Atiyah Riemann attack/ post mortem/ autopsy, Primes torch carried on

Is Sir Michael Atiyah giving lecture on Monday Sept. 25 @ #HLF18? Yes.
Will he presenenting a proof of the Riemann Hypothesis? Yes, that is what his abstract says. pic.twitter.com/v1dJhUUUEk

— Heidelberg Laureate Forum (@HLForum) September 20, 2018

😳 😥 hi all. am writing this with some heavy heart. in some ways there are a lot of angles to this story, in another way there are none at all. (to paraphrase one cold/ sarcastic respondent, nothing to see here, move along.)

the Riemann hypothesis is one of the worlds most premiere and probably hardest open problems, unsolved after 1½ century, and has a $1M Claymath prize now almost 2 decades old.[31]

coincidentally, just did a brief survey on links between Riemann and physics/ quantum mechanics, a connection that was noticed decades ago by Dyson and has gotten further attention over the years.[39] this crosscutting nature is somewhat rare of deep math/ physics problems, and esp wrt number theory. it even has significant connections to TCS.[13][14]

Atiyah recently announced a “simple” proof presented at the Heidelberg Laureate Forum one of the premiere scientific conferences in the world.[1] preprint is available on google drive (although these links are at risk to rot quickly, rats).[20] he is one of the worlds premiere living mathematicians.[34] he will be 90 years old in less than a year. theres an outstanding series of youtube lectures by him ranging over wide topics including physics + math crossover.[9] wrt the proof “attack” theres substantial reaction across multiple blogs [4][12][26][29][30][44] and mainstream media including quick-to-react New Scientist.[42][2]

there was mass buzz and excitement in cyberspace, eg across social media such as reddit quickly accruing hundreds of comments, some superficial, some substantial.[5][7][10][11] (many people from laymen to professionals are interested in this problem, with good reason, theres real scientific drama here, a relatively rare occurence in that world.) alas, even much to my own chagrin, a lot/ most of that seems to have collapsed quickly in the immediate case.

before the definitely far-from-funny analysis, the episode is evocative of some relevant cartoon humor, which display a kind of timeless wisdom wrt capturing certain deep archetypes definitely in play here. sydney harris famous cartoon I think you should be more explicit here, xkcd number theory/ collatz eventually all your friends will stop calling you, and smbc old physicists, we had a lot of good times.

Atiyah had already offered a proof of the 6-sphere problem within the last year, and it was not accepted by any other mathematicians.[6][8] and, digging deeper, its not exactly/ “merely” because of a “mistake”.

so there is apparently another “problem” going on. think this deserves some words, but am trying to be empathetic and sensitive at same time. sometimes, there are some situations that really have no emoticon to cover them…

insiders/ the mathematicial intelligentsia has been aware of Atiyahs “situation” for awhile but there is no open acknowledgement. there seems to be tiptoeing, whispering, innuendo, gossip. in short, they dont want to talk about it, openly that is.[3][10] aka some “beating around the bush”[37] and/ or “ostrichism”.[38] confronted directly about it (as just happened/ played out in chat room last wknd), the response might be defensive, grouchy, surly, biting, or worse. aka a 3rd rail. a diplomatic response might be “time for him to retire”.

in psychology and hollywood movies in addition to the classic “mid/ ½-life crisis” for 40somethings there is now recognized another concept of a ¼-life crisis for 20somethings. it looks like we are witnessing what might be delicately called a sunset crisis. its well understood that late life changes such as retirement can have substantial psychological impact ie be disorienting, upsetting, stressful, or worse. maybe for a living legend lifetime all-time high achiever like Atiyah, even more.

am going to try a bit )( to delicately-as-possible tiptoe around the landmines myself here. there is an old hindu parable about the Blind Men and the Elephant which any active research area or open problem can be related to. but here we have the Elephant in the Room.[35] here this necessarily is not so much an analysis as something leaning toward/ bordering on a diagnosis… “trigger warning” this analysis is nec going to say a lot about psychology!

