21. Riemann Hypothesis Information about Riemann Hypothesis. Link to complete Wikipedia entry is provided. http://www.campusprogram.com/reference/en/wikipedia/r/ri/riemann_hypothesis.html

MENU Home Universities Colleges Programs ... Store Web www.campusprogram.com Riemann Hypothesis Wikipedia Reference Information In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous, and important, 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 (RH) 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 zeros at the negative even integers (i.e. at s = -2, s = -4, s = -6, ...). These are called the trivial zeros. 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 ˝. 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. A polar graph of zeta, that is, Re(zeta) vs. Im(zeta), along the critical line s=it+1/2, with t running from to 34The Riemann hypothesis is one of the most important open problems of contemporary mathematics, mainly because a large number of deep and important other results have been proven under the condition that it holds. Most mathematicians believe the Riemann hypothesis to be true. (J. E. Littlewood and Atle Selberg have been reported as skeptical. Selberg's skepticism, if any, waned, from his young days. In a 1989 paper, he suggested that an analogue should hold for a much wider class of functions, the Selberg class). A $1,000,000 prize has been offered by the Clay Mathematics Institute for the first correct proof.

22. Article About "Riemann Hypothesis" In The English Wikipedia On 24-Jul-2004 The Riemann hypothesis, first formulated by Bernhard Riemann in 1859, is a conjecture about the distribution of the zeross of Riemann's zeta function ζ. It is one of the most http://july.fixedreference.org/en/20040724/wikipedia/Riemann_hypothesis

The Riemann hypothesis reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference : snapshots of Wikipedia from wikipedia.org) Riemann hypothesis Helping orphans the way you would do it The Riemann hypothesis , first formulated by Bernhard Riemann in , is a conjecture about the distribution of the zeross 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. In June 2004, Louis De Branges de Bourcia claimed to have proved the Riemann hypothesis but this has not yet been confirmed (see below). 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 showTocToggle("show","hide")

23. Riemann Hypothesis - Exampleproblems TemplateMillenium Problems. In mathematics, the Riemann hypothesis (also called the Riemann zeta hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most http://www.exampleproblems.com/wiki/index.php/Riemann_hypothesis

Riemann hypothesis From Exampleproblems Jump to: navigation search Template:Millenium Problems In mathematics , the Riemann hypothesis (also called the 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 s s s 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. Image:Criticalline.png The real part (red) and imaginary part (blue) of the critical line Re( z ) = 1/2 of the Riemann zeta function. You can see the first non-trivial zeros at Im(

24. Riemann Hypothesis - Facts, Information, And Encyclopedia Reference Article facts and reference information, also Cool links, search engines and more http://www.startsurfing.com/encyclopedia/r/i/e/Riemann_hypothesis.html

Riemann hypothesis Categories Zeta and L-functions Conjectures Hilbert's problems ... The Poincaré conjecture The Riemann hypothesis Yang-Mills existence and mass gap Navier-Stokes existence and smoothness The Birch and Swinnerton-Dyer conjecture In mathematics , the Riemann hypothesis (also called the 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 s s s 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.

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

26. The Riemann Hypothesis Here we define, then discuss the Riemann hypothesis. We provide several related links. http://primes.utm.edu/notes/rh.html

The Riemann Hypothesis (Another of the Prime Pages ' resources) Our book " Prime Curios! The Dictionary of Prime Number Trivia " is now available on CreateSpace Amazon Home Search Site ... 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 he could calculate were symmetric about the line Re( s The Riemann hypothesis is that all nontrivial zeros are on this line. Proving the Riemann Hypothesis would allow us to greatly sharpen many number theoretical results. For example, in 1901 von Koch showed that the Riemann hypothesis is equivalent to: But it would not make factoring any easier! There are a couple standard ways to generalize the Riemann hypothesis.. 1. The Riemann Hypothesis:

27. Riemann A short article with some graphical 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:

28. The Riemann Hypothesis 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 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, …

29. The Riemann Hypothesis, Volume 50, Number 3 M ARCH 2003 N OTICES OF THE AMS 341 The Riemann Hypothesis J. Brian Conrey H ilbert, in his 1900 address to the Paris International Congress of Mathematicians, listed the Riemann http://www.ams.org/notices/200303/fea-conrey-web.pdf

30. ZetaGrid - Verification Of The Riemann Hypothesis Why is Riemann's Hypothesis so important? The verification of Riemann's Hypothesis (formulated in 1859) is considered to be one of modern mathematic's most important problems. http://www.zetagrid.net/zeta/rh.html

