1. AThe Riemann Hypothesis Here we define, then discuss the riemann hypothesis. We provide several relatedlinks. The riemann hypothesis (Another of the Prime Pages resources). 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

2. The Riemann Hypothesis The riemann hypothesis. Spectral Interpretation. One idea for proving the Riemannhypothesis is to give a spectral interpretation of the zeros. http://match.stanford.edu/rh/

The Riemann Hypothesis Spectral Interpretation One idea for proving the Riemann hypothesis is to give a spectral interpretation of the zeros. That is, if the zeros can be interpreted as the eigenvalues of 1/2+iT, where T is a Hermitian operator on some Hilbert space, then since the zeros of a Hermitian operator are real, the Riemann hypothesis follows. This idea was originally put forth by Polya and Hilbert, and serious support for this idea was found in the resemblence between the ``explicit formulae'' of prime number theory, which go back to Riemann and Von Mangoldt, but which were formalized as a duality principle by Weil, on the one hand, and the Selberg trace formula on the other. GUE The best evidence for the spectral interpretation comes from the theory of the Gaussian Unitary Ensemble , which shows that the local behavior of the zeros mimics that of a random Hamiltonian. The link gives a more extended discussion of this topic. Goldfeld Goldfeld gave two spectral interpretations of the zeros of the zeta function; neither of these seems to prove the Riemann hypothesis. For example, in one interpretation, the zeros are eigenvalues of an operator, but it is unclear why the operator should be Hermitian. Connes Abstract: We reduce the Riemann hypothesis for L-functions on a global field k to the validity (not rigorously justified) of a trace formula for the action of the idele class group on the noncommutative space quotient of the adeles of k by the multiplicative group of k.

3. Riemann The riemann hypothesis is currently the most famous unsolved problem in mathematics I did some playing around with the riemann hypothesis, and I'm convinced it is true 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:

4. Riemann Hypothesis -- From MathWorld riemann hypothesis. http://mathworld.wolfram.com/RiemannHypothesis.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Calculus and Analysis Complex Analysis General Complex Analysis ... Wedeniwski Riemann Hypothesis First published by Riemann (1859), the Riemann hypothesis states that the nontrivial Riemann zeta function zeros all lie on the " critical line " where denotes the real part of s . While it was long believes 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). The hypothesis has thus far resisted all attempts to prove it. In the late 1940s, H. Rademacher's erroneous proof of the falsehood of Riemann's hypothesis was reported in Time magazine, even after a flaw in the proof had been unearthed by Siegel (Borwein and Bailey 2003, p. 97; Conrey 2003). In 2000, the Clay Mathematics Institute (

5. Riemann Hypothesis In A Nutshell The riemann hypothesis in a Nutshell. The Riemann Zeta Function. image source. (in Unix/Linux anyway). Verifying the riemann hypothesis. Basic Strategy. 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

6. The Riemann Hypothesis The riemann hypothesis. This web page highlights some of the conjectures and open problems concerning The riemann hypothesis. Click on the subject to see a short article on that topic. http://www.aimath.org/WWN/rh

The Riemann Hypothesis This web page highlights some of the conjectures and open problems concerning The Riemann Hypothesis. Click on the subject to see a short article on that topic. If you would like to print a hard copy of the entire web page, you can download a dvi postscript or pdf version. What is an $L$-function? Terminology and basic properties Functional equation Euler product ... Anecdotes about the Riemann Hypothesis

7. 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 This essay is my diploma thesis and was presented on Thursday November 29th 2001. The supervisor was professor J.A. Antoniadis . The evaluation commitee consisted also of Alexis Kouvidakis and Aristides Kontogeorgis The purpose of this essay is to show the usefulness of studying algebraic curves over finite fields, as far as Number Theory problems and Coding Theory are concerned. It contains a thorough treatment of Manin's proof of Hasse's theorem, which is a special case of Riemann Hypothesis for finite fields, and also examples of constructing algebraic geometry codes. In the first chapter we discuss basic properties of the theory of algebraic curves. In the second chapter we study elliptic curves over finite fields. In the third chapter we state basic notions of coding theory, some additional elements of algebraic curves theory and the essay ends with the detailed presentation of two algebraic geometry codes. See full introduction and contents in English: [html] [doc] [ps] In Word documents: Full/ZIP Greek English [90 pages / 344 Kb] Introduction/Contents/Chapter 1 Greek English [26 pages / 284 Kb] Chapters 2-3/Bibliography/Index Greek English [64 pages / 1,272 Kb]

