41. 143-Year-Old Problem Still Has Mathematicians Guessing - New York Times Jul 2, 2002 The Riemann hypothesis, first tossed off by Bernhard Riemann in 1859 in Hardy, for example, rated the Riemann hypothesis less difficult http://www.nytimes.com/2002/07/02/science/143-year-old-problem-still-has-mathema

Home Page Today's Paper Video Most Popular ... Login Search All NYTimes.com Science COLLECTIONS MATHEMATICIANS 143-Year-Old Problem Still Has Mathematicians Guessing By BRUCE SCHECHTER Published: July 02, 2002 Correction Appended In the early years of the 20th century, the great British mathematician Godfrey Harold Hardy used to take out a peculiar form of travel insurance before boarding a boat to cross the North Sea. If the weather looked threatening he would send a postcard on which he announced the solution of the Riemann hypothesis. Hardy, wrote his biographer, Constance Reid, was convinced ''that God with whom he waged a very personal war would not let Hardy die with such glory.'' The Riemann hypothesis, first tossed off by Bernhard Riemann in 1859 in a paper about the distribution of prime numbers, is still widely considered to be one of the greatest unsolved problems in mathematics, sure to wreath its conqueror with glory and, incidentally, lots of cash. Two years ago, to celebrate the millennium, the Clay Mathematics Institute announced an award of a million dollars for a proof (or refutation) of the hypothesis. Whether in pursuit of glory, cash (''prizes attract cranks,'' one mathematician sniffed) or pure mental satisfaction, more than a hundred of the world's leading mathematicians came to New York City recently to attend an unusual conference at New York University's Courant Institute. While most math conferences are devoted to presenting completed work, this one was held for mathematicians to swap hunches, warn of dead ends and get new ideas that could ultimately lead to a solution.

42. Riemann Hypothesis Proved? - CNET News Jun 09, 2004 With eyes on $1 million prize, Purdue University professor publishes proof for cult math problem. A CNET article by Michael Kanellos, Staff Writer, CNET News. http://news.cnet.com/Riemann-hypothesis-proved/2100-7348_3-5229702.html

PageVars.set('title', "Riemann hypothesis proved? - CNET News"); PageVars.set('description', "With eyes on $1 million prize, Purdue University professor publishes proof for cult math problem."); PageVars.set('href', "http://news.cnet.com/Riemann-hypothesis-proved/2100-7348_3-5229702.html"); home reviews news downloads ... video On The Insider: Mario Arrested After Attacking His Mom log in join CNET welcome, my profile log out Latest News CNET River ... Markets Life Insurance From MetLife Home News June 9, 2004 1:07 PM PDT Riemann hypothesis proved? By Michael Kanellos Staff Writer, CNET News 5 comments cnet_news406:http://news.cnet.com/Riemann-hypothesis-proved/2100-7348_3-5229702.html Related Stories Cooperative project reports new top prime number May 18, 2004 Multinational team cracks crypto puzzle April 27, 2004 Cooperative computing finds top prime number December 2, 2003 A mathematician at Purdue University claims to have come up with a proof for the Riemann hypothesis, often called the greatest unsolved math problem, though the work has yet to be peer-reviewed. Louis de Branges de Bourcia, the Edward C. Elliott Distinguished Professor of Mathematics at Purdue's School of Science, this week posted a

43. APOLOGY FOR THE PROOF OF THE RIEMANN HYPOTHESIS Louis De Branges File Format PDF/Adobe Acrobat Quick View http://www.math.purdue.edu/~branges/apology.pdf

44. Proof Of Riemann's Hypothesis A claimed proof of Riemann s Hypothesis. http://www.coolissues.com/mathematics/Riemann/riemann.htm

PROOF OF RIEMANN'S HYPOTHESIS James Constant math@coolissues.com Riemann's hypothesis is proved using Riemann's functional equation This page is now subject to the author's counterexample at http://www.coolissues.com/mathematics/Riemann/disproof.htm Introduction The famous conjecture known as Riemann' s hypothesis is to classical analysis what Fermat's last theorem is to arithmetic. Euler (1737) noted that the formula the sum extending to all positive integers n , and the product to all positive primes p. The necessary conditions of convergence hold for complex values of s with real part Considering as a function of the complex variable s , Riemann (1859) proved that satisfies a functional equation which led Riemann to the theorem that all the zeros of , except those at s=-2,-4,-6, . . . , lie in the strip of the s-plane for which where x is the real part of s . Riemann conjectured that all the zeros in the strip should lie on the line Attempts to prove or disprove this conjecture have generated a vast and intricate department of analysis, especially since Hardy (1914) proved that has an infinity of zeros on The question is still open in 2008. A prize is available to prove or disprove Riemann's hypothesis.

