1. Clay Mathematics Institute Article by Enrico Bombieri (PDF) and video by Jeff Vaaler (.ram) from the Clay Mathematics Institute. http://www.claymath.org/millennium/Riemann_Hypothesis/

Clay Mathematics Institute Dedicated to increasing and disseminating mathematical knowledge HOME ABOUT CMI PROGRAMS AWARDS ... PUBLICATIONS Riemann Hypothesis Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern, however the German mathematician G.F.B. Riemann (1826 - 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function s s s called the Riemann Zeta function . The Riemann hypothesis asserts that all interesting solutions of the equation lie on a certain vertical straight line. This has been checked for the first 1,500,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers. The Millennium Problems Lecture by Jeff Vaaler at the University of Texas (video) Facsimile of Riemann's 1859 manuscript Return to top Contact Search Clay Mathematics Institute

2. Riemann Hypothesis - Wikipedia, The Free Encyclopedia Bollobas, Bela, foreword to Littlewood s Miscellany, Cambridge University Press, 1986. Retrieved from http//en.wikipedia.org/wiki/riemann_hypothesis http://en.wikipedia.org/wiki/Riemann_hypothesis

Riemann hypothesis From Wikipedia, the free encyclopedia Jump to: navigation search There is also the Riemann hypothesis for curves over finite fields The real part (red) and imaginary part (blue) of the Riemann zeta-function along the critical line Re( s ) = 1/2. You can see the first non-trivial zeros at Im( s ) = ±14.135, ±21.022 and ±25.011. This shows the values of ζ(1/2+it) in the complex plane for ≤ t ≤ 34. (For t=0, ζ(1/2) ≈ -1.460 corresponds to the leftmost point of the red curve.) The Riemann hypothesis (also called the Riemann zeta-hypothesis ), first formulated by Bernhard Riemann in 1859, is one of the most famous and important unsolved problems in mathematics . It has been an open question for almost 150 years, 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 (RH) is a conjecture about the distribution of the zeros of the Riemann zeta-function s ). The Riemann zeta-function is defined for all

3. Riemann Hypothesis - Wikipedia Riemann hypothesis. The Riemann hypothesis, first formulated by Bernhard Riemann in 1859, is a conjecture about the distribution of the zeros of Riemann s http://wikipedia.inetbridge.net/ri/Riemann_hypothesis.html

Wikipedia Wiki Content Riemann hypothesis [Your user agent does not support frames or is currently configured not to display frames. However, you may visit the related document. The Riemann hypothesis , first formulated by Bernhard Riemann in , is a conjecture about the distribution of the zeros of s . It is one of the most important open problems of contemporary mathematics ; a $1,000,000 prize has been offered by the Clay Mathematics Institute for a proof. Most mathematicians believe the Riemann hypothesis to be true. s ) is defined for all complex numbers s 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 1/2. Thus the non-trivial zeros should lie on the so-called critical line it with t a real number and i the imaginary unit This traditional formulation obscures somewhat the true importance of the conjecture. The zeta function has a deep connection to the distribution of prime numbers and Helge von Koch proved in that the Riemann hypothesis is equivalent to the following considerable strengthening of the prime number theorem x ) is the prime-counting function , ln( x ) is the natural logarithm of x , and the O-notation is the Landau symbol The zeros of the Riemann zeta function and the prime numbers satisfy a certain duality property, known as the

4. Science : Math : Number_Theory : Analytic : Riemann_Hypothesis : - NoMoz.org Quick Button Reference Learn about NoMoz.org. close. Options. Directory, Entire Web. Top Science Math Number_Theory Analytic riemann_hypothesis http://www.nomoz.org/Top/Science/Math/Number_Theory/Analytic/Riemann_Hypothesis/

Arts Business Computers Games ... ABOUT Options: Directory Entire Web Top Science Math Analytic Riemann Hypothesis Top/Science/Math/Number Theory/Open Problems The Riemann Hypothesis - A prime pages article by Chris K. Caldwell. http://primes.utm.edu/notes/rh.html Computation of Zeros of the Zeta Function - Verification of RH up to the 10^13-th zero, with details of the computations and further results, by Xavier Gourdon with the help of Patrick Demichel. http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeroscompute.html Louis de Branges de Bourcia - Purdue University. Proposed solution for the Riemann Hypothesis (pdf format). http://www.math.purdue.edu/~branges/ Proof of Riemann's Hypothesis - A claimed proof of Riemann's Hypothesis. http://www.coolissues.com/mathematics/Riemann/riemann.htm Easy Proof of Riemann's Hypothesis - Generalisations of the zeta function might provide a proof of Riemann's hypothesis. http://www.coolissues.com/mathematics/Zeta/zeta.htm Zetagrid - Numerical verification of the Riemann Hypothesis by a collaborative computing effort, with downloadable software. http://www.zetagrid.net/

