me with questions I'm not able to answer, but the following pieces from George Dvorsky and a Quora thread will help make clear why the announcement is given little credence:

http://gizmodo.com/sorry-the-riemann-hypothesis-has-almost-certainly-not-1743014788

https://www.quora.com/Has-the-Riemann-Hypothesis-been-solved-by-a-Nigerian-professor?redirected_qid=5880926

What remains unclear to me is whether the individual involved (claiming the proof) is some sort of charlatan or a bonafide mathematician in error. Mistaken and crackpot Riemann Hypothesis proofs have been common over the decades and there's simply no basis for thinking this story is anything other. But I'll certainly update if, incredibly, anything more positive arises from the story.


image via Wikipedia 

Riemann...

Yesterday, Peter Woit passed along some interesting Riemann Hypothesis links here:
http://www.math.columbia.edu/~woit/wordpress/?p=7996 

Recommended to everyone is the freely downloadable book (pdf) on RH by Barry Mazur and William Stein. Get it!
==> UGHH, looks like link for download no longer works, so consider yourself lucky if you already got it; otherwise look forward to the book when eventually published. I understand the publisher not wishing free downloads to be available; on-the-other-hand I suspect most of those downloading will eventually want a hard copy of the final version anyway.

Primes, Zeta, Riemann, Oh My (from Marcus du Sautoy)

Love this podcast (~28 mins.) with Marcus du Sautoy on prime numbers, the zeta function, Riemann Hypothesis, etc.:

Contemplating Riemann

"In [his 1859 paper], Riemann made an incidental remark -- a guess, a hypothesis. What he tossed out to the assembled mathematicians that day has proven to be almost cruelly compelling to countless scholars in the ensuing years...

"...it is that incidental remark -- the Riemann Hypothesis -- that is the truly astonishing legacy of his 1859 paper. Because Riemann was able to see beyond the pattern of the primes to discern traces of something mysterious and mathematically elegant at work -- subtle variations in the distribution of those prime numbers. Brilliant for its clarity, astounding for its potential consequences, the Hypothesis took on enormous importance in mathematics. Indeed, the successful solution to this puzzle would herald a revolution in prime number theory. Proving or disproving it became the greatest challenge of the age...

"It has become clear that the Riemann Hypothesis, whose resolution seems to hang tantalizingly just beyond our grasp holds the key to a variety of scientific and mathematical investigations. The making and breaking of modern codes, which depend on the properties of the prime numbers, have roots in the Hypothesis. In a series of extraordinary developments during the 1970s, it emerged that even the physics of the atomic nucleus is connected in ways not yet fully understood to this strange conundrum. ...Hunting down the solution to the Riemann Hypothesis has become an obsession for many -- the veritable 'great white whale' of mathematical research. Yet despite determined efforts by generations of mathematicians, the Riemann Hypothesis defies resolution."

 -- John Derbyshire, from "Prime Obsession"


[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]

Prime Synchrony

Sunday reflection today, courtesy of Freeman Dyson and Hugh Montgomery (this reflection actually ties nicely into last week's Sunday offering as well)...:

"[Freeman] Dyson helped bring together the continuous and the discrete understandings of subatomic behavior. Similarly, by fusing his love of number theory with his expertise in creating the mathematical tools of physics, he would make the initial observation that would reinforce the connections between the discrete world of the integers and the continuous world of analysis, and thus galvanize research on the Riemann hypothesis.
"As Dyson recalls it, he and [Hugh] Montgomery [number theorist] had crossed paths from time to time at the [Princeton] Institute [for Advanced Study] nursery when picking up and dropping off their children. Nevertheless, they had not been formally introduced. In spite of Dyson's fame, Montgomery hadn't seen any purpose in meeting him. 'What will we talk about?' is what Montgomery purportedly said when brought to tea. Nevertheless, Montgomery relented and upon being introduced, the amiable physicist asked the young number theorist about his work. Montgomery began to explain his recent results on the pair correlation, and Dyson stopped him short -- 'Did you get this?' he asked, writing down a particular mathematical formula. Montgomery almost fell over in surprise: Dyson had written down the sinc-infused pair correlation function.
"Dyson had the right answer, but until that moment he had associated this formula with understanding a phenomenon that seemed completely unrelated to the primes and the Riemann hypothesis. In a flash he had drawn the analogy between the sinc-described structured repulsion of the zeta zeros and a similar tension seemingly exhibited by the different levels of energy displayed by atomic nuclei. Whereas Montgomery had traveled a number theorist's road to a 'prime picture' of the pair correlation, Dyson had arrived at this formula through the study of these energy levels in the mathematics of matrices. This connection is the source of most of the current excitement surrounding the Riemann hypothesis..."

