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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

Extractions: 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:

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/

Extractions: 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

Extractions: 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).

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

Extractions: 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

Extractions: Home Z(t) Plotter Verifying RH ... More Applets 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 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

Extractions: 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

Extractions: 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]

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/

Extractions: 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.

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

Extractions: This converges for complex s such that the real part of s is greater than 1, but for s 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

Extractions: Real and imaginary part of Zeta[1/2+I*y]. Both curves intersect precisely at the y-axis The same zeros as a "spectrum". 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

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

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

Extractions: 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

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

Extractions: 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: 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 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

Extractions: 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

Extractions: Hadamard Riemann de la Vallee Poussin Schedule of Talks Hotel and Dorm information Banquet information Registration information ... Transportation from the Airport 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 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

Extractions: 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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