61. Pantheon | Catalog | Stalking The Riemann Hypothesis By Dan Rockmore Stalking the riemann hypothesis The Quest to Find the Hidden Law of Prime NumbersWritten by Dan Rockmore Mathematics Advanced Pantheon Hardcover http://www.randomhouse.com/pantheon/catalog/display.pperl?isbn=037542136X

62. Chaitin, Thoughts On The Riemann Hypothesis the riemann hypothesis concerning the location of the complex zeroes of the Riemann In spite of the fact that I regard the riemann hypothesis as an http://www.maa.org/features/chaitin.html

Search MAA Online MAA Home Thoughts on the Riemann Hypothesis G. J. Chaitin, IBM Research The simultaneous appearance in May 2003 of four books on the Riemann hypothesis (RH) provoked these reflections. We briefly discuss whether the RH should be added as a new axiom, or whether a proof of the RH might involve the notion of randomness. New pragmatically-justified mathematical axioms that are not at all self-evident Are there important mathematical propositions for which there is a considerable amount of computational evidence, evidence that is so persuasive that a physicist would regard them as experimentally verified? Yes, I believe there are. Currently, the two best candidates* for useful new axioms of the kind that that are justified pragmatically as in physics are: the P NP hypothesis in theoretical computer science that conjectures that many problems require an exponential amount of work to resolve, and the Riemann hypothesis concerning the location of the complex zeroes of the Riemann zeta-function z s n n s p p s (Here n ranges over positive integers and p ranges over the primes.)**

63. Read This: The Riemann Hypothesis Read This! The MAA Online book review column review of The riemann hypothesisThe Greatest Unsolved Problem of Mathematics, by Karl Sabbagh. http://www.maa.org/reviews/sabbaghRH.html

Search MAA Online MAA Home Read This! The MAA Online book review column The Riemann Hypothesis: The Greatest Unsolved Problem of Mathematics by Karl Sabbagh Reviewed by S. W. Graham Within the past few months, we have seen the publication of three popular books on the Riemann Hypothesis (RH). I have reviewed the book by Derbyshire for MAA Online. I have not read the book by du Sautoy , but it too is being reviewed on MAA Online. The books by Derbyshire and Sabbagh focus on different aspects of the Riemann Hypothesis story. Derbyshire writes at length about the mathematics behind the Riemann zeta-function, and he makes a considerable effort to explain it to a non-expert reader. Sabbagh's discussion of the mathematics is fairly cursory, though he does have some nice graphs of the zeta-function in Chapter 5. Derbyshire writes extensively about Riemann's life, and he puts it in the context of the political and educational landscape of 19th century Europe. Sabbagh has little discussion of Riemann's life or times. Instead, he concentrates on contemporary mathematicians, and he attempts to explain what contemporary mathematical research is like. Overall, Sabbagh imparts a reasonably accurate view of what mathematicians are like, although he probably overemphasizes some of the stranger aspects of our behaviour. The joys and frustrations of mathematical research are well portrayed. There are many good quotes in the book, and it will probably be widely quoted. Indeed, one can already find numerous passages from the book on the internet.

64. Riemann Hypothesis What the riemann hypothesis says is that the nontrivial zeros of the Riemann If the riemann hypothesis turns out to be true, what do these harmonics in http://www.daviddarling.info/encyclopedia/R/Riemann_hypothesis.html

Return to The Worlds of David Darling INDEX LATEST NEWS ARCHIVE ... Z entire Web this site Riemann hypothesis The most important open question in number theory and, possibly, in the whole of mathematics. David Hilbert said: "If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?" A $1 million prize has been offered by the Clay Mathematics Institute for a proof. The hypothesis was first formulated by Bernhard Riemann in 1859, was included in David Hilbert's list of challenging problems for twentieth-century mathematicians, and is widely believed to be true. Yet a proof remains tantalizingly out of reach. What the Riemann hypothesis says is that the non-trivial zeros of the Riemann zeta function all have real part equal to 1/2. In plain language, the hypothesis asserts that there is an underlying order, akin to musical harmonics, in the way prime numbers are distributed. It's known that for any given number n there are approximately

65. D2r: Sunday Entertainment: The Riemann Hypothesis Going back to the riemann hypothesis. Proving it would be interesting for Okay, given that function, the riemann hypothesis is saying that all the http://www.dynamicobjects.com/d2r/archives/002340.html