so, caveat, rushing in where angels fear to tread™, from the earliest days here now over ½ decade ago its long considered a sterling virtue, now essentially a tradition, to pull no punches™, tell it like it is™… but ofc please dont shoot the messenger™…

one must be careful about Ageism[40] in this situation. but there is something worse than Ageism (a human discriminatory urge/ prejudice) that is real, it is called Age, and sometimes it involves deterioration of mental faculties, that is a statistical/ biological/ medical fact. this is similar to mental illness, although that can strike the young. is senescence leading to senility, dementia, alzheimers, parkinsons, considered mental illness(es)? it seems like a quibble to argue they arent. an “illness” is sometimes a temporary situation but while many mental illnesses are “treatable” they are often not “curable”. and lets face it, aging is the ultimate incurable condition.

one is reminded of the concept of entropy in thermodynamics and the 2nd law that says there is unavoidable increasing entropy in the universe. the manifestation of that in the human realm is Age. a simple example nearly everyone can identify/ relate to is tooth decay. something physical. a much more heavy aspect to ponder is mental decay. cartesian dualism notwithstanding— or the illusion pierced— our minds are run by our brains, right? “something physical”…

we discussed this at length on a chat line (among two other physics Phds, math Phd, etc), and a remark was made: “its like a skeleton popped out of the closet”.[41] in another word, re the title:

DOA

⭐ ⭐ ⭐

there are proofs that are mistaken, and there are proofs that are incoherent. in physics this is known as “not even wrong” as quipped almost a century ago by the sometimes caustic quantum pioneer Pauli.[36] (quantum pioneers faced very similar problems of separating signal from noise in research/ academia at the time.) definitions dont (really) define. etc. for example can anyone state definitely what a Todd function is invoked in his proof?[27] it all may even be written in impeccable/ flawless latex, but if an expert looks at it, they find that its similar to a “(technical) word salad” in places. “word salad” is a another symptom of various mental deteriorations.

years ago saw a similar case alluded to in this blog in earlier days. a Purdue Phd in CS wrote up a complicated ~75pg proof on P vs NP using my favorite, monotone circuit theory. it took me 6 weeks of back-and-forth with the author to figure out what other experts probably realized within a few minutes of scanning the paper. it seems like an oxymoron when stated outright, but there it is: highly (superficially!) polished gobbledygook… so “lesson learned”! the hard way! not exactly the one expected/ intended/ hoped for! and contrary to my own approach, within the scientific community one can forget the idea of professional courtesy, camraderie, or fraternity: its tossed out the window in these “outlier” cases as a perceived waste of time…. possibly fully justifiably so…

notice, Phds are never revoked, but some proofs are “retracted”!

another element. dislike using the word “crank” but sometimes theres no alternative. an aspect of crank physics is to work on deriving deep physics constants. this has been done successfully in the past by the worlds greatest physicists. however, a new derivation of a physical constant is an extremely rare breakthrough, it has happened a few times in a century (it would be really neat to see a historical survey of this by an expert, am not aware of any, the typical focus would tend to be on 1-at-a-time). crackpots directly or intuitively understand this. they are drawn like moths to the flame. and Riemann is a very big flame. so you get many “amateur” attempts/ attacks.[17]

Atiyah is lately turning to physics, analyzed/ “deriving” the fine structure constant in physics, preprints available.[28][15][21][22]

this leads to analysis/ topic of crank psychology. the diplomatic version is in Aaronsons “ten signs a claimed math breakthrough is wrong”[19] and the less diplomatic, more harsh version is Baez “crackpot index”.[18] alas multiple items can now be ticked off in both for Atiyah. (might go into more detail later, its in the chats, for now “you do the math”)[23][24][25]

have been looking for P vs NP angles over decades, and one runs into this phenomenon there. one key aspect of Crankery that have personally noticed, maybe not widely reported by others, is that they dont seem to understand/ grasp logical refutations. more accurately, they adamantly reject careful refutations. in a word, denial. the words “uncomprehending or oblivious” come to mind although this can border on delusional in worst/ extreme cases.