45. Riemann Hypothesis Scrolling graph of RiemannSiegel Z(t) with sound effects, Animated representation of the main sum in the Riemann-Siegel formula, 1993 Big Science Catastrophe Theory, Riemann http://www.dipity.com/timeline/Riemann-Hypothesis/list

46. Mathematical Constants Notes by Steven Finch. http://algo.inria.fr/bsolve/constant/apery/riemhyp.html

47. Ueber Die Anzahl Der Primzahlen Unter Einer Gegebenen Grösse Translate this page File Format PDF/Adobe Acrobat - Quick View http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/Zeta.pdf

48. Stalking The Riemann Hypothesis By Dan Rockmore - Hardcover - Random House Stalking the Riemann Hypothesis by Dan Rockmore, Category Mathematics Advanced, Format Hardcover , 304 pages, ISBN 9780375421365, On Sale April 5, 2005, Price $25.00, Item http://www.randomhouse.com/catalog/display.pperl?isbn=9780375421365

49. The Prime Glossary: Riemann Hypothesis Welcome to the Prime Glossary a collection of definitions, information and facts all related to prime numbers. This pages contains the entry titled 'Riemann hypothesis.' http://primes.utm.edu/glossary/xpage/RiemannHypothesis.html

Riemann hypothesis (another Prime Pages ' Glossary entries) Glossary: Index Search Home Previous ... Mail Editor Prime Pages: Home Search Index FAQ Top 5000: Primes Provers Curios Riemann noted that his zeta function had trivial zeros at -2, -4, -6, ... and that all nontrivial zeros were symmetric about the line Re( s The Riemann hypothesis is that all nontrivial zeros are on this line. In fact the classical proofs of the prime number theorem require an understanding of the zero free regions of this function, and in 1901 von Koch showed that the Riemann hypothesis is equivalent to: Because of this relationship to the prime number theorem, Riemann's hypothesis is easily one of the most important conjectures in prime number theory. See Also: RiemannZetaFunction Related pages (outside of this work) The Riemann hypothesis (expanded information) How the Riemann zeta function 'encodes' the primes ZetaGrid a distributed project to test the hypothesis Prime Time a simple article Chris K. Caldwell

50. Riemann Hypothesis Zeros Proof Function Zeta Number Non-trivial Riemann Hypothesis Zeros Proof Function Zeta Number Nontrivial Economy. http://www.economicexpert.com/a/Riemann:hypothesis.html

51. Zeta.html By Andrew Odlyzko. http://www.dtc.umn.edu/~odlyzko/doc/zeta.html

Andrew Odlyzko: Papers on Zeros of the Riemann Zeta Function and Related Topics (see also "Tables of zeros of the zeta function" and "Some unpublished materials" on the main home page) The zeta function on the critical line: Numerical evidence for moments and random matrix theory models , G. A. Hiary and A. M. Odlyzko. [PDF] The 10^22-nd zero of the Riemann zeta function , A. M. Odlyzko. Dynamical, Spectral, and Arithmetic Zeta Functions , M. van Frankenhuysen and M. L. Lapidus, eds., Amer. Math. Soc., Contemporary Math. series, no. 290, 2001, pp. 139-144. [Abstract] [PostScript] [PDF] [LaTeX] An improved bound for the de Bruijn-Newman constant , A. M. Odlyzko, Numerical Algorithms , 25 (2000), pp. 293-303. [Abstract] [PostScript] [PDF] [LaTeX] A nonlinear equation and its application to nearest neighbor spacings for zeros of the zeta function and eigenvalues of random matrices , P. J. Forrester and A. M. Odlyzko, in Organic Mathematics , J. Borwein, P. Borwein, L. Jorgenson, and R. Corless, eds., Amer. Math. Soc. 1997, pp. 239-250. Electronic version available at http://www.cecm.sfu.ca/projects/OMP/. [PostScript] [PDF] [LaTeX] A condensed version

52. The Music Of The Primes A popular article by Marcus du Sautoy on the Riemann Hypothesis; Science Spectra, Issue 11. http://www.dpmms.cam.ac.uk/~dusautoy/2soft/music.htm