Home Search/Archives Links About diego's weblog Main sunday entertainment: the Riemann Hypothesis Working on some stuff on n-dimensional topologies I remembered the Riemann Hypothesis. I'm not sure when I first read about it... but it keeps popping up in my head (simply because it's a challengenot that I have any delusions that I can actually solve it! :)). So as a way of relaxing a bit from work, here's some background on that. Not that it's something useful, but it should be at least entertaining. A couple of years ago, a $1,000,000 prize was created for whoever came up with the proof. What I didn't know was that they actually had seven "millenium prize problems" of which the Riemann Hypothesis was one. Here is the list of all seven problems. One million per proof. Not bad! Going back to the Riemann Hypothesis. Proving it would be interesting for number theory and the distribution of prime numbers, but it would have no direct practical applications whatsoever. Otherwise, I can see the headlines (maybe for The Onion UN Security Council Declares Understanding the Distribution of Prime Numbers is Priority One - Troops To Be Deployed in the Real Region of C with R > 1 - President declares "we will hunt down these prime number folks and we will characterize them. If you ask me, there's something evil about a number being divisible only by itself and one."

66. The Riemann Hypothesis PlanetMath.org; riemann hypothesis And Superconformal Invariance The Extendedriemann hypothesis and its Application to Computation PDF http://mathews.ecs.fullerton.edu/c2003/RiemannHypothesisBib/Links/RiemannHypothe

Internet Resources for the Riemann Hypothesis Riemann Hypothesis Eric Weisstein, World of Mathematics, Wolfram Res., Inc., Champaign, IL Riemann Hypothesis Lecture Jeff Vaaler, Clay Mathematics Institute, Cambridge, MA Artin's Conjecture Eric Weisstein, World of Mathematics, Wolfram Res., Inc., Champaign, IL The Riemann hypothesis Louis de Branges de Bourcia, Math. Dept., Purdue Univ., Lafayette IN Proposed proofs of the Riemann Hypothesis Matthew R. Watkins, Math. Sciences, Exeter Univ., U. K. The Riemann Hypothesis Chris K. Caldwell, Math. Dept., University of Tennessee, Martin, TN The Riemann Hypothesis in a Nutshell Glen Pugh, Math. Dept., Univ. of British Columbia, Vancouver, B.C., Canada Riemann Hypothesis E. Bombieri, Clay Mathematics Institute, Cambridge, MA The Zeta-Function of Riemann Krzysztof Maslanka, Astronomical Observatory, Jagiellonian Univ., Cracow, Poland Verification of the Riemann Hypothesis The Riemann Hypothesis Kimon-Elpidoforos Spiliopoulos, ICAP S.A., Athens, Greece Riemann hypothesis Wikipedia, the free encyclopedia, www.wikipedia.org

67. [physics/0503141] Comment On The Riemann Hypothesis The riemann hypothesis is identified with zeros of ${\cal N}=4$ supersymmetricgauge theory fourpoint amplitude. The zeros of the $\zeta(s)$ function are http://arxiv.org/abs/physics/0503141

Physics, abstract physics/0503141 From: Gordon Chalmers Dr [ view email ] Date: Wed, 16 Mar 2005 20:21:46 GMT (7kb) Comment on the Riemann Hypothesis Authors: Gordon Chalmers Comments: 7 pages, LaTeX Subj-class: General Physics Full-text: PostScript PDF , or Other formats References and citations for this submission: CiteBase (autonomous citation navigation and analysis) Which authors of this paper are endorsers? Links to: arXiv physics find abs

68. [math-ph/0504035] Riemann Hypothesis And Short Distance Fermionic Green's Functi We formulate the riemann hypothesis physically as a nonzero condition on thetransition amplitude between two special states associated with the point of http://arxiv.org/abs/math-ph/0504035

Mathematical Physics, abstract math-ph/0504035 From: Michael McGuigan [ view email ] Date: Sun, 10 Apr 2005 00:06:19 GMT (958kb) Riemann Hypothesis and Short Distance Fermionic Green's Functions Authors: Michael McGuigan Comments: 96 pages, 20 figures Subj-class: Mathematical Physics Full-text: PostScript PDF , or Other formats References and citations for this submission: CiteBase (autonomous citation navigation and analysis) Which authors of this paper are endorsers? Links to: arXiv math-ph find abs

69. Seminary Co-op Bookstore The riemann hypothesis The Greatest Unsolved Problem in Mathematics In Theriemann hypothesis, acclaimed author Karl Sabbagh interviews some of the http://semcoop.booksense.com/NASApp/store/Product?s=showproduct&isbn=0374529353