this relates to long/ deep psychological research on a concept called (mental) rigidity. just doing a quick google search this 2002 scientific study finds not-so-surprisingly that rigidity is curvilinearly related [correlated!] with age! (and ofc associated with various mental illnesses such as OCD (obsessive compulsive disorder) and schizophrenia incl a core feature of ASD (autism spectrum disorder)! …)

do others think reference to this type of Crankery is in poor taste? but even with my own distaste for the term, have found it to be unavoidable in some relatively rare but unmistakable/ clearcut cases. ie here, the insiders are saying this is exactly what is going on with Atiyah: one might say that there is a difference between “gaps vs incoherencies”, and experts have personally, apparently to some degree even face-to-face, pointed out the incoherencies of his papers, without coherent response, so to speak. (surprise! “[anti-]? reciprocity”!) 😮

in scientific circles, a retraction is a public embarrassment and humiliation, but sometimes this still honorable simple action separates the lucid from the cranks— its an admission of error. to err is human, to forgive is divine. in the case of a recent ~1yr old P vs NP proof attack by Blum covered in this blog, it was retracted within about a week after the withering-but-correct cyber peer analysis that quickly ensued as it went viral (various parallels to present circumstances). as my grandfather said in a phrase that rings thru my head whenever asking an attractive woman out on a date, cant get hung for trying™…!

is there some tendency of logicians to become mentally ill esp in late life? mental deterioration is sometime milder than so-called “craziness” and other times one blends into the other. there are many (semi-) famous cases: Nash, Goedel, Cantor, Post, Grothendieck (Genius or Crank—Both!). they seem to stick out. whether there are more mathematicians who lose their mind at the end of their life than the general population is very much open to question.[16]

this blog is titled an “attack” as hard problems involve. here maybe somewhat less in the sense of “problem attack” than “heart attack”. which brings up the topic of quixotism and tilting at windmills, a 4-century old observed phenomenon showing in the classic fiction of Cervantes.[32][33] this is a whimsical account of a serious issue: (late life) mental deterioration or illness. Don Quixote is harmless maybe to others but maybe not to himself, he is also mentally ill. has anyone analyzed this novel from the pov of mental illness?

another old expression seems to come into play: better to be silent and thought a fool than to speak up and remove all doubt. this has a double meaning here in the context of the community silence/ “omerta” on their now-tarnished/ fallen leader/ hero…

Don Quixote is portrayed as a sort of poignant/ humorous fool by Cervantes showing some blurring of the idea. apparently mental illness might take the guise of a fool. the Fool is a nearly universal archetype eg seen in Tarot cards (a ripe time to ponder its meaning). but of course mistaking or misattributing mental illness for “mere” foolishness is a sort of foolishness of its own… presumably many fools are not mentally ill and for that they might be grateful (if they had enough sense)…

looking of the definition of Quixotism refers to “enemies or adversaries”. it might not seem to hold here, but wait! a great living TCS theoretician Jukna has talked about circuits as “the (worthwhile/ dangerous) adversary” in his writing. so in our inevitable anthropological tendences, there can be a personification of an inanimate object or even concept as adversary eg Hillary vs Everest.

(realize am being sketchy here myself, and there are a key few quotes out there in cyberspace that were compelling for me and think surely will be for others, they are buried in reddit/ chat threads cited here and may dig them up and cite them here, but thats very timeconsuming. hint: might do this on request + it would help if anyone replies to this post in a respectful or at least noninsulting way.)

⭐ ⭐ ⭐

but there can be some silver lining in this cloud/ “consolation prize”. the primes are a 2millenia old fascination dating to the greeks, and this renews attention to the “atoms of numbers” (that old metaphor is becoming increasingly/ presciently/ delightfully apt these days with the Riemann-QM link). there are eg 5 excellent popsci books on Primes/ Riemann and hope that some new people will be drawn to read/ learn about it.[a2.3][a2.6-9]

theres some outstanding new work on analyzing the statistical nature of primes by physicist-chemists Torquato et al, there was very impressive/ loud press releases and coverage.[a2.1][a2.2][a2.5] none of it mentioned the word “fractal” but think it is highly applicable. came up decades ago with the idea that primes have a fractal nature in college but wasnt able to fully justify it. think it is increasingly accepted across different refs, but havent done a survey on that (really wish someone would). easily a book could be written on it. any takers? maybe Atiyah? (lol!) more seriously, maybe his students… the torch will be carried on at dawn even as the sun sets on one generation…

(speaking of cranks that dont respond to reasoned logic, some ugly troll(s) have been out lambasting/ hounding my innocuous/ totally harmless empirical approach to collatz over quite a few comments now in this blog, possibly picked up some gadflies from reddit it seems. shoo! didnt your mommies teach you, if you cant say anything nice, dont say anything at all? on edge of my seat to hear their erudite verdict on Torquatos unequivocal breakthru assuming they ever get around to noticing it… oh wait nevermind their shallow uninformed knee-jerk devaluation/ denigration/ dismissal is wholly predictable so let me just save you the trouble here… be careful what you wish for, you might get it, so on 2nd thought hope they remain speechless…!)

yet nevertheless (ignoring the naysayers/ anklebiters/ luddites/ troglodytes etc) empirical-physics approaches to number theory/ math finally “officially” introduced/ vindicated under bright lights in late 2018… more to come, stay tuned… at least reddit science noticed it, ~8k votes and ~½k comments!)

