Geometry.Net - Theorems_And_Conjectures: Riemann Hypothesis

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

Extractions: 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.
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 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.
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/

Extractions: 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. Contact Search Clay Mathematics Institute
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/
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
Riemann
A short article with some grahpical and numerical evidence in the critical strip.
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).
The Riemann Hypothesis
A short article by Kimon Spiliopoulos.
http://users.forthnet.gr/ath/kimon/Riemann/Riemann.htm

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

Extractions: 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 Riemanns paper is very obscure and confusing because of its excessive brevity. Fortunately
Mathematical Constants
Notes by Steven Finch.
http://pauillac.inria.fr/algo/bsolve/constant/apery/riemhyp.html

Extractions: 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 Inequalities and Approximation Real and Complex Analysis Probability and Stochastic Processes
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
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

Extractions: 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.
The Riemann Hypothesis
A prime pages article by Chris K. Caldwell.
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:
Riemann Hypothesis In A Nutshell
riemann hypothesis
http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126
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

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
Millennium Prize Problems
The seven problems proposed by the Clay Mathematics Institute P versus NP; Hodge Conjecture; PoincarÃ© 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/

Extractions: 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.
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
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

Extractions: 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 â 1. It has certain so-called "trivial" zeros for s s s = â6, ... 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 . 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.
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
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

Extractions: 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 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 Personal tools Navigation 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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