1. Riemann Hypothesis -- From MathWorld Article with links to other resources from MathWorld. http://mathworld.wolfram.com/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.

2. ZetaGrid Homepage An open source and platform independent grid system that uses idle CPU cycles from participating computers. ZetaGrid solves one problem in practice numerical verification of the riemann hypothesis. http://www.zetagrid.net/

ZetaGrid ZetaGrid Acknowledgement Performance characteristics Riemann Hypothesis Prizes ... Links This site is owned by Sebastian Wedeniwski Sponsors IBM Deutschland Entwicklung GmbH Webhosting powered by EDIS.at Do you need help? Try the ZetaGrid forum What is ZetaGrid? ZetaGrid is a platform independent grid system that uses idle CPU cycles from participating computers. Grid computing can be used for any CPU intensive application which can be split into many separate steps and which would require very long computation times on a single computer. ZetaGrid can be run as a low-priority background process on various platforms like Windows, Linux, AIX, Solaris, HP-UX, and Mac OSX. On Windows systems it may also be run in screen saver mode. ZetaGrid in practice: Riemann's Hypothesis is considered to be one of modern mathematics most important problems. This implementation involves more than 11,000 workstations and has a peak performance rate of about 7056 GFLOPS. More than 1 billion zeros for the zeta function are calculated every day. To learn more about ZetaGrid, you have two options:

3. Clay Mathematics Institute Article by Enrico Bombieri (PDF) and video by Jeff Vaaler (.ram) from the ClayMathematics 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

4. Louis De Branges Professor of Mathematics at Purdue University. Contact information, papers on the Bieberbach Conjecture, the riemann hypothesis, and related topics. http://www.math.purdue.edu/~branges/

Louis de Branges de Bourcia Purdue Mathematics Department Math Room 800 1395 Mathematics Building Lafayette IN 47907-1395 E-Mail branges@math.purdue.edu Phone 011-765-494-6057 This is web page is under construction now, but is made available to allow access to currently available papers. They are: Apology for the proof of the Riemann hypothesis (in pdf format). Hahn-Banach Theorem for Weierstrass Algebras (in pdf format). Proof of the Riemann Hypothesis (in pdf format). Complex Analysis (in pdf format). ... Staff Directory Problems or questions, contact the Webmaster? Webmaster.

5. 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

6. Riemann A short article with some grahpical and numerical evidence in the critical strip. http://www.mathpuzzle.com/riemann.html

The Riemann Hypothesis is currently the most famous unsolved problem in mathematics. Like the Goldbach Conjecture (all positive even integers greater than two can be expressed as the sum of two primes), it seems true, but is very hard to prove. I did some playing around with the Riemann Hypothesis, and I'm convinced it is true. My observations follow. The Zeta Function Euler showed that z p 6 , and solved all the even integers up to z (26). See the Riemann Zeta Function in the CRC Concise Encyclopedia of Mathematics for more information on this. It is possible for the exponent s to be Complex Number ( a + b I). A root of a function is a value x such that f x The Riemann Hypothesis : all nontrivial roots of the Zeta function are of the form (1/2 + b I). Mathematica can plot the Zeta function for complex values, so I plotted the absolute value of z b I) and z b I). z b I) for b = to 85. Note how often the function dips to zero. z b I) for b = to 85. Note how the function never dips to zero. The first few zeroes of z b I) are at b = 14.1344725, 21.022040, 25.010858, 30.424876, 32.935062, and 37.586178. Next, I tried some 3D plots, looking dead on at zero. The plot of the function looked like this:

7. The Riemann Hypothesis A short article by Kimon Spiliopoulos. http://users.forthnet.gr/ath/kimon/Riemann/Riemann.htm

The Riemann Hypothesis Riemann's Hypothesis was one of the 23 problems - milestones that David Hilbert suggested in 1900, at the 2nd International Conference on Mathematics in Paris, that they should define research in mathematics for the new century (and indeed, it is not an exaggeration to say that modern mathematics largely come from the attempts to solve these 23 problems). It is the most famous open question today, especially after the proof of Fermat's Last Theorem The Riemann zeta function is of central importance in the study of prime numbers. In its first form introduced by Euler, it is a function of a real variable x: This series converges for every x > 1 (for x=1 it is the non-corvergent harmonic series). Euler showed that this function can also be expressed as an infinite product which involves all prime numbers p n , n=1, Riemann studied this function extensively and extended its definition to take complex arguments z. So the function bears his name. Of particular interest are the roots of Trivial zeros are at z= -2, -4, -6,

8. Riemann Hypothesis A short article by Krzysztof Maslanka with numerical examples and graphics. http://www.oa.uj.edu.pl/~maslanka/zeros.html

