21. A Directory Of All Known Zeta Functions A directory of websites maintained by Matthew Watkins on the topic of The riemann hypothesis and Zeta Function. http://www.maths.ex.ac.uk/~mwatkins/zeta/directoryofzetafunctions.htm

a directory of all known zeta functions [This page is under continual construction! Any contributions would be welcome.] Over the years striking analogies have been observed between the Riemann zeta-function and other zeta- or L-functions. While these functions are seemingly independent of each other, there is growing evidence that they are all somehow connected in a way that we do not fully understand. In any event, trying to understand, or at least classify, all of the objects which we believe satisfy the Riemann hypothesis is a reasonable thing to do." J. Brian Conrey, "The Riemann Hypothesis" Notices of the AMS (March, 2003) p.347 In this essay I will give a strictly subjective selection of different types of zeta functions. Instead of providing a complete list, I will rather try to give the central concepts and ideas underlying the theory... Whenever entities are counted with some mathematical structure on them it is likely that a zeta function can be set up and often enough it will extend to a meromorphic function. Zeta functions show up in all areas of mathematics and they encode properties of the counted objects which are well hidden and hard to come by otherwise. They easily give fuel for bold new conjectures and thus drive on mathematical research. It is a fairly safe assertion to say that zeta functions of various kinds will stay in the focus of mathematical attention for times to come. A. Deitmar

22. ZetaGrid Homepage Numerical verification of the riemann hypothesis by a collaborative computing effort, with downloadable software. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

23. 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) 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

24. Purdue Mathematician Claims Proof For Riemann Hypothesis Purdue mathematician claims proof for riemann hypothesis. WEST LAFAYETTE, Ind. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

25. The Prime Glossary: Riemann Hypothesis Welcome to the Prime Glossary a collection of definitions, information and factsall related to prime numbers. This pages contains the entry titled http://primes.utm.edu/glossary/page.php?sort=RiemannHypothesis

26. "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.

27. The Riemann Hypothesis 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 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

28. Riemann Theory of functions, nonEuclidian geometry, relativity theory, Famous for the riemann hypothesis about the behavior of zeros of the Riemann zeta function, which he showed determines the distribution of prime numbers. http://turnbull.dcs.st-and.ac.uk/history/Mathematicians/Riemann.html

Georg Friedrich Bernhard Riemann Born: 17 Sept 1826 in Breselenz, Hanover (now Germany) Died: 20 July 1866 in Selasca, Italy Click the picture above to see three larger pictures Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Version for printing Bernhard Riemann 's father, Friedrich Bernhard Riemann, was a Lutheran minister. Friedrich Riemann married Charlotte Ebell when he was in his middle age. Bernhard was the second of their six children, two boys and four girls. Friedrich Riemann acted as teacher to his children and he taught Bernhard until he was ten years old. At this time a teacher from a local school named Schulz assisted in Bernhard's education. In 1840 Bernhard entered directly into the third class at the Lyceum in Hannover. While at the Lyceum he lived with his grandmother but, in 1842, his grandmother died and Bernhard moved to the Johanneum Gymnasium Legendre 's book on the theory of numbers and Bernhard read the 900 page book in six days. Gauss Gauss did lecture to Riemann but he was only giving elementary courses and there is no evidence that at this time he recognised Riemann's genius. Stern, however, certainly did realise that he had a remarkable student and later described Riemann at this time saying that he:-

29. Millennium Prize Problems The seven problems proposed by the Clay Mathematics Institute P versus NP; Hodge Conjecture; Poincar Conjecture; riemann hypothesis; YangMills http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

30. The Riemann Hypothesis A collection of links relating to the riemann hypothesis, the proof of which hasbeen 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

31. Guardian Unlimited The Guardian 'There Is More Chance Of Him 'There is more chance of him proving Riemann's hypothesis than wearing a sarong' Wayne Rooney will face Portugal tonight as England's new http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

32. Proposed Proofs Of The Riemann Hypothesis abstract A proof of the Riemann s hypothesis (RH) about the nontrivial zeroes Maybe the riemann hypothesis was simply unprovable within our current 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."