70. PlanetMath: Generalized Riemann Hypothesis generalized riemann hypothesis is owned by mathcam. This is version 2 ofgeneralized riemann hypothesis, born on 200308-28, modified 2003-08-29. http://planetmath.org/encyclopedia/GeneralizedRiemannHypothesis.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About generalized Riemann hypothesis (Definition) This generalization of the Riemann hypothesis to arbitrary Dedekind zeta functions states that for any number field , the only zeroes of the Dedekind zeta function that lie in the strip satisfy "generalized Riemann hypothesis" is owned by mathcam view preamble View style: HTML with images page images TeX source This object's parent Cross-references: number field Dedekind zeta functions Riemann hypothesis There is 1 reference to this object. This is version 2 of generalized Riemann hypothesis , born on 2003-08-28, modified 2003-08-29. Object id is 4662, canonical name is GeneralizedRiemannHypothesis. Accessed 942 times total. Classification: AMS MSC Pending Errata and Addenda None.

71. PlanetMath: Riemann Zeta Function The celebrated riemann hypothesis asserts that all nontrivial zeros $ s$ of thezeta function Also defines, Euler product formula, riemann hypothesis http://planetmath.org/encyclopedia/EulerProductFormula.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About Riemann zeta function (Definition) Definition The Riemann zeta function is defined to be the complex valued function given by the series which is valid (in fact, absolutely convergent ) for all complex numbers with . We list here some of the key properties ] of the zeta function. For all with , the zeta function satisfies the Euler product formula where the product is taken over all positive integer primes , and converges uniformly in a neighborhood of The zeta function has a meromorphic continuation to the entire complex plane with a simple pole at , of residue , and no other singularities. The zeta function satisfies the functional equation for any (where denotes the Gamma function Distribution of primes The Euler product formula ( ) given above expresses the zeta function as a product over the primes , and consequently provides a link between the analytic properties of the zeta function and the distribution of primes in the integers. As the simplest possible illustration of this link, we show how the properties of the zeta function given above can be used to prove that there are infinitely many primes.

72. Jascha Hoffman: Prime Time The riemann hypothesis Karl Sabbagh Farrar, Straus Giroux, $25 (cloth) The riemann hypothesis is, roughly speaking, a 150year-old guess about how the http://www.bostonreview.net/BR29.2/hoffman.html

CURRENT ISSUE table of contents FEATURES new democracy forum new fiction forum poetry fiction ... archives ABOUT US masthead mission rave reviews contests ... advertising SERVICES bookstore locator literary links subscribe Search this site or the web Powered by FreeFind Site Web Prime Time Jascha Hoffman Prime Obsession John Derbyshire Joseph Henry Press, $28 (cloth) The Riemann Hypothesis Karl Sabbagh The Music of the Primes Marcus du Sautoy HarperCollins, $25 (cloth) Elements not prime, then it must be expressible as a product of smaller whole numbers, and we may continue factoring until it is written as a product of primes. None of these factors can be 2, 3, 5, or 7, so you have still produced primes not in your original set. Either way, your list must be incomplete. Since this argument works for any finite list of primes, the number of primes cannot be finite. This is a fantastic argument: without having any idea how to come up with the next prime, Euclid was able to prove that the supply is inexhaustible. Music of the Primes . Struck by their unpredictability as a teenager in 1792, Gauss decided to count up the primes in the first 10, 100, 1,000, and 10,000 numbers to see if a pattern emerged. He noticed that the primes got thinner and thinner the further you counted: a full 40 percent of the first ten were prime, but only 25 percent of the first 100 and 16.8 percent of the first thousand were. He chanced on a surprisingly effective formula for estimating these proportions: the number of primes smaller than some number n was approximately n/(log

73. Riemann Hypothesis -- From MathWorld Kids.net.au Encyclopedia riemann hypothesis -Most mathematicians believe the riemann hypothesis to be true. The Riemannhypothesis is concerned with the non-trivial zeros, and states that http://www.astro.virginia.edu/~eww6n/math/RiemannHypothesis.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Calculus and Analysis Special Functions Riemann Zeta Function ... Wedeniwski Riemann Hypothesis First published by Riemann (1859), the Riemann hypothesis states that the nontrivial Riemann zeta function zeros , i.e., the values of other than -2, -4, -6, ... such that (where is the Riemann zeta function ) all lie on the " critical line " (where denotes the real part of ). While it was long believed that Riemann's hypothesis was the result of deep intuition on the part of Riemann , an examination of his papers by C. L. Siegel showed that Riemann had made detailed numerical calculations of small zeros of the Riemann zeta function to several decimal digits (Granville 2002; Borwein and Borwein 2003, p. 68). A more general statement known as the generalized Riemann hypothesis conjectures that neither the Riemann zeta function nor any Dirichlet L-series has a zero with real part larger than 1/2.