-- from "Stalking the Riemann Hypothesis" by Dan Rockmore

Explaining Riemann

Barry Mazur and William Stein are working on a volume entitled "Primes" to explain the Riemann Hypothesis to a larger audience. A working (very lay-friendly) draft of it is available for free download here!:

http://modular.math.washington.edu/rh/

On the same page, you can also view an hour-long Fora/TV talk of Mazur explaining, how to go about explaining the RH.

Riemann Hype or Hypothesis?

h/t to William Wu for pointing out a recent (April) submission to arXiv.org from Mingchun Xu (professor from South-China Normal University), claiming a proof of the Riemann Hypothesis (…in 7 pages no less!):

http://arxiv.org/pdf/1402.5952v2.pdf

Such Riemannian proofs show up from time-to-time... my impression is that they are usually sincere attempts, but dispatched with in fairly short order. Have seen no commentary or other news about this particular one on the Web, so don't know how seriously it's being taken -- but not really expecting to see $1 million from the Clay Institute change hands anytime soon. Also, don't know anything about the author, nor understand the paper's content, but Gary Davis tweets me that "such weak tools will not prove" the Riemann Hyp.

This all makes me a bit curious to know if there is any consensus as to which RH "proof" from the past might be considered to have been the most promising, or alternatively, which proof withstood the longest scrutiny before succumbing to a discovered flaw? Any ideas?

In other matters, h/t to Colin Beveridge for passing along this 3 7-month-old Jeremy Kun mini-rant on high school math teaching (it again fleshes out the same idea Keith Devlin and others have pushed):

http://j2kun.svbtle.com/you-never-did-math-in-high-school

A Weekend Potpourri… (including Tegmark, Riemann, JMM14)

1) Not being an educator, I often ignore tweets appearing in my math feeds which seem geared strictly for teachers, but a recent one was getting RE-tweeted so often I finally caved and had to look at the link being passed along to see what it was. Indeed it was a delightful, simple idea most any primary school could employ… check it out if you've not seen it:

https://twitter.com/wterral/status/423755360512790528/photo/1

2) Below, a quick note on the packed talk covering recent twin-prime gap work at the Joint Math Meetings in Baltimore, MD.:

http://blogs.ams.org/jmm2014/2014/01/17/mind-the-gap/

3) I'm currently about half-way through Max Tegmark's new book, "Our Mathematical Universe."  Probably won't do a full review of it here since a) it's far more physics than mathematics (despite the title) and b) there are already many reviews of it around, so no need to add to the cacophony of publicity (good and bad) it's receiving.  Having said that, I will say I'm immensely enjoying it as a popular science volume -- in fact, it's one of the BEST cosmology reads I've yet come across (and I've read many)… which is not an endorsement of its views but simply of its readability… as cosmology books go, I'd call it a lively romp! For a more critical look though, see multiverse-phobic ;-) Peter Woit's review here, and be sure to peruse the comments:

http://www.math.columbia.edu/~woit/wordpress/?p=6551

A New Scientist review below (and Dr. Tegmark has been all over the Web promoting the volume as well, just google the book's title):

http://tinyurl.com/jwdfjbv

and more quick takes here:
http://space.mit.edu/home/tegmark/mathematical.html

4) Finally, a fantastic overview (in under 20 mins.) of the Riemann Hypothesis from that gregarious explainer James Grime (if anyone can make the Riemann Hypothesis fun, James can!):



That should fill your weekend for awhile.

Tao Tackles Riemann

Honk if you can read and understand this:

http://terrytao.wordpress.com/2013/07/19/the-riemann-hypothesis-in-various-settings/

...I'm pretty much lost from the word "meromorphically" on… :-(

But seriously, Terry Tao has put up a verrrry long entry on the Riemann hypothesis, which no doubt, serious number theorists will be scrutinizing.
For me the most interesting bit comes at the end, with the first comment posted, invoking a dose of Andrew-Wiles-deja-vu! I quote it in its entirety:

"So Terence Tao has posted a blog regarding the Riemann Hypothesis. He notes that his blog is one that makes 'no new progress' on the hypothesis, but later refers to the entry as one in which he is 'simply arranging existing facts together.'
"And THAT, ladies and gentlemen, would be the perfect way for someone as amazing as Tao to provide a subtle, suave introduction to a long series of posts, the culmination of which could be an actual proof to Riemann’s actual hypothesis.
"Has the already-brilliant Terry Tao solved the Riemann Hypothesis? Stay tuned to his quadrant of internet space to find out…."