Atiyah is like the captain of a ship going down with it, not unlike Don Quixote fighting galliantly to the bitter end, and there is both some real noble honor, and indefatiguable human spirit/ tragedy in that, but it appears his entire crew has now deserted him (at least in defense of his math + physics). and few dare to speak of it. aka lost marbles[43]

borrowing from physics metaphors, a reputational slow/ fast-motion train wreck. a star collapsing into reputational black hole.

speaking of mental illness/ health… from mania to depression… with some anxiety mixed in… 😮 😥 👿

hey didnt really want to write all that, but the lurking/ slinking-away mathematicians left me no choice. really wished someone else had written something like this before diving into the pandoras box myself instead of taking many hours to piece together the full/ real story. sometimes blogging is a dirty job but somebodys gotta do it. caveat emptor to the next guy.

whew! that was a lot to write. yeah, this is very triggering for me (“too”) can you tell? heres a mental note of some key dramatic stuff from chat dialogue maybe for next time:

• retired elderly/ outright hostile curmudgeon TS telling me crossly that Atiyah has been informed about his errors by other experts, is unresponsive, and “go take your math Phd and write your own analysis” after asking for more tangible peer review than a rumor…
• cohort/ sidekick MM calling discussion of the topic spam and proposing to (temporary) block anyone discussing it
• reference to Nash giving an incoherent announced talk in front of aghast mathematicians! dont think that was in the movie, but what a scene that would have been, cant believe they skipped it!
• reddit comment by an insider saying that eg other fields medalists have talked to Atiyah and hes not listening and Atiyah has the hots for physics lately against mathematicians…

“until next time”…

⭐ ⭐ ⭐

(9/28) some more fallout developments (whew, hard to keep up) many from the usually brilliant RJL P vs NP blog + further copious surfing (some on a free day off).[45] math SE direct question about definition of Todd function but not really a direct answer, same on mathoverflow.[47][48] oh just found this on mathoverflow, a meta question struggling with what to do, official mathoverflow mod statement, and mod Trimble saying I think this one is for the blogs, if anyone wants to take it up… (yeah wouldnt that be nice)[51]

more MSM coverage, the science article mentions Atiyah (privately) circulating a 12-page “proof” of Feit-Thompson (which normally takes ~350 pgs). nobody is talking about that much, but this is more red flag crank math.[46] new scientist reports of “skepticism”.[49] it cites another 2010 article suggesting that fine structure “constant” may vary over the universe, something other physicists are pointing out.[50]

and speaking of Big Archetypes In Play… some other english phrases/ idioms/ vernacular now come to mind, there are so many applicable to this situation, and have to maybe find some consolation/ distraction/ diversion in etymological origins research? (1) screw loose (2) off ones rocker (rocking chairs were also seen historically as more appropriate for the elderly… my grandparents had one!) oh, (3) lol! which reminds me, how about this one for the mathematicians reactions?

this is 3 messed-up proofs in less than a year: (1) Feit-Thompson, (2) 6-spheres, (3) Riemann. as a physics Phd cohort remarked/ quipped in chat/ discussion with others shortly on reviewing the fine-structure constant angle,

because if you want to tackle one mystery, you might as well tackle them all at once…[x]

speaking of elephant in the room… that reminds me of Einstein. hes in a class by himself. he did attack multiple key physics mysteries in his life, successfully. rare geniuses like him seem to come around about max 1 per century and honestly Atiyah with his Fields medal + Abel prize (maybe only both held by 1 other living mathematician, Milnor) is definitely in that class. Einstein did tend to become more isolated in the end of his life as he rejected some fundamental QM findings/ foundations (later followed up/ explored in the them of nonlocality of Bohms/ Bells work) and pursued an idiosyncratic grand unified theory that no other physicists considered viable or possibly even sensible. some extreme accounts might have even called him a pariah… speaking of math, “more parallels”! however in contrast Einstein was apparently very reserved about publishing/ circulating his unfinished/ provisional ideas, ie he knew they were incomplete (in english vernacular ½-baked!) but here again theres a distinction between eccentricity and incoherence…