74. Riemann Hypothesis The riemann hypothesis is a conjecture about the distribution of the The riemann hypothesis is concerned with the nontrivial zeros, and states that http://www.algebra.com/algebra/about/history/Riemann-hypothesis.wikipedia

Riemann hypothesis Regular View Dictionary View (all words explained) Algebra Help my dictionary with pronunciation , wikipedia etc Riemann hypothesis In mathematics , the Riemann hypothesis (aka Riemann zeta hypothesis ), first formulated by Bernhard Riemann in , is one of the most famous of all unsolved problems. It has been an open question for well over a century, despite attracting concentrated efforts from many outstanding mathematicians. Unlike some other celebrated problems, it is more attractive to professionals in the field than to amateurs. The Riemann hypothesis is a conjecture about the distribution of the zeros of the Riemann zeta function s ). The Riemann zeta function is defined for all complex numbers s ≠ 1. It has certain so-called "trivial" zeros for s s s = −6, ... The Riemann hypothesis is concerned with the non-trivial zeros, and states that: The real part of any non-trivial zero of the Riemann zeta function is ½. Thus the non-trivial zeros should lie on the so-called critical line it with t a real number and i the imaginary unit . The Riemann zeta function along the critical line is sometimes studied in terms of the Z function , whose real zeros correspond to the zeros of the zeta function on the critical line.

75. Generalized Riemann Hypothesis -- Facts, Info, And Encyclopedia Article The generalized riemann hypothesis was probably formulated for the first time by Like the original riemann hypothesis, it has far reaching consequences http://www.absoluteastronomy.com/encyclopedia/g/ge/generalized_riemann_hypothesi

Generalized Riemann hypothesis [Categories: Conjectures, Algebraic geometry, Algebraic number theory, Analytic number theory] The (Click link for more info and facts about Riemann hypothesis) Riemann hypothesis is one of the most important (Reasoning that involves the formation of conclusions from incomplete evidence) conjecture s in (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics . It is a statement about the zeros of the (Click link for more info and facts about Riemann zeta function) Riemann zeta function . Various geometrical and arithmetical objects can be described by so-called global (Click link for more info and facts about L-functions) L-functions , which are formally similar to the Riemann zeta function. One can then ask the same question about the zeros of these L-functions, yielding various generalizations of the Riemann hypothesis. None of these conjectures have been proven or disproven, but many mathematicians believe them to be true. Global L-functions can be associated to (Click link for more info and facts about elliptic curve) elliptic curve s

76. Riemann Hypothesis -- Facts, Info, And Encyclopedia Article The riemann hypothesis is a (Reasoning that involves the formation of The riemann hypothesis is concerned with the nontrivial zeros, and states that http://www.absoluteastronomy.com/encyclopedia/r/ri/riemann_hypothesis.htm

Riemann hypothesis [Categories: Conjectures, Zeta functions, Analytic number theory] (Any of several hormones produced in the hypothalamus and carried by a vein to the anterior pituitary gland where they stimulate the release of anterior pituitary hormones; each releasing hormone causes the anterior pituitary to secrete a specific hormone) RH directs here. RH is also the common abbreviation for the (A serialized program usually dealing with sentimentalized family matters that is broadcast on radio or television (frequently sponsored by a company advertising soap products)) soap opera (Click link for more info and facts about Ryan's Hope) Ryan's Hope In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics , the Riemann hypothesis (aka Riemann zeta hypothesis ), first formulated by (Pioneer of non-Euclidean geometry (1826-1866)) Bernhard Riemann in 1859, is one of the most famous of all unsolved problems. It has been an open question for well over a century, despite attracting concentrated efforts from many outstanding mathematicians. Unlike some other celebrated problems, it is more attractive to professionals in the field than to amateurs. The Riemann hypothesis is a (Reasoning that involves the formation of conclusions from incomplete evidence) conjecture about the distribution of the (The quantity that registers a reading of zero on a scale) zero s of the (Click link for more info and facts about Riemann zeta function) Riemann zeta function s

77. Crooked Timber » » Riemann Hypothesis Proved ? Purdue University news release reports a proof of the riemann hypothesis by The riemann hypothesis is not the last significant mathematical conjecture. http://crookedtimber.org/2004/06/11/riemann-hypothesis-proved

