1. AThe Riemann Hypothesis A prime pages article by Chris K. Caldwell.Category Science Math Number Theory Analytic 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 Finding ... 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 Papers and other links related to RH compiled by Dan Bump.Category Science Math Number Theory Analytic 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 Hypothesis This simple statement is the famous riemann hypothesis. Nobody knows for certain if this is true. 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

4. Riemann Hypothesis -- From MathWorld Article with links to other resources from MathWorld.Category Science Math Number Theory Analytic riemann hypothesis......riemann hypothesis, André Weil proved the riemann hypothesis to be truefor field functions (Weil 1948, Eichler 1966, Ball and Coxeter 1987). http://mathworld.wolfram.com/RiemannHypothesis.html

Calculus and Analysis Complex Analysis General Complex Analysis Calculus and Analysis ... Sondow Riemann Hypothesis First published in 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 . The Riemann hypothesis is also known as Artin's conjecture. The Riemann hypothesis is equivalent to the statement that all the zeros of the Dirichlet eta function (a.k.a. the alternating zeta function) falling in the critical strip lie on the critical line Wiener showed that the prime number theorem is literally equivalent to the assertion that the Riemann zeta function has no zeros on (Hardy 1999, pp. 34 and 58-60). In 1914, Hardy proved that an infinite number of values for s can be found for which and . However, it is not known if all nontrivial roots s satisfy roots must lie on the critical line (Le Lionnais 1983), a result which has since been sharpened to 40% (Vardi 1991, p. 142). It is known that the zeros are symmetrically placed about the line . This follows from the fact that, for all complex numbers

5. 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 What is ZetaGrid? ZetaGrid is an open source and platform independent grid system that uses idle CPU cycles from participating computers. It can be used for any CPU intensive application which can be split into many separate steps and which would run very long 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 mathematic's most important problems. This implementation involves more than 4,000 workstations and has a peak performance rate of about 530 GFLOPS. More than 1 billion zeros for the zeta function are calculated every day. To learn more about ZetaGrid, you have two options: view grid monitoring data and statistics of the current implementation on our performance page.

6. ZetaGrid - Verification Of The Riemann Hypothesis Verification of the riemann hypothesis. ZetaGrid. Acknowledgement 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

7. The Riemann Hypothesis A collection of links relating to the riemann hypothesis, the proof of which has been described as the 'holy grail' of modern mathematics. riemann hypothesis. " Hilbert included the problem of proving the riemann hypothesis in his list of the most important 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 Basic introduction to the Riemann Hypothesis (C. Caldwell)

8. Clay Mathematics Institute - Riemann Hypothesis riemann hypothesis. The riemann hypothesis asserts that all interestingsolutions of the equation. z(s) = 0. lie on a straight line. http://www.claymath.org/Millennium_Prize_Problems/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, 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 “ z (s)” called the Riemann Zeta function . The Riemann hypothesis asserts that all interesting solutions of the equation z (s) = lie on a 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. Lecture by Jeff Vaaler at UT.ram Official Problem Description.pdf Privacy Statement Help

9. Georg Friedrich Bernhard Riemann (1826-1866) Riemann's inaugural lecture on the foundations of geometry. The Riemann Zeta Function and the riemann hypothesis http://www.maths.tcd.ie/pub/HistMath/People/Riemann

Georg Friedrich Bernhard Riemann (1826-1866) The Life of Bernhard Riemann The Mathematical Papers of Bernhard Riemann Riemann's inaugural lecture on the foundations of geometry The Riemann Zeta Function and the Riemann Hypothesis Back to: Mathematicians and Philosophers in the History of Mathematics archive The History of Mathematics David R. Wilkins dwilkins@maths.tcd.ie ... Trinity College, Dublin

10. Riemann A short article with some grahpical and numerical evidence in the critical strip.Category Science Math Number Theory Analytic riemann hypothesis......The riemann hypothesis is currently the most famous unsolved problemin mathematics. Like the Goldbach Conjecture (all positive 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:

11. Riemann's Hypothesis A beginners guide by Jon Perry.Category Science Math Number Theory Analytic riemann 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] 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 a 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

12. The Riemann Hypothesis A short article by Kimon Spiliopoulos.Category Science Math Number Theory Analytic riemann hypothesis......The riemann hypothesis Riemann's The fact that riemann hypothesis holdsfor billions of nontrivial zeros does not guarantee anything. As 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,

13. Register At NYTimes.com The riemann hypothesis is currently the most famous unsolved problem in mathematics. http://www.nytimes.com/2002/07/02/science/physical/02MATH.html

Welcome to The New York Times on the Web! For full access to our site, please complete this simple registration form. As a member, you'll enjoy: In-depth coverage and analysis of news events from The New York Times FREE Up-to-the-minute breaking news and developing stories FREE Exclusive Web-only features, classifieds, tools, multimedia and much, much more FREE Please enter your Member ID: Please enter your password: Remember my Member ID and password on this computer. Forgot your password? Choose a Member ID: Choose a password: (Five character minimum) Re-enter your password for verification: E-Mail Address: Remember my Member ID and password on this computer We'll keep your information private. The following fields are required. NYTimes.com respects your privacy , so we will never share any personal information without your consent. Gender: Year of Birth: Male Female (Click here if you are under 13) Zip Code: Country of Residence: United States Afghanistan Albania Algeria American Samoa Andorra Angola Anguilla Antarctica Antigua and Barbuda Argentina Armenia Aruba Australia Austria Azerbaijan Bahamas Bahrain Bangladesh Barbados Belarus Belgium Belize Benin Bermuda Bhutan Bolivia Bosnia and Herzegowina Botswana Bouvet Island Brazil British Indian Ocean Territory Brunei Darussalam Bulgaria Burkina Faso

