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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/

Extractions: 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

2. Riemann Hypothesis - Wikipedia, The Free Encyclopedia
http://en.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

3. 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. It's best to look ahead to the future and not to dwell on the past. * 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 the front cover and some reviews Here are errata and addenda to the book (last updated 5/25/2004), 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

4. The Prime Glossary: Riemann Hypothesis
Welcome to the Prime Glossary a collection of definitions, information and facts all related to prime numbers. This pages contains the entry titled 'riemann hypothesis.' Come explore a new prime
http://primes.utm.edu/glossary/page.php/RiemannHypothesis.html

Extractions: (another Prime Pages ' Glossary entries) Glossary: Prime Pages: 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 fact the classical proofs of the prime number theorem require an understanding of the zero free regions of this function, and in 1901 von Koch showed that the Riemann hypothesis is equivalent to:  Because of this relationship to the prime number theorem, Riemann's hypothesis is easily one of the most important conjectures in prime number theory. See Also: Riemann zeta function Related pages (outside of this work) The Riemann hypothesis (expanded information) How the Riemann zeta function 'encodes' the primes ZetaGrid a distributed project to test the hypothesis Prime Time a simple article The Riemann hypothesis

5. Clay Mathematics Institute
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/Riemann_Hypothesis/

Extractions: Clay Mathematics Institute Dedicated to increasing and disseminating mathematical knowledge 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. The Millennium Problems Official Problem Description Lecture by Jeff Vaaler at the University of Texas (video)

6. 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

7. Riemann Hypothesis - Statistics
1 J. van de Lune, HJJ te Riele, DT Winter,On the Zeros of the Riemann Zeta Functionin the Critical Strip IV,Mathematics of Computation 46 (1986), 667681.
http://www.hipilib.de/zeta/statistic.html

Extractions: A previous version of my program was organized in such a way that in case the value of Z(t), obtained with method A, was too small for a rigorous sign determination, a few small shifts of the arguments were tried before method B was involved. Therefore, my program uses for j n J(1, n) J(2, n) J(3, n) J(4, n) J(5, n) J(6, n) J(7, n) J(8, n) J(9, n) J(10, n) Number of Gram intervals containing excatly m zeros n m = m = 1 m = 2 m = 3 m = 4 First occurrences and number of Gram blocks of various types Gram block of type T First occurrences of a Gram block of type T Number of Gram blocks of type T

8. ZetaGrid - Verification Of The Riemann Hypothesis
Verification of the riemann hypothesis. Why is Riemann s Hypothesis soimportant? The verification of Riemann s Hypothesis (formulated
http://www.zetagrid.net/zeta/rh.html

Extractions: 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

9. 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/

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

10. The Riemann Hypothesis
of the problem and the million dollar prize offered by the Clay Institute.
http://www.claymath.org/prize_problems/riemann.htm

11. The Riemann Hypothesis
Article by Enrico Bombieri (PDF) and video by Jeff Vaaler (.ram) from the Clay Mathematics Institute.
http://www.claymath.org/Millennium_Prize_Problems/Riemann_Hypothesis/

here. Prime Obsession is a nonfiction book on the riemann hypothesis,a famous unsolved problem in higher mathematics. The book
http://olimu.com/Riemann/Riemann.htm

Extractions: Paperback Note The paperback edition of Prime Obsession will be available May 25, 2004. It is being published by Plume, a division of Penguin Books. I have established a separate set of pages for the paperback edition, which you can view 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. A paperback edition will appear in July 2004. In these pages (see "Navigate down" at the left there) I offer a brief description of the book, all the reviews I am aware o f, a list of

13. The Riemann Hypothesis
Some of the conjectures and open problems concerning RH, compiled by the AIM.
http://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

14. THE RIEMANN HYPOTHESIS
Web article by Aldo Peretti.
http://www.peretti.da.ru

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 Riemanns paper is very obscure and confusing because of its excessive brevity. Fortunately

15. The Music Of The Primes
A popular article by Marcus du Sautoy on the riemann hypothesis; Science Spectra, Issue 11.
http://www.dpmms.cam.ac.uk/~dusautoy/2soft/music.htm