anyway, for those still not talking about it/ addressing it/ confronting directly, what more signs do you want of some kind of mild unravelling and/ or mental instability? and not even counting the derivation of the fine structure constant. this is now like the boy who cried wolf.[53] fool me once, shame on you. fool me twice, shame on me. fooled 3x, yeah, shame on everyone (eg for sticking their collective heads in the sand). now fear that some lurid tabloid story is going to be written on Atiyah, like those written years ago on Perelman, suspect its really just a matter of time. from the science article, somewhat conspicuously still avoiding the general issue of lucidity, seemingly harmless quotes from the Emperor himself,[52] but in larger context, maybe some further evidence of quixoticism/ windmill tilting/ outright crankery:

“I am throwing myself to the lions. I hope to emerge unscathed.” “People will complain and grumble,” Atiyah says, “but that’s because they’re resistant to the idea that an old man might have come up with an entirely new method.”

(9/29) 😀 ❗ 💡 ❤ holy @#%& cow sure am getting a lot of hits, record/ viral for this blog, thx everyone! more latebreaking news, from a brief/ bit )( vague comment/ tip on RJLs blog about “number crunching Riemann”, tracked it down and found Terence Tao/ Polymath15 are working on a very cool numerical attack. more world class vindication for numerical approaches into cutting edge math research/ number theory! (by the winner of the worlds most elite/ valuable $3M math prize) … reddit likes it and thats saying a lot! take that @#$& nasty vicious badmouthing trashtalking luddite troglodyte naysaying detracting denigrating anklebiting hatemonger empirical math troll LOSERS! IN YOUR FACE! 🙄 😛 👿 😈

(10/1) still high views, just adding more links for completeness, other news outlets, mostly patriotic/ generally repetitive british reports, breathlessly talking about the $1M awarded as soon as its verified, none mentioning his other “attacks” context. ScienceAlert[54], Independent[55], Times[56], Mirror[57], NBCNews Mach[58], Azcarraga interview with lots of physics analysis[59], APeriodical linked to Heidelberg Laureate Forum[60] including a comment by Tim Gowers and other critical comments. reddit peanut gallery valvino, jkl9184 et al rain on the Tao/ Polymath parade of attacking Riemann Hypothesis via numerical work as “not pioneering” and “doubtful the project is even tackling the Riemann Hypothesis” lol! rigidly defiant until the end! just like Atiyah and my own beloved blog trolls! 😮