5. CiteULike: Tag Riemann_hypothesis [3 Articles] Recent papers classified by the tag riemann_hypothesis. posted to riemann_hypothesis primes by NitinCR on 200705-29 051219 as http://212.69.38.158/tag/riemann_hypothesis

Register Log in FAQ Sponsored link CiteULike News CiteGeist Discussion Journals Browse current issues Groups Search groups Tag riemann_hypothesis [3 articles] Recent papers classified by the tag riemann_hypothesis. Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings (Springer Monographs in Mathematics) (10 August 2006) by Michel L Lapidus , Machiel van Frankenhuijsen posted to by NitinCR on 2008-03-02 21:24:09 as Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (25 May 2004) by John Derbyshire posted to primes by NitinCR on 2007-05-29 05:12:19 as Thoughts on the Riemann hypothesis (22 Sep 2003) by GJ Chaitin posted to by NitinCR on 2006-07-18 18:30:50 as Note: You may cite this page as: http://www.citeulike.org/tag/riemann_hypothesis RIS BibTeX RSS Related Tags Tags related to: riemann_hypothesis Filter: primes CiteULike organises scholarly (or academic) papers or literature and provides bibliographic (which means it makes bibliographies) for universities and higher education establishments. It helps undergraduates and postgraduates. People studying for PhDs or in postdoctoral (postdoc) positions. The service is similar in scope to EndNote or RefWorks or any other reference manager like BibTeX, but it is a social bookmarking service for scientists and humanities researchers.

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8. 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 http://www.daviddarling.info/encyclopedia/R/Riemann_hypothesis.html

NUMBER THEORY A B C ... CONTACT 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 20th-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. Broadly speaking, 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 n /log n prime numbers that are less than n . The formula is not exact: sometimes it is a little high and sometimes a little low. Riemann looked at these deviations and found that they contain periodicities. His hypothesis quantifies and formalizes this discovery, positing that the zeros of the zeta function can be regarded as the harmonic frequencies in the distribution of primes. If the Riemann hypothesis turns out to be true, what do these harmonics in the "music" of the primes mean?

9. Riemann Hypothesis - Wikipedia Retrieved from http//nostalgia.wikipedia.org/wiki/riemann_hypothesis . This page was last modified 1636, 26 September 2001. Content is available under http://nostalgia.wikipedia.org/wiki/Riemann_hypothesis

Riemann hypothesis From Wikipedia HomePage Recent changes View source Page history ... Log in Special pages Double redirects Broken redirects Disambiguation pages Log in Preferences My watchlist Recent changes Upload file File list Gallery of new files User list Statistics Random page Orphaned pages Uncategorized pages Uncategorized categories Uncategorized files Uncategorized templates Unused categories Unused files Wanted pages Wanted categories Most linked to pages Most linked to categories Most linked-to templates Pages with the most categories Most linked to files Pages with the most revisions Pages with the fewest revisions Short pages Long pages New pages Oldest pages Dead-end pages Protected pages Protected titles All pages Prefix index List of blocked IP addresses and usernames User contributions What links here Book sources Categories Export pages Version System messages Logs MIME search Search for duplicate files List redirects Unused templates Random redirect Pages without language links File path Search List of Wikimedia wikis Expand templates CategoryTree Gadgets Parser diff test Cross-namespace links Wikimedia Board of Trustees election Search web links Printable version The Riemann hypothesis , first formulated by Bernhard Riemann in 1859, is a

10. Riemann Hypothesis - RationalWiki Retrieved from http//rationalwiki.com/wiki/riemann_hypothesis . Categories Articles needing expansion Articles needing RWification Very short article http://rationalwiki.com/wiki/Riemann_Hypothesis

Riemann Hypothesis From RationalWiki Jump to: navigation search The Riemann Hypothesis states that a cute lolcat can cover for an empty page. edit See also Dead end pages S T U B The Pet Goat ate it) Retrieved from " http://rationalwiki.com/wiki/Riemann_Hypothesis Categories Articles needing expansion Articles needing RWification ... Very short article Views Article Talk Edit fossil record Personal tools Log in / create account Navigation Main Page Recent changes Random page Help Files ... Forums Search Toolbox What links here Related changes Upload file Special pages ... Site support This page was last modified 02:39, 25 February 2008. This page has been accessed 63 times. All text, unless explicitly noted otherwise, is licensed under the GNU Free Documentation License 1.2 About RationalWiki

11. AUMTP656f07 » Georg Bernhard Riemann » Code (the following explanationg of the Riemann Zeta function is partially taken from the http//www.claymath.org/millennium/riemann_hypothesis/ Clay http://aumtp656f07.wikispaces.com/page/code/Georg Bernhard Riemann