1.-When the British mathematician Andrew Wiles told the world about his proof of the Last Theorem of the seventeenth century French lawyer, Pierre de Fermat, it looked as if the Holy Grail had been grasped. Fermat's Last Theorem has often been called the greatest unsolved riddle of mathematics. But many mathematicians would argue that this name belongs rather to an idea first put forward in the middle of the nineteenth century by the German mathematician Bernhard Riemann: The Riemann Hypothesis. 2.-PRIME NUMBERS It remains unresolved but, if true, the Riemann Hypothesis will go to the heart of what makes so much of mathematics tick: the prime numbers. These indivisible numbers are the atoms of arithmetic. Every number can be built by multiplying prime numbers together. The primes have fascinated generations of mathematicians and non-mathematicians alike, yet their properties remain deeply mysterious. Whoever proves or disproves the Riemann Hypothesis will discover the key to many of their secrets and this is why it ranks above Fermat as the theorem for whose proof mathematicians would trade their soul with Mephistopheles. 3.-Although the Riemann Hypothesis has never quite caught on in the public imagination as Mathematics' Holy Grail, prime numbers themselves do periodically make headline news. The media love to report on the latest record for the biggest prime number so far discovered. In November 1996 the Great Internet Prime Search announced their discovery of the current record, a prime number with 378,632 digits. But for mathematicians, such news is of only passing interest. Over two thousand years ago Euclid proved that there will be infinitely many such news stories, for the primes never run dry.

53. Riemann Hypothesis (mathematics) -- Britannica Online Encyclopedia Riemann hypothesis (mathematics), in number theory, hypothesis by German mathematician Bernhard Riemann concerning the location of solutions to the Riemann zeta function, which http://www.britannica.com/EBchecked/topic/503221/Riemann-hypothesis

document.write(''); Search Site: Advanced Search With all of these words With the exact phrase With any of these words Without these words Home SHOP BROWSE BLOG ... HELP Username: Password: Remember me Forgot your password? Help Email " is the e-mail address you used when you registered. Password " is case sensitive. If you need additional assistance, please contact customer support Enter the e-mail address you used when enrolling for Britannica Premium Service and we will e-mail your password to you. Riemann-hypothesis My Britannica CREATE MY Riemann hypo... NEW ARTICLE ... SAVE Riemann hypothesis Table of Contents: Riemann hypothesis Article Article Related Articles Related Articles External Web sites External Web sites Citations Article Related Articles External Web sites Citations ARTICLE from the Riemann hypothesis in number theory , hypothesis by German mathematician Bernhard Riemann concerning the location of solutions to the Riemann zeta function , which is connected to the prime number theorem and has important implications for the distribution of prime numbers . Riemann included the hypothesis in a paper

54. Some Extended Zeta Functions Provide Easy Proofs 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

Some Extended Zeta Functions Provide Easy Proofs of Riemann's Hypothesis This page is now subject to the author's counterexample at http://www.coolissues.com/mathematics/Riemann/disproof.htm James Constant math@coolissues.com While extended zeta functions support investigations of Riemann's hypothesis and estimates for the Prime Number Theorem, some zeta functions offer better prospects for providing easy proofs. Definitions A first zeta function is defined by oo z(s)= s=x+jy n=1 A second zeta function is defined by oo z(1-s)= s=x+jy n=1 In 1859, Riemann had the idea to define z(s) for all complex numbers s by analytic extension. This extension is important in number theory and plays a central role in the distribution of prime numbers. One way of extending is by using the first f function alternating series defined by oo f(s)= s=x+jy n=1 1 by means of the formula f(s)=(1-2 )z(s) A second f function is defined by oo f(1-s)= s=x+jy n=1 1 by means of the formula f(1-s)=(1-2 )z(1-s) Equations (1) through (6) are analytic. Riemann's Extended Zeta Function and Functional Equation Euler (1737) noted that the formula n

55. Riemann Hypothesis -- Math Fun Facts From the Fun Fact files, here is a Fun Fact at the Advanced level Riemann Hypothesis If you know about complex numbers, you will be able to appreciate one of the great http://www.math.hmc.edu/funfacts/ffiles/30002.5.shtml

hosted by the Harvey Mudd College Math Department Francis Su The Math Fun Facts App is now in the App Store Subscribe to our RSS feed or Any Easy Medium Advanced The Math Fun Facts App! List All List Recent List Popular ... Francis Edward Su From the Fun Fact files, here is a Fun Fact at the Advanced level: Riemann Hypothesis If you know about complex numbers, you will be able to appreciate one of the great unsolved problems of our time. The Riemann zeta function is defined by Zeta(z) = SUM k=1 to infinity (1/k z This is the harmonic series for z=1 and Sums of Reciprocal Powers if you set z equal to other positive integers. The function can be extended to the entire complex plane (with some poles) by a process called "analytic continuation", although what that is won't concern us here. It is of great interest to find the zeroes of this function. The function is trivially zero at the negative even integers, but where are all the other zeroes? To date, the only other zeroes known all lie on the line in the complex plane with real part equal to 1/2. This has been checked for several hundred million zeroes! No one knows, however, if