Verification of the Riemann Hypothesis ZetaGrid Acknowledgement Performance characteristics Riemann Hypothesis Prizes Motivation News Statistics ... Links Why is Riemann's Hypothesis so important? The verification of Riemann's Hypothesis (formulated in ) is considered to be one of modern mathematic's most important problems. The last 140 years did not bring its proof, but a considerable number of important mathematical theorems which depend on the Hypothesis being true, e.g. the fastest known primality test of Miller. The Riemann zeta function is defined for Re( s )>1 by and is extended to the rest of the complex plane (except for s =1) by analytic continuation. The Riemann Hypothesis asserts that all nontrivial zeros of the zeta function are on the critical line (1/2+ it where t is a real number). To verify empirically the Riemann Hypothesis for certain regions and make it usable, in the first fifteen zeros of Riemann's zeta function t Participate in the verification of Riemann's Hypothesis! Today, we have better resources to verify or falsify Riemann's Hypothesis. First the high-speed computers, then the networks have increased the capacity of calculations. Now we want to go one step further by bundling up the resources into a grid network. Therefore, I invite all interested people to participate in the verification of the zeros of the Riemann zeta function for a new record. Before I have started with the computation on August 28, 2001, the hypothesis has been checked for the first 1,500,000,001 zeros. On October 27, 2001, J. van de Lune checked the hypothesis for the first 10 billion zeros. Up to now, it has been extended to the first 100 billion zeros which required more than 1.3

31. Riemann Hypothesis In A Nutshell Jun 20, 2008 The Riemann Hypothesis (RH) is that all nontrivial zeros of the zeta function lie on the critical line. Let s say that again http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html

Glen's Home VIU Math VIU Home email me The Riemann Hypothesis in a Nutshell Jump to section: [ Z(t) Plotter Verifying RH More Applets 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

32. The Riemann Hypothesis The Riemann Hypothesis This web page highlights some of the conjectures and open problems concerning The Riemann Hypothesis. If you would like to print a hard copy of the whole http://www.aimath.org/WWN/rh/

The Riemann Hypothesis This web page highlights some of the conjectures and open problems concerning The Riemann Hypothesis. If you would like to print a hard copy of the whole outline, you can download a dvi postscript or pdf version. What is an $L$-function? Terminology and basic properties Functional equation Euler product ... Examples Dirichlet series associated with Maass forms Higher rank L-functions The Selberg class Dirichlet series Analytic Continuation Functional Equation ... Selberg Conjectures Analogues of zeta-functions Dynamical zeta-functions Spectral zeta functions Riemann Hypotheses Riemann Hypotheses for global L-functions The Riemann Hypothesis The Generalized Riemann Hypothesis The Extended Riemann Hypothesis ... The vertical distribution of zeros The Lindelof hypothesis and breaking convexity Perspectives on RH Analytic number theory Physics Probability Fractal geometry Equivalences to RH Primes The error term in the PNT More accurate estimates ... The Farey series Mikolas functions Amoroso's criterion Weil's positivity criterion Li's criterion Bombieri's refinement Complex function theory Speiser's criterion Logarithmic integrals An inequality for the logarithmic derivative of xi Function spaces ... Salem's criterion Other analytic estimates M. Riesz series

33. Sign In To Read: Has The Riemann Hypothesis Finally Been Proven? - Physics-math Mar 22, 2008 These are from people claiming to have cracked a longstanding conundrum known as the Riemann hypothesis. At any given moment we probably http://www.newscientist.com/article/mg19726481.500-has-the-riemann-hypothesis-fi

@import "/css/article.css"; @import "/css/barrier.css"; Log in Email Password Remember me Your login is case sensitive Forgotten your password? Subscriber? Activate now! Subscribe now Institutional Subscribers ... SCIENCE IN SOCIETY This is a preview of the full article New Scientist full online access is exclusive to subscribers as part of their subscription package. Registered users are given limited access to content, find out more Gain instant online access - Including 20 years of archive Receive New Scientist magazine delivered to you every week Access the most comprehensive round up of the latest news in science Home In-Depth Articles Has the Riemann hypothesis finally been proven? 22 March 2008 by Eric Kvaalen Magazine issue Subscribe and save AS ONE of the editors of the Annals of Mathematics , Peter Sarnak sees his fair share of mathematical proofs. Yet there is one unsolved problem for which proofs keep on turning up in his mailbox. These are from people claiming to have cracked a long-standing conundrum known as the Riemann hypothesis. "At any given moment we probably have 10 claimed proofs submitted," says Sarnak, a mathematician at Princeton University. Perhaps that is not so surprising. First put forward in 1859 by German mathematician Bernhard Riemann, the hypothesis is one of mathematics's most beguiling problems. Its allure lies in the fact that it holds the key to the primes, those numbers that underpin so much of today's mathematics. The

34. Riemann Hypothesis: Definition From Answers.com The conjecture that the only zeros of the Riemann zeta function with positive real part must have their real part equal to . http://www.answers.com/topic/riemann-hypothesis

35. ZetaGrid Homepage Numerical verification of the Riemann Hypothesis by a collaborative computing effort, with downloadable software. http://www.zetagrid.net/

ZetaGrid ZetaGrid Acknowledgement Performance characteristics Riemann Hypothesis Prizes ... Links This site is owned by Sebastian Wedeniwski Sponsors restaurant reviews london, travel reports root server, dedicated server The ZetaGrid activities must come to a final end! Now all services are down and this domain will be closed soon. The official last update note was sent on December 1, 2005. Thanks for your interest to this community. 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.