Nontrivial zeros of the zeta-function of Riemann Real and imaginary part of Zeta[1/2+I*y]. Both curves intersect precisely at the y-axis The same zeros as a "spectrum". Numerical values of the zeros computed using Mathematica Imaginary values of the first hundred of nontrivial zeros of the zeta-function of Riemann. Their number and accuracies are rather modest, especially when compared to the recent spectacular computational achievement of Andrew M. Odlyzko from Bell Labs. Nevertheless, in the literature I have never seen any tables of these larger some twenty zeros. All real parts of the non-trivial zeros of zeta are supposed to be exactly 1/2. This simple statement is the famous Riemann hypothesis. Nobody knows for certain if this is true. Many suspect that it is. However, everybody would like to know. Everybody would also agree that this is the most important unsolved mathematical problem today. There exists simple numerical fit to these points (red line; s0[i] - denotes i -th zero, hence Zeta[s0[i]]=0). Im[s0[i]] = 6,5662*(i-1)^0,76511 + 14,720

9. THE RIEMANN HYPOTHESIS Web article by Aldo Peretti. http://www.cuatrovientos.com.ar/_vti_z/

THE RIEMANN HYPOTHESIS By Aldo Peretti Download all the text 1– Introduction In 1859, G.F.B.Riemann published a most famous paper concerning the distribution of prime numbers, with the title: “On the quantity of prime numbers below a given quantity”, where, for the first time were used the methods of complex variable functions in order to determine (x) x. His starting formula was the product decomposition that Euler had found for the zeta function i.e. the formula where p stands for the prime numbers. (Riemann used the letter s to denote the variable, s + it ; and this way of notation was unanimously used after him) In the first part of the memoir, he proves the functional equation of the zeta function, and after this he deduces the formula valid for f(x) This formula had been obtained formerly in 1848 by Tchebychev (whose work on the subject very likely was known to Riemann). But he was unable to make the inversion of this formula, that Riemann succeeded to do, obtaining thus: f(x) = The remaining part of Riemann’s paper is very obscure and confusing because of its excessive brevity. Fortunately

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

Mathematical Constants by Steven R. Finch Clay Mathematics Institute Book Fellow My website is smaller than it once was. Please visit again, however, since new materials will continue to appear occasionally. * My book Mathematical Constants is now available for online purchase from Cambridge University Press (in the United Kingdom and in North America ). It is far more encompassing and detailed than my website ever was. It is also lovingly edited and beautifully produced - many thanks to Cambridge! - please support us in our publishing venture. Thank you. (If you wish, see several very kind reviews . You can also search the book via Amazon and Google by keyword.) Here are errata and addenda to the book (last updated 8/11/2005), as well sample essays from the book about integer compositions optimal stopping and Reuleaux triangles . Here also are recent supplementary materials, organized by topic: Number Theory and Combinatorics Bipartite, k-colorable and k-colored graphs Transitive relations, topologies and partial orders Series-parallel networks Two asymptotic series ... Quadratic Dirichlet L-series Elliptic curves over Q Inequalities and Approximation Hardy-Littlewood maximal inequalities Bessel function zeroes Nash's inequality Uncertainty inequalities ... Moduli of continuity Real and Complex Analysis Radii in geometric function theory Numerical radii of linear operators Coefficient estimates for univalent functions Planar harmonic mappings ... Constant of interpolation Probability and Stochastic Processes

11. The Riemann Hypothesis Here we define, then discuss the riemann hypothesis. We provide several related links. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

12. Extended Riemann Hypothesis -- From MathWorld Extended riemann hypothesis. COMMENT On this Page SEE ALSO Generalizedriemann hypothesis, riemann hypothesis. Pages Linking Here http://mathworld.wolfram.com/ExtendedRiemannHypothesis.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 Number Theory Reciprocity Theorems Extended Riemann Hypothesis The first quadratic nonresidue mod of a number is always less than (Wedeniwski 2001). SEE ALSO: Generalized Riemann Hypothesis Riemann Hypothesis [Pages Linking Here] REFERENCES: Bach, E. Analytic Methods in the Analysis and Design of Number-Theoretic Algorithms. Cambridge, MA: MIT Press, 1985. Wagon, S. Mathematica in Action. New York: W. H. Freeman, p. 295, 1991. Wedeniwski, S. "Primality Tests on Commutator Curves." Dissertation. Tübingen, Germany, 2001. http://www.hipilib.de/prime/primality-tests-on-commutator-curves.pdf CITE THIS AS: Eric W. Weisstein. "Extended Riemann Hypothesis." From MathWorld A Wolfram Web Resource. http://mathworld.wolfram.com/ExtendedRiemannHypothesis.html Wolfram Research, Inc.