8. 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 an open source and 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, 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 10,000 workstations and has a peak performance rate of about 5649 GFLOPS. More than 1 billion zeros for the zeta function are calculated every day. To learn more about ZetaGrid, you have two options:

9. Riemann's Hypothesis Riemann s Hypothesis. Riemann s Hypothesis. Euler s zeta function This new zetafunction has zeroes, and these form the basis for the riemann hypothesis. 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

10. 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

11. Register At NYTimes.com Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Riemann s paper includes the celebrated riemann hypothesis . Further informationconcerning the riemann hypothesis is to be found on the following Web pages http://www.nytimes.com/2002/07/02/science/physical/02MATH.html

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12. Riemann Hypothesis From MathWorld riemann hypothesis from MathWorld First published by Riemann (1859), the riemann hypothesis states that the nontrivial Riemann zeta function zeros all lie on the "critical line" \sigma=\Res=1 http://rdre1.inktomi.com/click?u=http://mathworld.wolfram.com/RiemannHypothesis.

13. 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/prize_problems/

14. Extended Riemann Hypothesis -- From MathWorld Extended riemann hypothesis. The first quadratic nonresidue mod p of a number isalways less than . Generalized riemann hypothesis, riemann hypothesis. search. http://mathworld.wolfram.com/ExtendedRiemannHypothesis.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Number Theory Reciprocity Theorems Extended Riemann Hypothesis The first quadratic nonresidue mod p of a number is always less than Generalized Riemann Hypothesis Riemann Hypothesis search 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. Eric W. Weisstein. "Extended Riemann Hypothesis." From MathWorld A Wolfram Web Resource. http://mathworld.wolfram.com/ExtendedRiemannHypothesis.html Wolfram Research, Inc.

15. The Riemann Zeta Function Formulae for the Riemann zetas function, its analytic continuation and functional equation, and the riemann hypothesis. http://numbers.computation.free.fr/Constants/Miscellaneous/zeta.html

The Riemann Zeta-function z (s) (pdf version : pdf ; (postscript version : ps The Riemann Zeta-function is defined as z (s) = n=1 n s for complex values of s. While converging only for complex numbers s with z n=1 n but it was Riemann who, in the 1850's, generalized its use and showed that the distribution of primes is related to the location of the zeros of z (s). Riemann conjectured that the non trivial zeros of z (s) are located on the critical line (s)=1/2. This conjecture, known as the Riemann hypothesis , has never been proved or disproved, and is probably the most important unsolved problem in mathematics. The Riemann hypothesis makes the zeta function so famous, and numerical computation have been made to check it for various sets of zeros. We expose here the most classical results about the zeta-function, together with some computational aspects. Our presentation is divided into several parts that are listed here : The Riemann Zeta-function z (s) : generalities Numerical evaluation of the Riemann Zeta-function ... Distribution of the zeros of the Riemann Zeta function In preparation : Computation of zeros of the Riemann Zeta-function. Back to numbers, constants and computation

16. Riemann Hypothesis - Wikipedia, The Free Encyclopedia The riemann hypothesis, first formulated by Bernhard Riemann in 1859, is a conjecture about the distribution of the Most mathematicians believe the riemann hypothesis to be true http://www.wikipedia.org/wiki/Riemann_hypothesis

Riemann hypothesis From Wikipedia, the free encyclopedia. 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. ( J. E. Littlewood and Atle Selberg have been reported as skeptical.) 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 Table of contents 1 History 2 The Riemann hypothesis and primes 3 Possible connection with operator theory 4 External links ... edit History Riemann mentioned the conjecture that became known as the Riemann hypothesis in his 1859 paper On the Number of Primes Less Than a Given Magnitude , but as it was not essential to his central purpose in that paper, he did not attempt a proof. Riemann knew that the non-trival zeros of the zeta function were symmetrically distributed about the line

17. Clay Mathematics Institute riemann hypothesis. Some numbers have the special property that they cannot be expressed as the s) called the Riemann Zeta function. The riemann hypothesis asserts that all http://www.claymath.org/prizeproblems/riemann.htm

