101. Marginal Revolution: Has The Riemann Hypothesis Been Solved? Has the riemann hypothesis been solved? Maybe. Of course there have been false The riemann hypothesis, often considered the greatest unsolved problem in http://www.marginalrevolution.com/marginalrevolution/2004/06/has_the_riemann.htm

hostName = '.marginalrevolution.com'; Marginal Revolution Small steps toward a much better world Recent Posts Ironic direct mail solicitations The debate over think tanks How to teach driving Vampires who understand the Hotelling rule ... Our poverty statistics Our Web Pages Tyler Cowen's Personal Web Page. Alex Tabarrok's Home Page Email Us Email Tyler Cowen Email Alex Tabarrok Our Guests September S M T W Th F St Archives September 2005 August 2005 July 2005 ... Syndicate this site (XML) Google Search Blogs We Like Statistical Modeling (Andrew Gelman) GeekPress Freakonomics Macroblog ... Bradford DeLong Resources The Iowa Electronic Markets Museum of Communism Library of Economics and Liberty FDAReview.org ... National Bureau of Economic Research Interesting People Nick Bostrom InvisibleHeart.com Steven Levitt Chris Sciabarra ... Main Has the Riemann hypothesis been solved? Maybe . Of course there have been false alarms before. The Riemann hypothesis, often considered the greatest unsolved problem in mathematics, concerns whether the distribution of prime numbers has a particular order. Here is one summary of the problem; here is

102. Riemann Hypothesis Proved? | Tech News On ZDNet riemann hypothesis proved? With eyes on $1 million prize, Purdue Universityprofessor publishes proof for cult math problem. http://news.zdnet.com/2100-3513_22-5229702.html

@import url(http://news.zdnet.com/css/z/m.css); @import url(http://news.zdnet.com/css/z/ads/hs.css); @import url(/css/z/s.css); News Blogs White Papers Downloads ... RSS Feeds GO Riemann hypothesis proved? By Michael Kanellos, CNET News.com Published on ZDNet News: June 9, 2004, 1:07 PM PT Forward in EMAIL Format for PRINT ZDNet Tags: Security 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 23-page paper detailing his attempts at a proof. Usually, mathematicians announce such breakthroughs at conferences or in scientific journals. Finding a solution to the Riemann hypothesis, however, carries a $1 million prize, so he decided to publish early. "I invite other mathematicians to examine my efforts," de Branges said in a prepared statement. "While I will eventually submit my proof for formal publication, due to the circumstances, I felt it necessary to post the work on the Internet immediately."

103. Atlas: Is The Riemann Hypothesis Necessary ? By Eric Bach Is the riemann hypothesis Necessary ? by Eric Bach come close to empiricaldata without assuming the riemann hypothesis or one of its generalizations. http://atlas-conferences.com/c/a/c/f/50.htm

Atlas home Conferences Abstracts about Atlas Turku Symposium on Number Theory in Memory of Kustaa Inkeri May 31 - June 4, 1999 University of Turku Turku, Finland Organizers Matti Jutila, Tauno Metsänkylä View Abstracts Conference Homepage Is the Riemann Hypothesis Necessary ? by Eric Bach University of Wisconsin To one unacquainted with number theory, we might explain our interest in the zeroes of the zeta function by saying that we can thereby estimate the error incurred by using logarithm integral to count the primes up to a given bound. As a practical matter, however, we know enough about these zeroes that numerical estimates for prime number sums, accurate enough for most purposes, can be easily obtained. There are other situations in computational number theory, though, where it seems impossible to obtain results that come close to empirical data without assuming the Riemann hypothesis or one of its generalizations. One example is the least witness required to prove a number composite using the strong pseudoprime test, and there are many others. For this reason, Riemann hypotheses have now become a standard tool of the algorithm designer. This talk will provide an introduction to and survey of these matters.

104. Nat' Academies Press, Prime Obsession: Bernhard Riemann And The Greatest Unsolve 24 chapters found with a hit on riemann hypothesis . All chapters aredisplayed; show in context, n hits is displayed beside the chapters with matches. http://www.nap.edu/booksearch.php?term=riemann hypothesis&isbn=0309085497

105. Riemann Hypothesis | Jef's Web Files riemann hypothesis. 143Year-Old Problem Still Has Mathematicians Guessing Theriemann hypothesis, first tossed off by Bernhard Riemann in 1859 in a paper http://www.jefallbright.net/node/279

@import "misc/drupal.css"; @import "modules/sidecontent/sidecontent.css"; @import "themes/jwftheme/style.css"; home misc about empathy ... extropy Home Subcategories: Empathy Energy Efficiency Extropy ... Riemann hypothesis 143-Year-Old Problem Still Has Mathematicians Guessing 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. "One of the things we hope to do is to consolidate the approaches," said Dr. Brian Conrey, a professor of mathematics at Oklahoma State University and executive director of the American Institute of Mathematics, a private group that organized the meeting with support from the Courant Institute and the National Science Foundation. "We're looking for brand-new ideas with which to open the door."