Extractions: 1.-When the British mathematician Andrew Wiles told the world about his proof of the Last Theorem of the seventeenth century French lawyer, Pierre de Fermat, it looked as if the Holy Grail had been grasped. Fermat's Last Theorem has often been called the greatest unsolved riddle of mathematics. But many mathematicians would argue that this name belongs rather to an idea first put forward in the middle of the nineteenth century by the German mathematician Bernhard Riemann: The Riemann Hypothesis. 2.-PRIME NUMBERS It remains unresolved but, if true, the Riemann Hypothesis will go to the heart of what makes so much of mathematics tick: the prime numbers. These indivisible numbers are the atoms of arithmetic. Every number can be built by multiplying prime numbers together. The primes have fascinated generations of mathematicians and non-mathematicians alike, yet their properties remain deeply mysterious. Whoever proves or disproves the Riemann Hypothesis will discover the key to many of their secrets and this is why it ranks above Fermat as the theorem for whose proof mathematicians would trade their soul with Mephistopheles. 3.-Although the Riemann Hypothesis has never quite caught on in the public imagination as Mathematics' Holy Grail, prime numbers themselves do periodically make headline news. The media love to report on the latest record for the biggest prime number so far discovered. In November 1996 the Great Internet Prime Search announced their discovery of the current record, a prime number with 378,632 digits. But for mathematicians, such news is of only passing interest. Over two thousand years ago Euclid proved that there will be infinitely many such news stories, for the primes never run dry.

16. Georg Friedrich Bernhard Riemann (1826-1866)
The Riemann Zeta Function and the riemann hypothesis. Back to Mathematicians andPhilosophers in the History of Mathematics archive The History of Mathematics.
http://www.maths.tcd.ie/pub/HistMath/People/Riemann/

17. The Reimann Hypothesis -- The Greatest Unsolved Problem In Mathematics -- Karl S
of that paper to the present day, the world s mathematicians have been fascinated,infuriated, and obsessed with proving the riemann hypothesis, and so great
http://www.semcoop.com/detail/0374250073

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18. Riemann Hypothesis
riemann hypothesis. The riemann hypothesis, first Institute for a proof.Most mathematicians believe the riemann hypothesis to be true.
http://www.fact-index.com/r/ri/riemann_hypothesis.html

Extractions: Main Page See live article Alphabetical index 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. 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 This traditional formulation obscures somewhat the true importance of the conjecture. The zeta function has a deep connection to the distribution of prime numbers and Helge von Koch proved in that the Riemann hypothesis is equivalent to the following considerable strengthening of the prime number theorem x ) is the prime-counting function , ln( x ) is the natural logarithm of x , and the O-notation is the Landau symbol The zeros of the Riemann zeta function and the prime numbers satisfy a certain duality property, known as the

19. Generalized Riemann Hypothesis
Generalized riemann hypothesis. The riemann hypothesis is one of the most importantconjectures in mathematics. Generalized riemann hypothesis (GRH).
http://www.fact-index.com/g/ge/generalized_riemann_hypothesis.html

Extractions: Main Page See live article Alphabetical index The Riemann hypothesis is one of the most important conjectures in mathematics . It is a statement about the zeros of the Riemann zeta function . Various geometrical and arithmetical objects can be described by so-called global L-functions , which are formally similar to the Riemann zeta function. One can then ask the same question about the zeros of these L-functions, yielding various generalizations of the Riemann hypothesis. None of these conjectures have been proven or disproven, but many mathematicians believe them to be true. Global L-functions can be associated to elliptic curves, number fields (in which case they are called Dedekind zeta functions ), Maass waveforms, and Dirichlet characters (in which case they are called Dirichlet L-functions ). When the Riemann hypothesis is formulated for Dedekind zeta functions, it is known as the extended Riemann hypothesis and when it is formulated for Dirichlet L-functions, it is known as the generalized Riemann hypothesis . These two statements will be discussed in more detail below. Table of contents 1 Generalized Riemann Hypothesis (GRH)

20. Mudd Math Fun Facts: Riemann Hypothesis
riemann hypothesis. If you know about complex numbers, you will be ableto appreciate one of the great unsolved problems of our time.
http://www.math.hmc.edu/funfacts/ffiles/30002.5.shtml

Extractions: Francis Edward Su From the Fun Fact files, here is a Fun Fact at the Advanced level: If you know about complex numbers, you will be able to appreciate one of the great unsolved problems of our time. The Riemann zeta function is defined by Zeta(z) = SUM k=1 to infinity (1/k z This is the harmonic series for z=1 and Sums of Reciprocal Powers if you set z equal to other positive integers. The function can be extended to the entire complex plane (with some poles) by a process called "analytic continuation", although what that is won't concern us here. It is of great interest to find the zeroes of this function. The function is trivially zero at the negative even integers, but where are all the other zeroes? To date, the only other zeroes known all lie on the line in the complex plane with real part equal to 1/2. This has been checked for several hundred million zeroes! No one knows, however, if

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