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14. Porydd Thumbnail of http//numbers.computation. Computation of Zeros of the Zeta Function Verification of RH up to the 10^13th zero, with details of the http://odp.rpfuller.com/porydd/?cat=Science/Math/Number_Theory/Analytic/Riemann_

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var SERVERDOM = ".stumbleupon.com"; StumbleUpon Discover new sites Home Websites ... Videos Have an account? Login Website review: Riemann hypothesis - Wikipedia, the... Bradley discovered this in Mathematics reviews since Jul 19, 2005 mathematics riemann en.wikipedia.org/wiki/Riemann_hypothesis People who like this website Show me more (5) People online now Top stumblers Please Login or Join StumbleUpon to see other people who like this site. It takes less than 2 mins to join. kancerman themasterchief Colorado muscmussell Charleston ossamasyed Strongsville xiist Buffalo chrismeer Mount Airy lastsalvo Pasadena New York AChronicArgonaut Berkeley Bradley San Francisco StumbleUpon is the best way to discover great web sites, videos, photos, blogs and more - based on your interests. Everything is submitted and rated by the community. Discover, share and review the best of the web! Reviews of this website Add Your Review Please Login or Join StumbleUpon to add a review. It takes less than 2 mins to join. Share this with a friend Take a tour Sign up for free Meet Bradley Bradley discovered this site. He is a 36 year old guy from

17. SAGE Worksheet: Sato-tate Traceback (most recent call last) g40000 = delta_dist(40,40000) File /home/was/papers/riemann_hypothesis/sage_notebook/worksheets/sato_tate/cells/code13\ http://sage.math.washington.edu/sato-tate/

You can also download the live interactive SAGE worksheet [sato-tate.sws]. You'll need the free program SAGE to use it. Alternatively, check out this live worksheet """ AUTHOR: William Stein, 2006-08-20 Create graph of (1) the frequently histogram (distribution) of rounded values acos(a_p/2*sqrt(p)) for all good primes p up to the second argument (e.g., 50000). """ #%auto def sato_tate(E, N): return [acos(E.ap(p)/(2*sqrt(p))) for p in prime_range(N+1) if N%p != 0] g = graph_ellcurve(EllipticCurve('11a'),200,150000) show(g) def sato_tate_noacos(E, N): return [E.ap(p)/(2*sqrt(p)) for p in prime_range(N+1) if N%p != 0] def graph_ellcurve(E, b=10, num=5000): v = sato_tate_noacos(E, num) d, total_number_of_points = dist(v,b,-1,1) return graph(d, b, total_number_of_points,-1,1) show(graph_ellcurve(EllipticCurve('11a'),50,num=100000)) show(g,dpi=200) f.integrate('x',0,pi) %pi/2 %pi/2 #%auto def delta_dist(b=10, num=5000): D = delta_qexp(num) print "got delta" w = [float(D[p])/(2*float(p)^(5.5)) for p in prime_range(num+1)] print "normalized" v = [acos(x) for x in w] print "arccos" d, total_number_of_points = dist(v,b) print "distributed" return graph(d, b, total_number_of_points) g = delta_dist(30,100000) show(g)

18. Bulletin Of The American Mathematical Society .pdf....... The Riemann Hypothesis, The Clay Mathematics Institute, 2000. http//www.claymath.org/millennium/riemann_hypothesis/ Official_Problem_ http://www.ams.org/bull/2006-43-02/S0273-0979-06-01084-6/home.html

Journals Home Search Author Info Subscribe ... Help ISSN 1088-9485(e) ISSN 0273-0979(p) Previous issue Table of contents Next issue Articles in press ... Next Article Book Review The AMS does not provide abstracts of book reviews. You may download the entire review from the links below. Retrieve article in: PDF DVI PostScript Book Information Author(s): Henryk Iwaniec and Emmanuel Kowalski Title: Analytic number theory Additional book information: Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004, xii+618, US$99.00, 0-8218-3633-1 References: E. Bombieri, Problems of the millennium: The Riemann Hypothesis , The Clay Mathematics Institute, 2000. http://www.claymath.org/millennium/Riemann_Hypothesis/ Official_Problem_Description.pdf. H. Davenport, Multiplicative Number Theory , Graduate Studies in Mathematics, vol. 74, 3rd ed. (H. L. Montgomery, ed.), Springer-Verlag, New York, 2000. MR D. Goldfeld

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20. Top/Science/Math/Number_Theory/Analytic - Anypages Business Directory riemann_hypothesis (19). Sites about the Riemann Hypothesis, that the zeroes of the Riemann zeta function within the critical strip lie entirely on the http://www.anypages.com/WebListing.aspx?lang=0&ccode=sg&classid=60204

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