• a. atiyah
• a2. primes

a. atiyah

• 1. 6th HLF – Lecture: Sir Michael Francis Atiyah – YouTube
• 2. A Mathematician Claims He’s Found A “Simple Proof” Of This Million-Dollar Problem | IFLScience
• 3. Sir Michael Atiyah’s conference on the Riemann Hypothesis – MathOverflow
• 4. The Specter of Simple(r) Proofs | Gödel’s Lost Letter and P=NP
• 5. Sir Michael Atiyah announced a proof of the Riemann Hypothesis for monday. The abstract says it is „simple“ : math
• 6. dg.differential geometry – Atiyah’s May 2018 paper on the 6-sphere – MathOverflow
• 7. Is there an error in the preprint published by Atiyah with his proof of the Riemann hypothesis? : math
• 8. dg.differential geometry – What is the current understanding regarding complex structures on the 6-sphere? – MathOverflow
• 9. Michael Atiyah (Mathematician) – YouTube
• 10. Atiyah’s lecture on the Riemann Hypothesis : math
• 11. “The Riemann Hypothesis” by Michael Atiyah – Preprint | Hacker News
• 12. The Reference Frame: Nice try but I am now 99% confident that Atiyah’s proof of RH is wrong, hopeless
• 13. Riemann Hypothesis and Complexity Theory – Theoretical Computer Science Stack Exchange
• 14. cc.complexity theory – Implications of Riemann Hypothesis variants in TCS – Theoretical Computer Science Stack Exchange
• 15. Atiyah’s computation of the fine structure constant (pertinent to RH preprint) : math
• 16. Logic and Madness? | Richard Zach
• 17. Proposed (dis)proofs of the Riemann Hypothesis
• 18. Crackpot index
• 19. Shtetl-Optimized » Blog Archive » Ten Signs a Claimed Mathematical Breakthrough is Wrong
• 20. 2018-The_Riemann_Hypothesis.pdf – Google Drive
• 21. 2018-The_Fine_Structure_Constant (2).pdf – Google Drive
• 22. 2018-The_Fine_Structure_Constant.pdf – Google Drive
• 23. Sir Michael Atiyah’s Proof of the Riemann Hypothesis | chat.stackexchange.com
• 24. Mathematics – 2018-09-24 (page 2 of 5)
• 25. The h Bar – 2018-09-22 (page 1 of 2)
• 26. Atiyah’s problem | Complex Projective 4-Space
• 27. Google search frequency for “Todd function”, “Todd function mathematics” : math
• 28. Abel Lecture ICM 2018 — The future of mathematical physics: new ideas in old bottles, Michael Atiyah – YouTube
• 29. Riemann hypothesis, fine structure constant, Todd function
• 30. Preview of the Atiyah Talk | Gödel’s Lost Letter and P=NP
• 31. HLF Blogs: What is the Riemann Hypothesis? | The Aperiodical
• 32. Quixotism – Wikipedia
• 33. Tilting at windmills – Wikipedia
• 34. Michael Atiyah – Wikipedia
• 35. Elephant in the room – Wikipedia
• 36. Not even wrong – Wikipedia
• 37. etymology – Where does the idiom “beating around the bush” come from? – English Language & Usage Stack Exchange
• 38. ostrichism – Wiktionary
• 39. math highlights 2018¾ | Turing Machine
• 40. Ageism – Wikipedia
• 41. Skeleton in the closet (idiom) – Wikipedia
• 42. Famed mathematician claims proof of 160-year-old Riemann hypothesis | New Scientist
• 43. This is why you’re losing your marbles
• 44. Atiyah and the Fine-Structure Constant
• 45. Reading Into Atiyah’s Proof/ RJLipton blog
• 46. Skepticism surrounds renowned mathematician’s attempted proof of 160-year-old hypothesis/ Science
• 47. What is the Todd’s function in Atiyah’s paper? / Math stackexchange
• 48. What is the definition of the function T used in Atiyah’s attempted proof of the Riemann Hypothesis? / Mathoverflow
• 49. Riemann hypothesis likely remains unsolved despite claimed proof/ new scientist
• 50. Laws of physics may change across the universe/ New scientist
• 51. Is there a way to discuss the correctness of the proof of the RH by Atiyah in MO?/ Mathoverflow
• 52. Emperors new clothes/ Wikipedia
• 53. Boy who cried wolf/ Wikipedia
• 54. Top Mathematician Says He’s Solved a 160-Year-Old Maths Problem Worth $1 Million/ ScienceAlert
• 55. Mathematician claims to have solved 160-year-old Reimann hypothesis/ The Independent
• 56. Sir Michael Atiyah: Age is just a number, says 89-year-old maths genius/ The Times
• 57. Mathematician reveals ‘simple’ solution to a problem that has remained unsolved for 160 YEARS/ The Mirror
• 58. Retired mathematician rocks math world with claim that he’s solved $1 million problem/ NBCNews Mach
• 59. ON MATHEMATICS AND PHYSICS: A CONVERSATION WITH SIR MICHAEL ATIYAH/ J. Adolfo de Azcárraga
• 60. Atiyah Riemann Hypothesis proof: final thoughts/ Steckles, Lawson-Perfect/ The APeriodical

a2. primes

• 1. A Chemist Shines Light on a Surprising Prime Number Pattern | Quanta Magazine
• 2. Surprising hidden order unites prime numbers and crystal-like materials
• 3. Closing the Gap: The Quest to Understand Prime Numbers: Vicky Neale: 9780198788287: Amazon.com: Books
• 4. Prime Numbers and the Riemann Hypothesis
• 5. Uncovering multiscale order in the prime numbers via scattering – IOPscience
• 6. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics: John Derbyshire: 9780452285255: Amazon.com: Books
• 7. The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics: Marcus du Sautoy: 9780062064011: Amazon.com: Books
• 8. The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics: Karl Sabbagh: 9780374529352: Amazon.com: Books
• 9. Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers: Dan Rockmore: 9780375727726: Amazon.com: Books