Ipotèz Riemann

Se yon ipotèz matematik, yon konjekti fèt pa Bernhard Riemann.
Kesyon l poze ap prezante konsa : èske tout zero fonksyon Zeta Riemann an genyen 1/2 pou pati reyèl ?

Li twouve yon fomilasyon pli senp pa :

Pou tout ε > 0, nou genyen

lè π(x) ap deziyen kantite nonm premye ki anvan x, ln(x) se fonksyon logaritm, evalye nan x, epi a dwat ou ap twouve notasyon Landau.

Yon vèsyon ki pa asenptotik pa Lowell Schoenfeld, ap di ke ipotèz Riemann an ekivalan a :

 Referans 
 Referans istorik 
 Bernhard Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, (1859) Monatsberichte der Berliner Akademie. (This site provides both a facsimile of the original manuscripts, as well as English translations.)
 Jacques Hadamard, Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques, Bulletin Société Mathématique de France 14 (1896) pp 199-220.

 Teknik jodi jou 
 H. M. Edwards, Riemann's Zeta Function, Academic Press, 1974. (Reprinted by Dover Publications, 2001 ISBN 0-486-41740-9)
 E. C. Titchmarsh, The Theory of the Riemann Zeta Function, second revised (Heath-Brown) edition, Oxford University Press, 1986
  (A relationship in terms of Harmonic numbers.)
 (no author credited), Computation of zeros of the Zeta function (2004). (Reviews the GUE hypothesis, provides an extensive bibliography as well).
 Schoenfeld, Lowell. "Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II." Mathematics of Computation 30 (1976), no. 134, 337--360.
 Conrey, J. Brian. "the Riemann Hypothesis" Notices of the American Mathematical Society, March 2003, 341-353. Available free http://www.ams.org/notices/200303/fea-conrey-web.pdf

 Referans popilè 
 Clay Mathematics Institute, Millennium Problems, (2000) (Announcement of the million dollar rewards for solutions to famous problems in mathematics)
 Marcus du Sautoy, The Music of the Primes, HarperCollins, 2003
 Marcus du Sautoy, "Prime Numbers Get Hitched", Seed Magazine" (03/27/2006)
 Daniel Rockmore, Stalking the Riemann Hypothesis : The Quest to Find the Hidden Law of Prime Numbers, Pantheon Books, New York, 2005, ISBN 0-375-42136-X
 John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Joseph Henry Press (April 23, 2003), ISBN 0-309-08549-7. 448 page book at a non-specialist level, can be read online for free
 Zetagrid (2002) A distributed computing project that attempted to disprove Riemann's hypothesis; closed in November 2005
 A Beautiful Mind (2001, produced by Ron Howard). In the film, the mathematician John Nash attempts to prove the Riemann Hypothesis
 Law & Order: Criminal Intent: In season 2, episode 2, "Bright Boy", the detective played by Vincent D'Onofrio tricks a father into confessing to a murder by telling him his son has solved the Riemann Hypothesis
 Numb3rs, (CBS), season 1 episode 5, "Prime Suspect", Criminals kidnap a mathematician's daughter and demand his allegedly complete proof of Riemann's Hypothesis as ransom
 Ed Pegg, Jr., Ten Trillion Zeta Zeros, (2004) Math Games website. A discussion of Xavier Gourdon's calculation of the first ten trillion non-trivial zeros
 de Vries, The Graph of the Riemann Zeta function ζ(s) (2004), a simple animated Java applet
 Erica Klarreich, "Prime Time", New Scientist - November 11, 2000, p. 32. A simple introduction to the Riemann Hypothesis, and its connection to prime numbers and quantum systems
 QEDen, A wiki dedicated to solving the millennium problems
 Perplex City, an Alternate Reality Game soliciting answers to various puzzles from its players, is offering 60 points to any player to submit a proof of the Riemann Hypothesis on card 238 of Season 1, entitled "Riemann"
 Proof, a film starring Gwynneth Paltrow and Anthony Hopkins, in which a character is purported to have solved a problem that sounds like the Riemann Hypothesis
 PopCo, a novel wherein the main character's grandmother used to spend all her free time trying to prove the Riemann Hypothesis
 Thomas Pynchon, the novel Against The Day, wherein the Riemann Hypothesis and the question of non-trivial zeros are recurring tropes and preoccupations of the students and spies gathered at Göttingen University in the years preceding World War I
 Karl Sabbagh, The Riemann Hypothesis: the greatest unsolved problem in mathematics, (2003) Farrar, Straus and Giroux, ISBN 0-374-25007-3. Also (2004) First American paperback edition. Conversations with mathematicians working on the problem. The reader should not expect to learn anything about the problem itself

 Lòt referans 
 Bollobas, Bela, foreword to Littlewood's Miscellany, Cambridge University Press, 1986

 Lyen deyò 
 The Millennium Grand Challenge in Mathematics
 The Millennium Prize Problems

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