I can't imagine that the above wishful scenario would be fulfilled… but hey, it never hurts to stay tuned. (And if Dr. Tao schedules a series of 3 lectures at an upcoming conference, well, the room will be packed to the rafters.)

A Little Bedtime Reading…

…okay, NOT really!

This is probably the most heavy, dense mathematical piece (on the Riemann Hypothesis, quasicrystals, Freeman Dyson, and more) I've ever linked to, and I don't comprehend it… but, all things Riemannian fascinate me, and even without following the mathematics here I think this very long piece from John Baez communicates the beauty and profundity of methodical, thoughtful, mathematical analysis. Simply put, it is, I think, a beautiful piece of mathematical exposition (not at all unusual for Baez)….

http://golem.ph.utexas.edu/category/2013/06/quasicrystals_and_the_riemann.html#more

It starts off thusly:

"Freeman Dyson is a famous physicist who has also dabbled in number theory quite productively.  If some random dude said the Riemann Hypothesis was connected to quasicrystals, I’d probably dismiss him as a crank.  But when Dyson says this, it’s a lot more interesting.  So I’ve been trying to understand his remarks on this.  And it’s been productive, in that I’ve learned some interesting things, and I now feel closer to seeing why the Riemann Hypothesis is a natural and important conjecture."

 ...On a side-note, I also find it interesting that Freeman Dyson advanced this original quasicrystal notion, which has generated a lot of interest (like so many things Dyson has advanced over the years), at the ripe young age of 85! Who says mathematics must be a young man's game!

I Gotta Love This!

see here:

http://math-fail.com/2013/05/riemann-hypothesis-bumper-sticker.html

On investigation I discovered that these have actually been around for awhile, and are given out at American Mathematics Society meetings, but I couldn't find if they were for sale anywhere either on the AMS website or elsewhere. If someone knows about general availability to the public let us know.

Riemann, Dyson, Montgomery, Quasi-crystals, Matrix Theory...

....and, quantum physics.

Below a longish and deep read from the Institute For Advanced Study on the mysterious linkage between the Riemann zeta function and physics (a lot to chew on or just try to comprehend):

https://www.ias.edu/about/publications/ias-letter/articles/2013-spring/primes-random-matrices


Just a couple of passages from the long piece:

 “ 'The Riemann zeta function remains one of the mysteries of modern mathematics. It is a function that we understand a lot about except for the most important question,' says Peter Sarnak, Professor in the School of Mathematics. 'It connects the theory of prime numbers, or encodes deep information about the theory of prime numbers, with the zeros. It controls the prime numbers in a way that nothing else we know does. While understanding prime numbers is an important problem, it is the generalizations of the Riemann zeta function and the objects associated with these that make it more significant.' ”....

"Quasi-crystals were discovered in 1984 and exist in spaces of one, two, or three dimensions. Dyson suggests mathematicians obtain a complete enumeration and classification of all one-dimensional quasi-crystals, the most prevalent type, with the aim of identifying one with a spectrum that corresponds to the Riemann zeta function and one that corresponds to the L-functions that resemble the Riemann zeta function. If it can be proved that a one-dimensional quasi-crystal has properties that identify it with the zeros of the Riemann zeta function, then the Riemann Hypothesis will have been proved."

I haven't had occasion to read the 2nd and 3rd volumes of Matthew Watkin's trilogy on prime numbers, but I can't help but think that the above article may overlap with or relate to some of the notions reached by Watkins who also writes of the 'vibrational' nature of the number system.