36. Chaitin, Thoughts On The Riemann Hypothesis The simultaneous appearance in May 2003 of four books on the Riemann hypothesis (RH) provoked these reflections. We briefly discuss whether the RH should be added as a new axiom http://www.maa.org/features/chaitin.html

Thoughts on the Riemann Hypothesis G. J. Chaitin, IBM Research The simultaneous appearance in May 2003 of four books on the Riemann hypothesis (RH) provoked these reflections. We briefly discuss whether the RH should be added as a new axiom, or whether a proof of the RH might involve the notion of randomness. New pragmatically-justified mathematical axioms that are not at all self-evident Are there important mathematical propositions for which there is a considerable amount of computational evidence, evidence that is so persuasive that a physicist would regard them as experimentally verified? Yes, I believe there are. Currently, the two best candidates* for useful new axioms of the kind that that are justified pragmatically as in physics are: the P NP hypothesis in theoretical computer science that conjectures that many problems require an exponential amount of work to resolve, and the Riemann hypothesis concerning the location of the complex zeroes of the Riemann zeta-function z s n n s p p s (Here n ranges over positive integers and p ranges over the primes.)**

37. Louis De Branges: Home Purdue University. Proposed solution for the Riemann Hypothesis (pdf format). http://www.math.purdue.edu/~branges/

Department of Mathematics About Us Faculty Emeritus Faculty ... Home Louis de Branges Home Papers Publications PhD: Cornell University 1957 Title: Edward C. Elliott Distinguished Professor of Mathematics Professional Interests: complex analysis, Fourier analysis, functional analysis Office: MATH 800 Phone: Email: Homepage: http://www.math.purdue.edu/~branges/ Home Papers Publications Department of Mathematics, Purdue University 150 N. University Street, West Lafayette, IN 47907-2067 Phone: (765) 494-1901 - FAX: (765) 494-0548 Contact the Webmaster West Lafayette, IN 47907 USA, 765-494-4600 An equal access/equal opportunity university Contact the Webmaster to address accessibility issues.

38. Stalking The Riemann Hypothesis « Not Even Wrong My friend Dan Rockmore has a new book out, entitled Stalking the Riemann Hypothesis, which is quite good. Dan had the misfortune of starting work on this book at the same time as http://www.math.columbia.edu/~woit/wordpress/?p=197

39. The Riemann Hypothesis Is 150 Years Old The Riemann Hypothesis is 150 years old The Riemann Hypothesis the most important unsolved problem in mathematics celebrates its 150th birthday in 2009. http://aimath.org/RH150/

The Riemann Hypothesis is 150 years old The Riemann Hypothesis the most important unsolved problem in mathematics celebrates its 150th birthday in 2009. Three special events will celebrate this landmark. Upcoming events RH Day: On November 18, 2009 the anniversary of the Riemann Hypothesis will be celebrated with a series of lectures all over the world. See the lecture schedule (additional lectures are welcome). Note: Riemann communicated his paper to the Monatsberichte der Berliner Akademie on October 19, 1859, and Kummer read the paper at the meeting of the academy on November 3. The specific date it appeared in print, which would be the "official" date of the paper, does not seem to be known. (Thank you to Juan Arias de Reyna for this information.) Problem Lists: Also on November 18, 2009, a new interactive tool for maintaining problem lists will be released. A list of problems related to the Riemann Hypothesis will be featured. ( Link available on November 18. Past events Advances in number theory and geometry : 150 years of Riemann Hypothesis, at RISM, Verbania, Italy. April 19 to 24, 2009. Zeta function days at Yonsei University, Seoul, South Korea. August 31 to September 5, 2009.

40. The Riemann Hypothesis Some of the conjectures and open problems concerning RH, compiled by the AIM. http://aimath.org/WWN/rh/

The Riemann Hypothesis This web page highlights some of the conjectures and open problems concerning The Riemann Hypothesis. If you would like to print a hard copy of the whole outline, you can download a dvi postscript or pdf version. What is an $L$-function? Terminology and basic properties Functional equation Euler product ... Examples Dirichlet series associated with Maass forms Higher rank L-functions The Selberg class Dirichlet series Analytic Continuation Functional Equation ... Selberg Conjectures Analogues of zeta-functions Dynamical zeta-functions Spectral zeta functions Riemann Hypotheses Riemann Hypotheses for global L-functions The Riemann Hypothesis The Generalized Riemann Hypothesis The Extended Riemann Hypothesis ... The vertical distribution of zeros The Lindelof hypothesis and breaking convexity Perspectives on RH Analytic number theory Physics Probability Fractal geometry Equivalences to RH Primes The error term in the PNT More accurate estimates ... The Farey series Mikolas functions Amoroso's criterion Weil's positivity criterion Li's criterion Bombieri's refinement Complex function theory Speiser's criterion Logarithmic integrals An inequality for the logarithmic derivative of xi Function spaces ... Salem's criterion Other analytic estimates M. Riesz series

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