18. The Riemann Hypothesis A collection of links relating to the riemann hypothesis, the proof ofwhich has been described as the holy grail of modern mathematics. http://www.maths.ex.ac.uk/~mwatkins/zeta/riemannhyp.htm

The Riemann Hypothesis Hilbert included the problem of proving the Riemann hypothesis in his list of the most important unsolved problems which confronted mathematics in 1900, and the attempt to solve this problem has occupied the best efforts of many of the best mathematicians of the twentieth century. It is now unquestionably the most celebrated problem in mathematics and it continues to attract the attention of the best mathematicians, not only because it has gone unsolved for so long but also because it appears tantalizingly vulnerable and because its solution would probably bring to light new techniques of far reaching importance. H.M. Edwards - Riemann's Zeta Function "Right now, when we tackle problems without knowing the truth of the Riemann hypothesis, it's as if we have a screwdriver. But when we have it, it'll be more like a bulldozer." P. Sarnak , from "Prime Time" by E. Klarreich ( New Scientist "The consequences [of the Riemann Hypothesis] are fantastic: the distribution of primes, these elementary objects of arithmetic. And to have tools to study the distribution of these of objects." H. Iwaniec, quoted in K. Sabbagh's

19. "AIM Sponsored Symposium On RH" A Symposium on the riemann hypothesis. University of Washington, Seattle, USA; 1215 August 1966. http://www.math.okstate.edu/~conrey/rh-conf.html

In Celebration of the Centenary of the Proof of the Prime Number Theorem A Symposium on the Riemann Hypothesis Sponsored by the American Institute of Mathematics Hadamard Riemann de la Vallee Poussin Conference Announcement Dates: August 12 - 15, 1996 Location: Seattle, Washington (Immediately following the MathFest) Schedule of Talks Hotel and Dorm information Banquet information Registration information ... Transportation from the Airport Tentative List of speakers: (Schedule is below) Michael Berry, University of Bristol Alain Connes, IHES William Duke, Rutgers University Dorian Goldfeld, Columbia University Roger Heath-Brown, Oxford University Dennis Hejhal, University of Minnesota and Uppsala University, Sweden Henryk Iwaniec, Rutgers University Nobushige Kurokawa, Tokyo Institute of Technology Hugh Montgomery, University of Michigan Samuel Patterson, Mathematisches Institut, Universitats Gottingen Peter Sarnak, Princeton University Atle Selberg, Institute for Advanced Study Financial support We have received funding from the National Science Foundation and from the National Security Agency to support some attendees, especially graduate students and other young mathematicians. To apply for this funding one should send a brief vita, the name of a reference, and a paragraph describing your interest in the conference. Send this information by e-mail to rh-conf.math.okstate.edu or to the Mathematics Department, Oklahoma State University, Stillwater, OK, 74078, care of Jennifer Gibson.

20. Proposed Proofs Of The Riemann Hypothesis proposed proofs of the riemann hypothesis. This is the latest C. Castro andJ. Mahecha, Final steps towards a proof of the riemann hypothesis . http://www.maths.ex.ac.uk/~mwatkins/zeta/RHproofs.htm

proposed proofs of the Riemann Hypothesis Without doubt it would be desirable to have a rigorous proof of this proposition; however I have left this research aside for the time being after some quick unsuccessful attempts, because it appears to be unnecessary for the immediate goal of my study... [B. Riemann] If you are a university mathematics lecturer who teaches analytic number theory, you might want to consider setting your students the task of deconstructing the more serious of these. They may otherwise never be given any serious attention, which would be a shame. C. Castro, A. Granik, and J. Mahecha, "On SUSY-QM, fractal strings and steps towards a proof of the Riemann hypothesis" Despite the humility of the title, this preprint does contain a (purported) proof of the RH. The following preprint examines the strategy proposed. E. Elizalde V. Moretti , and S. Zerbini "On recent strategies proposed for proving the Riemann hypothesis" (abstract) "We comment on some apparently weak points in the novel strategies recently developed by various authors aiming at a proof of the Riemann hypothesis. After noting the existence of relevant previous papers where similar tools have been used, we refine some of these strategies. It is not clear at the moment if the problems we point out here can be resolved rigorously, and thus a proof of the RH be obtained, along the lines proposed. However, a specific suggestion of a procedure to overcome the encountered difficulties is made, what constitutes a step towards this goal."

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