56. Riemann Hypothesis@Everything2.com The Riemann Hypothesis states conjectures that all nontrivial roots of the Riemann Zeta Function occur on the critical line on the argand plane Re(z) = 1/2, that is, the real part of http://www.everything2.com/title/riemann hypothesis

57. 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

Computation of zeros of the Zeta function New Riemann Hypothesis verification record Riemann Hypothesis verified until the 10 -th zero. (October 12th 2004), by Xavier Gourdon with the help of Patrick Demichel. Billions of zeros at very large height (around the 10 -th zero) have also been computed. Details can be found in The 10 first zeros of the riemann zeta function, and zeros computation at very large height The Riemann Hypothesis (RH) is one the most important unsolved problem in mathematics (see Zeta generalities for details about the RH). It has naturally been numerically checked threw the ages, thanks to techniques about the Zeta functions evaluations (see Numerical evaluation of the Riemann Zeta-function for details). History of numerical verification of the RH Numerical computations have been made threw the ages to check the RH on the first zeros. Computer age, starting with Turing computations, permitted to perform verification higher than billions of zeros. An history of the RH verification on the first n zeros is given below. Year n Author J. P. Gram [

58. Algebraic Curves, Riemann Hypothesis And Coding Marios Magioladitis, University of Crete, 2001. Introduction and text (DOC, PS). http://www.math.uoc.gr/~marios/essay.htm

Algebraic Curves, Riemann hypothesis and coding File moved here

59. Maths Holy Grail Could Bring Disaster For Internet | Technology | The Guardian Sep 7, 2004 The Riemann hypothesis would explain the apparently random pattern of prime numbers numbers such as 3, 17 and 31, for instance, http://www.guardian.co.uk/technology/2004/sep/07/highereducation.science

document.domain = "guardian.co.uk"; Turn autoplay off Turn autoplay on Please activate cookies in order to turn autoplay off Jump to content [s] Jump to site navigation [0] Jump to search [4] Terms and conditions [8] ... Register Text larger smaller About us About us Contact us Press office Guardian Print Centre ... Subscribe Today's paper The Observer Comment Sport New Review ... Observer Food Monthly Zeitgeist Today's hot topics guardian.co.uk Technology Web News Sport Comment Culture ... Technology Maths holy grail could bring disaster for internet Two of the seven million dollar challenges that have baffled for more than a century may be close to being solved Tweet this Tim Radford , science editor The Guardian Tuesday 7 September 2004 Mathematicians could be on the verge of solving two separate million dollar problems. If they are right - still a big if - and somebody really has cracked the so-called Riemann hypothesis, financial disaster might follow. Suddenly all cryptic codes could be breakable. No internet transaction would be safe. Both problems have stood for a century or more. Each is almost dizzyingly arcane: the problems themselves are beyond simple explanation, and the candidate answers published on the internet are so intractable that they could baffle the biggest brains in the business for many months.

60. A Directory Of All Known Zeta Functions A directory of websites maintained by Matthew Watkins on the topic of The Riemann Hypothesis and Zeta Function. http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/directoryofzetafunctio

a directory of all known zeta functions [This page is under continual construction! Any contributions would be welcome.] Over the years striking analogies have been observed between the Riemann zeta-function and other zeta- or L-functions. While these functions are seemingly independent of each other, there is growing evidence that they are all somehow connected in a way that we do not fully understand. In any event, trying to understand, or at least classify, all of the objects which we believe satisfy the Riemann hypothesis is a reasonable thing to do." J. Brian Conrey, "The Riemann Hypothesis" Notices of the AMS (March, 2003) p.347 In this essay I will give a strictly subjective selection of different types of zeta functions. Instead of providing a complete list, I will rather try to give the central concepts and ideas underlying the theory... Whenever entities are counted with some mathematical structure on them it is likely that a zeta function can be set up and often enough it will extend to a meromorphic function. Zeta functions show up in all areas of mathematics and they encode properties of the counted objects which are well hidden and hard to come by otherwise. They easily give fuel for bold new conjectures and thus drive on mathematical research. It is a fairly safe assertion to say that zeta functions of various kinds will stay in the focus of mathematical attention for times to come. A. Deitmar

Page 3     41-60 of 99    Back | 1  | 2  | 3  | 4  | 5  | Next 20