Crooked Timber Main Operation Bagration Riemann hypothesis proved ? Posted by John Quiggin According to this report It is also, I think, the last of the big and well-known unsolved problems in mathematics, and it would be nice to see the search ending in success. Some of the other big problems have been closed, rather than solved. The classic problems of the Greeks such as squaring the circle were shown to be insoluble in the 19th century, and the Hilbert program of formalisation was shown by Godel to be infeasible. And the four-colour problem (not a really important problem, but a big one because it was easily described, interesting and very tough) was dealt with by a brute-force computer enumeration. Almost instant update Commenter Eric on my blog points to Mathworld posted on Friday, June 11th, 2004 at 3:54 am comments Posted by Jason Kuznicki June 11th, 2004 at 4:34 am Concerning the purported proof of the twin prime conjecture, the following is from http://www.youngmath.net/concerns/story/2004/5/29/131042/934 Posted by Michael Kremer · June 11th, 2004 at 4:45 am

78. IFTF's Future Now: Riemann Hypothesis And The Future Of E-commerce The riemann hypothesis asserts that all interesting solutions of the equation If the riemann hypothesis is true, it won t produce a prime number http://future.iftf.org/2004/09/riemann_hypothe.html

hostName = '.iftf.org'; IFTF's Future Now Emerging technologies and their implications for the future. About the Institute for the Future The Institute for the Future is a non-profit research organization located in Palo Alto, California. Founded in 1968, the Institute conducts basic research on the futures of emerging technologies health , and global business trends . It also works with clients on targeted research projects. www. flick r .com This is a Flickr badge showing photos in a set called Work . Make your own badge here See all IFTF-tagged pictures on Flickr About IFTF's Future Now IFTF's Future Now draws research and forecasting at the Institute for the Future , a Palo Alto, CA think tank specializing in the future of technology, health, and organizational change. (Read more about the blog here Contributors IFTF's Future Now is a group weblog, founded by Institute research director Alex Soojung-Kim Pang in September 2003. Its contributors include IFTF researchers interested in emerging technologies, the future of Asia, and the social and economic impacts on new technologies; IFTF corporate affiliates; academic partners; and members of the Innovation Lab, a Danish futures group with offices in Aarhus and Copenhagen. A complete list of contributors is available here Blogdex Futures Sci/tech Digital culture Design Pervasive computing Subscribe to this blog's feed Colophon About IFTF Powered by TypePad Main September 13, 2004

79. The Riemann Hypothesis The riemann hypothesis. The famous riemann hypothesis is equivalent to theassertion that. $\displaystyle \pi(x) = \Li(x) +O http://modular.fas.harvard.edu/edu/Fall2001/124/lectures/lecture4/html/node5.htm

Next: About this document ... Up: How many primes are Previous: Counting Primes Today The Riemann Hypothesis The function is also a good approximation to The famous Riemann Hypothesis is equivalent to the assertion that (This is another $1000000 prize problem.) pi(10^22) = 201467286689315906290 Li(10^22) = 201467286691248261498.1505... (using Maple) Log(x)/(x-1) = 201381995844659893517.7648... (pari) William A Stein 2001-09-19

80. New Scientist Books - Stalking The Riemann Hypothesis: The Quest To Find The Hid Dan Rockmore s Stalking the riemann hypothesis is a fine addition. Much of thecontent may be familiar, but Rockmore s style, emphasis and language are all http://www.newscientist.com/article/mg18524912.700.html

Free E-Zine Subscribe to Magazine Customer Service 17 September 2005 HO M E N ... JOBS JOB OF THE WEEK University of California, Davis Davis, US California Faculty Positions in Cell and Molecular Biology Weill Medical College, Cornell University ... Change address Stalking the Riemann Hypothesis: The quest to find the hidden law of numbers by Dan Rockmore 19 March 2005 From New Scientist Print Edition. Subscribe and get 4 free issues. Ben Longstaff Book Details Stalking the Riemann Hypothesis by Dan Rockmore Pantheon Books ISBN 037542136X (published in April) "THE riddle does not exist. If a question can be put at all, then it can also be answered." Presumably this remark by philosopher Ludwig Wittgenstein is a sick joke at the expense of mathematicians, who frequently spend their whole careers clawing at unyielding riddles. And who can blame them for persevering with exquisite torments like the 150-year-old Riemann hypothesis? It may be an arcane puzzle about the array of prime numbers, but the Riemann hypothesis is also hugely important, and its story has spawned many excellent books. Dan Rockmore's Stalking the Riemann Hypothesis is a fine addition. Much of the content may be familiar, but Rockmore's style, emphasis and language are all admirably fresh, and it is fully accessible to non-specialists. Of course, we are still waiting for the final chapter in the story - that someone proves the hypothesis - but Rockmore is an excellent guide to take you right to the edge of the mathematical map, and he's bang up to date.

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