13. The Riemann Hypothesis A prime pages article by Chris K. Caldwell. http://www.utm.edu/research/primes/notes/rh.html

The Riemann Hypothesis (Another of the Prime Pages ' resources) Home Search Site Largest The 5000 ... Submit primes Summary: When studying the distribution of prime numbers Riemann extended Euler's zeta function (defined just for s with real part greater than one) to the entire complex plane ( sans simple pole at s = 1). 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 1901 von Koch showed that the Riemann hypothesis is equivalent to: The Riemann Hypothesis: Euler studied the sum for integers s >1 (clearly (1) is infinite). Euler discovered a formula relating k ) to the Bernoulli numbers yielding results such as and . But what has this got to do with the primes? The answer is in the following product taken over the primes p (also discovered by Euler): Euler wrote this as Riemann later extended the definition of s ) to all complex numbers s (except the simple pole at s =1 with residue one). Euler's product still holds if the real part of

14. Riemann Hypothesis In A Nutshell riemann hypothesis http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

15. Riemann Hypothesis In A Nutshell The riemann hypothesis (RH) is that all nontrivial zeros of the zeta function You can, however, verify the validity of the riemann hypothesis in large http://www.math.ubc.ca/~pugh/RiemannZeta/RiemannZetaLong.html

Home Z(t) Plotter Verifying RH ... More Applets The Riemann Hypothesis in a Nutshell The Riemann Zeta Function image source In his 1859 paper On the Number of Primes Less Than a Given Magnitude , Bernhard Riemann (1826-1866) examined the properties of the function for s a complex number. This function is analytic for real part of s greater than and is related to the prime numbers by the Euler Product Formula again defined for real part of s greater than one. This function extends to points with real part s less than or equal to one by the formula (among others) The contour here is meant to indicate a path which begins at positive infinity, descends parallel to and just above the real axis, circles the origin once in the counterclockwise direction, and then returns to positive infinity parallel to and just below the real axis. This function is analytic at all points of the complex plane except the point s = 1 where it has a simple pole. This last function is the Riemann Zeta Function ( the zeta function The Riemann Hypothesis The zeta function has no zeros in the region where the real part of s is greater than or equal to one. In the region with real part of

16. Millennium Prize Problems The seven problems proposed by the Clay Mathematics Institute P versus NP; Hodge Conjecture; Poincar© Conjecture; riemann hypothesis; YangMills Existence and Mass Gap; Navier-Stokes Existence and Smoothness; Birch and Swinnerton-Dyer Conjecture. Resources include articles on each problem by leading researchers. http://www.claymath.org/millennium/

Clay Mathematics Institute Dedicated to increasing and disseminating mathematical knowledge HOME ABOUT CMI PROGRAMS AWARDS ... PUBLICATIONS Millennium Problems In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) has named seven Prize Problems . The Scientific Advisory Board of CMI selected these problems, focusing on important classic questions that have resisted solution over the years. The Board of Directors of CMI designated a $7 million prize fund for the solution to these problems, with $1 million allocated to each. During the Millennium Meeting The Importance of Mathematics , aimed for the general public, while John Tate and Michael Atiyah spoke on the problems. The CMI invited specialists to formulate each problem. One hundred years earlier, on August 8, 1900, David Hilbert delivered his famous lecture about open mathematical problems at the second International Congress of Mathematicians in Paris. This influenced our decision to announce the millennium problems as the central theme of a Paris meeting. The rules for the award of the prize have the endorsement of the CMI Scientific Advisory Board and the approval of the Directors. The members of these boards have the responsibility to preserve the nature, the integrity, and the spirit of this prize.

17. Louis De Branges Professor of Mathematics at Purdue University. Contact information, papers on the Bieberbach Conjecture, the riemann hypothesis, and related topics. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

18. Riemann Hypothesis - Wikipedia, The Free Encyclopedia The riemann hypothesis is a conjecture about the distribution of the zeros of the The riemann hypothesis is one of the most important open problems of http://en.wikipedia.org/wiki/Riemann_hypothesis

Riemann hypothesis From Wikipedia, the free encyclopedia. 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.

19. Riemann Hypothesis From MathWorld First published by Riemann (1859), the riemann hypothesis states that the nontrivial Riemann zeta function zeros, i.e., the values of s other than http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

20. Grand Riemann Hypothesis - Wikipedia, The Free Encyclopedia In mathematics, the grand riemann hypothesis is a generalisation of the Riemann The modified grand riemann hypothesis is the assertion that the http://en.wikipedia.org/wiki/Grand_Riemann_hypothesis

Grand Riemann hypothesis From Wikipedia, the free encyclopedia. In mathematics , the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and Generalized Riemann hypothesis . It states that the nontrivial zeros of all automorphic L -functions lie on the critical line 1/2 + it with t a real number and i the imaginary unit. The modified grand Riemann hypothesis is the assertion that the nontrivial zeros of all automorphic L -functions lie on the critical line or the real line edit Notes It is widely believed that all global L -functions are automorphic L -functions. The Siegel zero , conjectured not to exist, is a possible real zero of a Dirichlet L-series , rather near s mathematics -related article is a stub . You can help Wikipedia by expanding it Retrieved from " http://en.wikipedia.org/wiki/Grand_Riemann_hypothesis Categories Mathematics stubs Analytic number theory ... Conjectures Views Article Discussion Edit this page History Personal tools Create account / log in Navigation Main Page Community portal Current events Recent changes ... Donations Search Toolbox What links here Related changes Upload file Special pages ... Permanent link This page was last modified 18:09, 22 March 2005.

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