106. The Riemann Hypothesis. - Anagrams The riemann hypothesis. anagrams. Rearranging the letters of The riemann hypothesis . Find more anagrams of the riemann hypothesis. (or any other text)! http://www.anagramgenius.com/archive/riemann-hypothesis.html

Archive The definitive site for fun anagrams! The Riemann Hypothesis. anagrams Rearranging the letters of The Riemann Hypothesis. (Famous conjecture in Maths) gives: Yeah! Hints on the primes. (by V.Rabin) See trophy in math shine? (by Mike Mesterton-Gibbons by hand) Ah, hype: isn't in theorems! (by Mike Mesterton-Gibbons by hand) See also: Godfrey Harold Hardy Sir Henry Peter Francis Swinnerton-Dyer. Download FREE anagram-generating software for your Windows computer Webmasters: make money from your website! Instructions for linking to this page! Learn about the Anagram Genius software (Windows/MacOS) Search the Archive Add YOUR anagrams to the Archive! League table of top contributors Find anagram aliases of the riemann hypothesis. (or any other text)! Find gold service anagrams of the riemann hypothesis. (or any other text)! Anagram Genius Archive Main Index www.anagramgenius.com home page Search the anagram archive (e.g. osama) Subject Author Anagram See this page for more relating to the riemann hypothesis.

107. Riemann's Hypothesis A beginners guide by Jon Perry. http://www.users.globalnet.co.uk/~perry/maths/riemannshypothesis/riemannshypothe

Riemann's Hypothesis [Home] [Other Maths] [Riemann's Hypothesis Part 2] My proof of Riemann's Hypothesis Riemann's Hypothesis Euler's zeta function Euler's zeta function, which forms the basis for Riemann's Hypothesis, is the sum of the integers from 1 to infinity raised to a complex power. It is written: This converges for complex s such that the real part of s is greater than 1, but for s <=1 it diverges, and is not considered to be valid on this region. Riemann's zeta function Riemann had the idea to extend this function into the whole complex plane, which he managed to do, except for a simple pole at s=1. He achieved this through a process called analytic continuation. Analytic continuation is whereby an alternative function is used that behaves exactly as the original function in the domain of the original function, and continues the function outside of the original domain. This is the idea in defining i =-1. The previous definition of square root did not allow for square root of negative numbers, and i is the analytic continuation of the square root function. With analytic continuation, we can have different expressions for the zeta function, but they all behave the same. This is similar to writing either sigma(1/n

108. Body 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 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, and the product to all positive primes p.

109. 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. 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 as a function of 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 x = ½. 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 x The question is still open in 2000. A prize is available to prove or disprove Riemann's hypothesis. Proof Using Riemann's Functional Equation It has already been shown that all zeros are in the critical strip and that they are symmetric about the critical line x Riemann's functional equation can be restated as =A( in which A( at all points in the critical strip. Since functions

110. Riemann's Hypothesis Part 2 Home Other Maths riemann s hypothesis Part 1. riemann s hypothesis Part 2.riemann s approximation to pi(x). The Prime Number Theorem states that http://www.users.globalnet.co.uk/~perry/maths/riemannshypothesis2/riemannshypoth

Riemann's Hypothesis Part 2 [Home] [Other Maths] [Riemann's Hypothesis Part 1] Riemann's Hypothesis Part 2 Riemann's approximation to pi(x) The Prime Number Theorem states that pi(x) ~ Li(x). This is a fair enough approximation, but it has fairly large error terms. The end result of Riemann's paper on pi(x) was to narrow this error term down significantly. Of course the validity of this result is pending the proof that all zeroes lie on the critical line. Although this is not the precise method Riemann used, it covers his general flow. We begin by defining a jump function S(x) as a function with an unusual technique for counting primes and prime powers. S(x) = +1 if x is prime +1/n if x is p n Riemann then determined a formula relating S(x) to pi(x): This result is obvious - S(x) is the sum of the primes to x, plus 1/2 the sum of the number of prime squares, which is identical to pi(x ), and so on. We require this equation with pi(x) as the main term, so we apply the Mobius Inversion formula: divides n = 1 if n is square-free with an even number of prime factors = -1 if n is square-free with an odd number of prime factors Making the exchange gives: The next step is to make a substitution for S(x). S(X) can be analytically defined (via a lengthy process) as:

111. AoPS Math Forum :: View Topic - Riemann's Hypothesis All times are GMT 7 Hours, riemann s hypothesis, Mark the topic unread Post Posted Thu Apr 28, 2005 301 pm Post subject riemann s hypothesis http://www.artofproblemsolving.com/Forum/topic-35163.html

Round 1 of the USA Mathematical Talent Search is now available. The postmark deadline for the first round is October 3 Font Size: The time now is Fri Sep 16, 2005 5:04 pm All times are GMT - 7 Hours Riemann's Hypothesis Mark the topic unread Math Forum Olympiad Section ... Number Theory Open Questions Moderators: amfulger Arne harazi Megus ... pbornsztein Page of [4 Posts] Printer View Tell a Friend View previous topic View next topic Author Message Anto Poincare Conjecture Joined: 13 Apr 2004 Posts: 119 Posted: Thu Apr 28, 2005 3:01 pm Post subject: Riemann's Hypothesis Source: Proofs from THE BOOK (Martin Aigner- Gunter M.Ziegler) Basicly i'm not gonna post a related problem. I'm just so excited with that i have just read and i would like to put it in the table. In the book of Martin Aigner- Gunter M.Ziegler it is written that Jeff Lagarias has very resently proved that Riemann's Hypothesis is equivalent to the following inequality : , for all and equality only if where Isn't it unbelievable ? Is it so hard to attact this inequality ? (obvious...

112. Notices Table Of Contents http://www.ams.org/notices/200303/200303-toc.html

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