Quote… Unquote

Yet again, I'm pulled back to prime numbers for a post… GREAT compendium of quotes pertaining to prime numbers, the zeta function, and Riemann Hypothesis here:

http://empslocal.ex.ac.uk/people/staff/mrwatkin/isoc/quotes.htm

The page comes from (Brit) Matthew Watkins, who co-authored "The Mystery of the Prime Numbers," a volume I haven't read myself, but have seen several uniformly positive reviews of. His webpage linking to prime number-related stuff is here:

http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics.htm

And here a small sampling of quotes from the initially-cited page:

"How can so much of the formal and systematic edifice of mathematics, the science of pattern and rule and order per se, rest on such a patternless, unruly, and disorderly foundation? Or how can numbers regulate so many aspects of our physical world and let us predict some of them when they themselves are so unpredictable and appear to be governed by nothing but chance?"
   -- H. Peter Aleff, from the 'e-book' Prime Passages to Paradise
-----------------------------

"We may - paraphrasing the famous sentence of George Orwell - say that "all mathematics is beautiful, yet some is more beautiful than the other." But the most beautiful in all mathematics is the zeta function. There is no doubt about it."   -- Polish cosmologist Krzysztof Maslanka
-----------------------------

"In [his 1859 paper], Riemann made an incidental remark - a guess, a hypothesis. What he tossed out to the assembled mathematicians that day has proven to be almost cruelly compelling to countless scholars in the ensuing years...

...it is that incidental remark - the Riemann Hypothesis - that is the truly astonishing legacy of his 1859 paper. Because Riemann was able to see beyond the pattern of the primes to discern traces of something mysterious and mathematically elegant at work - subtle variations in the distribution of those prime numbers. Brilliant for its clarity, astounding for its potential consequences, the Hypothesis took on enormous importance in mathematics. Indeed, the successful solution to this puzzle would herald a revolution in prime number theory. Proving or disproving it became the greatest challenge of the age...

It has become clear that the Riemann Hypothesis, whose resolution seems to hang tantalizingly just beyond our grasp holds the key to a variety of scientific and mathematical investigations. The making and breaking of modern codes, which depend on the properties of the prime numbers, have roots in the Hypothesis. In a series of extraordinary developments during the 1970s, it emerged that even the physics of the atomic nucleus is connected in ways not yet fully understood to this strange conundrum. ...Hunting down the solution to the Riemann Hypothesis has become an obsession for many - the veritable 'great white whale' of mathematical research. Yet despite determined efforts by generations of mathematicians, the Riemann Hypothesis defies resolution."  -- J. Derbyshire, from the dustjacket description of Prime Obsession (John Henry Press, 2003)
----------------------------- 

 "For many mathematicians working on it, $1m is less important than the satisfaction that would come from finding a proof. Throughout my researches among the mathematicians' tribe (I have interviewed 30 in the past year), Riemann's Hypotheis was often described to me in awed terms. Hugh Montgomery of the University of Michigan said this was the proof for which a mathematician might sell his soul. Henryk Iwaniec, a Polish-American mathematician, sounded as if he were already discussing terms with Lucifer"

'I would trade everything I know in mathematics for the proof of the Riemann Hypothesis. It's gorgeous stuff. I'm only worried that I'll be unable to understand it. That would be the worst...'"   -- K. Sabbagh, "Beautiful Mathematics", Prospect, January 2002
-----------------------------

"Proving the Riemann hypothesis won't end the story. It will prompt a sequence of even harder, more penetrating questions. Why do the primes achieve such a delicate balance between randomness and order? And if their patterns do encode the behaviour of quantum chaotic systems, what other jewels will we uncover when we dig deeper?

Those who believe mathematics holds the key to the Universe might do well to ponder a question that goes back to the ancients: What secrets are locked within the primes?"   -- E. Klarreich, "Prime Time" (New Scientist, 11/11/00)

--------------------------------------------------------------------------------

 There now exist a great many volumes related to prime numbers for lay folks (it's been a hot topic in recent years). Among my own favorites are (in no particular order):

"The Music of the Primes" -- Marcus du Sautoy
"The Riemann Hypothesis" -- Karl Sabbagh
"Stalking the Riemann Hypothesis" -- Dan Rockmore
"Prime Numbers" by David Wells, a more encyclopedic/academic (but still interesting) offering.

And finally, John Derbyshire's "Prime Obsession" was another hugely popular volume on the subject, though I didn't enjoy it quite as much as the preceding titles.

Crowdsourcing to Solve the Riemann Hypothesis

h/t to C.Pickover for this piece about an Indian mathematician's "ZetaTrek Project," looking at "one-dimensional quasi-crystals" to potentially crowd-source a solution to the Riemann Hypothesis:

http://blog.zombal.com/post/a-quest-to-solve-one-of-maths-great-puzzles

also here:

http://fadereu.posterous.com/knk103-the-crystals-of-mt-zeta

(I have no ability to judge how productive this approach may... or may not be???)