33. Donald L. Hitzl Home Page Research and other interests of Dr. Donald L. Hitzl, including a recent paper on the Zeta function which experimentally verifies the riemann hypothesis. http://www.donhitzl.com/

Don Hitzl's home page Resume Zeta Function Paper Poetry ... Proposed Book Donald Leigh Hitzl DOB: February 2, 1941 Address: 7 Candlestick Road Orinda, CA 94563-3701 Phone: (925) 253-0513 E-mail: domarltd@attbi.com Personal status: My wife, Marjorie and I plus three animals - Keoki, the Keeshond, Katie, the Sheltie and T-man, the tabby cat - live in Orinda, California. We live close to children, grandchildren, and other relatives which is a constant joy to us. In addition, as retirees, we devote a lot of our time to the Orinda Community Church - choir, committees, Council, to name a few. Also, I am an active Rotarian and presently am the Chair of Public Relations for the Rotary Club of Orinda Professional Experience Education Honors ... Most Recent Paper - title and excerpts Other Interests: Poetry Proposed book - History of Stanford Academics that immigrated before WW II At the end of each day, we just keep working away... 7650 visits Site design by Eyerarts Detected Browser: SecretBrowser/007 validate

34. Jean-Fran Ois Burnol Universit Lille. Number theory. Publications and research papers. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

35. The Riemann Hypothesis The Lindelof hypothesis and breaking convexity; Perspectives on RH Analytic numbertheory Physics; Probability; Fractal geometry. Equivalences to RH 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

36. 17a The Generalized riemann hypothesis(GRH) is the assertion that the riemann hypothesisis true, Back to the main index for The riemann hypothesis. http://www.aimath.org/WWN/rh/articles/html/17a/

The Generalized Riemann Hypothesis The Generalized Riemann Hypothesis(GRH) is the assertion that the Riemann Hypothesis is true, and in addition the nontrivial zeros of all Dirichlet $L$-functions lie on the critical line Equivalently, GRH asserts that the nontrivial zeros of all degree 1 -functions lie on the critical line. The Modified Generalized Riemann Hypothesis(MGRH) is the assertion that the Riemann Hypothesis is true, and in addition the nontrivial zeros of all Dirichlet $L$-functions lie either on the critical line or on the real axis. Back to the main index for The Riemann Hypothesis.

37. ZetaGrid - Verification Of The Riemann Hypothesis Why is Riemann s Hypothesis so important? The verification of Riemann sHypothesis (formulated in 1859) is considered to be one of modern mathematic s most 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

38. Riemann Hypothesis In A Nutshell An article by Glen Pugh with a Java applet for viewing zeta on the critical line. http://www.math.ubc.ca/~pugh/RiemannZeta/

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

39. 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

40. Home Page For Prime Obsession Prime Obsession is a nonfiction book on the riemann hypothesis, a famous unsolvedproblem in higher mathematics. The book was published April 16, http://olimu.com/Riemann/Riemann.htm

Prime Obsession Navigate up John Derbyshire's home page Navigate across Fire from the Sun 36 Great American Poems Seeing Calvin Coolidge in a Dream Print journalism ... Photographs Navigate down Description Reviews Promotional events FAQs ... Paperback Note The paperback edition of Prime Obsession was published in May, 2004 by Plume, a division of Penguin Books. I have established a separate set of pages for the paperback edition, which you can view here Newsflash Sept. 16, 2005: Researchers Roger Plymen and Kuok Fai Chao over in the U.K. have published a paper on the "Littlewood violation" (p. 236 of Prime Obsession) in which they claim to have improved the Bays-Hudson result. The paper is on the internet here Prime Obsession is a nonfiction book on the Riemann Hypothesis, a famous unsolved problem in higher mathematics. The book was published April 16, 2003 by Joseph Henry Press of Washington, D.C. It can be ordered on Amazon.com and BarnesAndNoble.com, and is available in bookstores and libraries.

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