14. Body A claimed proof of Riemann's Hypothesis.Category Science Math Number Theory Analytic riemann hypothesis......PROOF OF RIEMANN'S HYPOTHESIS. James Constant. math@coolissues.com. Riemann'shypothesis is proved using Riemann's functional equation. Introduction. http://www.coolissues.com/mathematics/riemann.htm

PROOF OF RIEMANN'S HYPOTHESIS James Constant math@coolissues.com Riemann's hypothesis is proved using Riemann's functional equation. Introduction The famous conjecture known as Riemann' s hypothesis is to classical analysis what Fermat's last theorem is to arithmetic. Euler (1737) noted that the formula the sum extending to all positive integers n, and the product to all positive primes p. The neces- sary conditions of convergence hold for complex values of s as a function of of the complex variable s , Riemann (1859) proved that satisfies a functional equation which led Riemann to the theorem that all the zeros of , except those at s=-2,-4,-6, . . . , lie in the strip of the s -plane for which where x is the real part of s . Riemann conjectured that all the zeros in the strip should lie on the line x = ½. Attempts to prove or disprove this conjecture have generated a vast and intricate department of analysis, especially since Hardy (1914) proved that has an infinity of zeros on x The question is still open in 2000. A prize is available to prove or disprove Riemann's hypothesis. Proof Using Riemann's Functional Equation It has already been shown that all zeros are in the critical strip and that they are symmetric about the critical line x Riemann's functional equation can be restated as =A( in which A( at all points in the critical strip. Since functions

15. Body Generalisations of the zeta function might provide a proof of Riemann's hypothesis.Category Science Math Number Theory Analytic riemann hypothesis......Proofs of Riemann's Hypothesis. James Constant. math@coolissues.com. ProvingRiemann's Hypothesis. My theory for proving Riemann's hypothesis is simple. http://www.coolissues.com/mathematics/zeta.htm

Some Extended Zeta Functions Provide Easy Proofs of Riemann's Hypothesis James Constant math@coolissues.com While extended zeta functions support investigations of Riemann's hypothesis and estimates for the Prime Number Theorem, some zeta functions offer better prospects for providing easy proofs. Definitions A first zeta function is defined by oo z(s)= s=x+jy n=1 A second zeta function is defined by oo z(1-s)= s=x+jy n=1 In 1859, Riemann had the idea to define z(s) for all complex numbers s by analytic extension. This extension is important in number theory and plays a central role in the distribution of prime numbers. One way of extending is by using the first f function alternating series defined by oo f(s)= s=x+jy n=1 1 by means of the formula f(s)=(1-2 )z(s) A second f function is defined by oo f(1-s)= s=x+jy n=1 1 by means of the formula f(1-s)=(1-2 )z(1-s) Equations (1) through (6) are analytic. Riemann's Extended Zeta Function and Functional Equation Euler (1737) noted that the formula n, and the product to all positive primes p.

16. 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's Zeta function z (s) Definition The Zeta function was first introduced by Euler and is defined by z (s) = n = 1 n s The series is convergent when s is a complex number with z (s) are well known, for example the values z p z p /90, were obtained by Euler. In 1859, Riemann had the idea to define z (s) for all complex number s by analytic continuation. This continuation is very important in number theory and plays a central role in the study of the distribution of prime numbers. Several techniques permit to extend the domain of definition of the Zeta function (the continuation is independent of the technique used because of uniqueness of analytic continuation). One can for example start from the Zeta alternating series z a (s) n = 1 n-1 n s defining an analytic function for z a (s) = n = 1 n s n = 1 s z (s) - s z (s). In other words, we have z (s) = z a (s) 1-s Since z a (s) is defined for ) permits to define the Zeta function for all complex number s with positive real part, except for s = 1 for which we have a pole. The extension of the Zeta function to the domain (s) can also be done (a different technique should be used).

17. 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/

18. Extended Riemann Hypothesis -- From MathWorld Extended riemann hypothesis, The first quadratic nonresidue mod p of anumber is always less than . riemann hypothesis. References. http://mathworld.wolfram.com/ExtendedRiemannHypothesis.html

Number Theory Reciprocity Theorems Extended Riemann Hypothesis The first quadratic nonresidue mod p of a number is always less than Riemann Hypothesis 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. Author: Eric W. Weisstein Wolfram Research, Inc.

19. 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

20. Bernhard Riemann Theory of functions, nonEuclidian geometry, relativity theory, His profound conjecture (the riemann hypothesis) about the behavior of the zeta (or Riemann) function, which he showed determines the distribution of the prime numbers. http://userwww.sfsu.edu/~rsauzier/Riemann.html

Riemann, (Georg Friedrich) Bernhard (1826-66) Riemann, (Georg Friedrich) Bernhard (1826-66) Euclid ian geometry, important in modern physics and relativity theory. His profound conjecture (the Riemann hypothesis ) about the behavior of the zeta (or Riemann) function, which he showed determines the distribution of the prime numbers, has resisted proof since its publication in 1857. Return to the biography index . Back to my index , the school , the bookstore , or the Underground student pipeline

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